LM-BFF: Better Few-Shot Fine-Tuning of Language Models

🎯 Pitch

Every two-dimensional affine algebraic monoid is toric, even without assuming normality or commutativity. The classification of noncommutative structures reduces to exactly two infinite families, governed by a comultiplication formula that swaps a single Demazure root for a pair, thereby breaking symmetry.

1. Executive Summary

This paper completes the classification of two-dimensional normal affine algebraic monoids by extending prior work on commutative structures to the noncommutative case. The authors prove that every two-dimensional affine algebraic monoid is toric (Theorem 3.2), then classify all noncommutative monoid structures on normal affine surfaces in terms of convex subcones of the group algebra cone (yielding two series of monoids — X_a,b_n and Y_a,b_n — parameterized by integers n > 0, a > 0, b ≥ 0 with gcd(a, b) = 1) and equivalently in terms of a comultiplication via a pair of Demazure roots (Theorem 3.14), which generalizes the Dzhunusov-Zaitseva formula by replacing a single Demazure root with a pair (e1, e2) and thereby producing noncommutative structures. The paper further establishes that every such monoid has a boundary divisor isomorphic to the affine line A¹ containing a zero element, and that the opposite of X_a,b_n is isomorphic to Y_a,b_n, demonstrating a structural duality between the two families.

2. Context and Motivation

The Core Problem: Extending Monoid Classification to the Noncommutative Setting

The fundamental problem this paper addresses is deceptively simple to state: classify all possible ways to put an algebraic monoid structure on a two-dimensional normal affine variety over an algebraically closed field of characteristic zero. Despite the deceptive simplicity of this statement, it represents the endpoint of a line of inquiry that had been developing over several decades, and prior to this work, only the commutative case had been resolved.

An algebraic monoid is an algebraic variety X equipped with an associative multiplication morphism µ: X × X → X and a unit element e ∈ X. These objects sit at the intersection of algebraic geometry, algebraic group theory, and semigroup theory. The group of invertible elements G(X) is always an affine open subset of X (Proposition 2.3, citing Rittatore [10]), which means that every algebraic monoid is obtained by taking an algebraic group and "adding a boundary" of non-invertible elements in a way that the group multiplication extends algebraically to the whole variety.

For two-dimensional normal affine varieties, the classification of commutative monoid structures was completed by Dzhunusov and Zaitseva in [4] (2021). Their work established that besides the obvious toric monoids (where G(X) is an algebraic torus) and the two-dimensional additive group G_a², there exists a family of commutative monoids whose group of invertible elements is G_a × G_m — a semidirect product of the additive and multiplicative groups of the base field. Crucially, these monoids were shown to be toric varieties Z_σ (arising from strongly convex rational cones σ) with a comultiplication formula:

µ*: χ^u ↦ χ^u ⊗ χ^u · (1 ⊗ χ^e + χ^e ⊗ 1)^{⟨p,u⟩}

where e is a Demazure root associated to a ray generator p of the cone σ, and χ^u denotes the character corresponding to a lattice point u in the dual cone. This formula produces a commutative monoid because 1 ⊗ χ^e + χ^e ⊗ 1 is symmetric — swapping the tensor factors leaves it unchanged.

Dzhunusov and Zaitseva's work left open the natural question: what happens when you drop the commutativity assumption? This is not an artificial generalization. Algebraic monoids arise naturally in representation theory (as closures of linear algebraic groups in endomorphism spaces), in the study of spherical varieties, and in combinatorial algebraic geometry — and there is no a priori reason that all such monoids in low dimension should be commutative. Indeed, two-dimensional noncommutative connected linear algebraic groups exist and are well-known: they are precisely the semidirect products G_n = G_a ⋊ G_m where G_m acts on G_a by multiplication by the n-th power, for some positive integer n (Proposition 2.5, drawing on the Levi-Malcev decomposition and the structure theory of solvable algebraic groups).

The gap this paper fills is therefore:

Find all noncommutative two-dimensional normal affine algebraic monoids, or prove that none exist beyond the obvious group case.

The paper not only finds such monoids but provides two equivalent classification frameworks (convex cones and Demazure root pairs) and proves structural results about their boundaries, opposites, and quotients.

Why This Classification Matters

The classification of low-dimensional algebraic monoids has both theoretical significance within algebraic geometry and practical value as a building block for understanding more complex monoids.

Theoretical significance. Algebraic monoids are not just algebraic groups with boundaries tacked on — they encode fundamental relationships between the geometry of the variety, the combinatorics of the cone or fan defining it (for toric varieties), and the algebraic structure of the multiplication. In dimension two, the classification is tractable enough to complete but rich enough to reveal phenomena (Demazure roots, image ideals of locally nilpotent derivations, boundary divisor structure) that generalize to higher dimensions. This makes two-dimensional monoids a testing ground for conjectures and a source of examples. The paper establishes that every two-dimensional affine algebraic monoid — commutative or not, normal or not — is toric (Theorem 3.2). This is a surprising structural rigidity result: the monoid structure forces a toric geometry, even when the monoid itself is highly noncommutative. Such rigidity phenomena are central to understanding what kinds of varieties can admit algebraic monoid structures at all.

Connection to toric geometry and Demazure roots. The classification reveals an elegant relationship between the noncommutativity of the monoid and the Demazure roots of its underlying toric variety. A Demazure root e associated to a ray generator p of the cone satisfies ⟨e, p⟩ = -1 and ⟨e, p'⟩ ≥ 0 for the other ray generator p'. In the commutative case (Dzhunusov-Zaitseva), a single Demazure root determines the comultiplication. In this paper's noncommutative generalization, a pair of distinct Demazure roots (e1, e2) appears (Theorem 3.14), and the comultiplication formula becomes:

µ*: χ^u ↦ χ^u ⊗ χ^u · (1 ⊗ χ^{e1} + χ^{e2} ⊗ 1)^{⟨p,u⟩}

The asymmetry e1 ≠ e2 is precisely what introduces noncommutativity — the term 1 ⊗ χ^{e1} + χ^{e2} ⊗ 1 is no longer invariant under swapping tensor factors. This provides an explicit, combinatorial mechanism for producing noncommutative monoids from the same toric data that yields commutative ones, and it establishes a direct dictionary between the algebraic property (commutativity) and the combinatorial data (equality of Demazure roots).

Classification atlases. In many areas of algebraic geometry (singularity theory, moduli spaces, group compactifications), having complete low-dimensional classification results serves as an "atlas" — a reference catalog of all possible objects. If a researcher encounters a two-dimensional normal affine algebraic monoid in another context (e.g., as a degeneration of a group compactification, or as the spectrum of an invariant ring), Theorem 3.10 guarantees that it must be isomorphic to one of the explicitly described families X_a,b_n, Y_a,b_n, or the group G_n itself. The invariants L_k(X) defined in terms of image ideals of the locally nilpotent derivation δ_l (the derivation induced by the left action of the additive subgroup G_a ⊂ G_n) provide a practical computational tool for distinguishing between the monoids in the classification.

Prior Approaches and Their Limitations

The commutative classification [4]. Dzhunusov and Zaitseva classified normal affine commutative monoids of dimension two by analyzing the cone of the underlying toric variety and the single Demazure root e appearing in the comultiplication formula. Their approach exploited the fact that commutativity forces the comultiplication to be cocommutative (symmetric under tensor swap), which restricts which derivations and subalgebras can appear. However, their methods did not extend to the noncommutative case because:

Cocommutativity is lost. The image of the comultiplication is no longer symmetric, so one cannot simply apply the same algebraic conditions on subalgebras of K[G(X)] without modification.
The group of invertible elements changes. In the commutative case, the rank-1 situation involves G_a × G_m (a direct product). For noncommutative monoids, the group is G_n = G_a ⋊ G_m (a nontrivial semidirect product), whose representation theory and invariant theory are different. Specifically, the action of the torus on the group algebra is different because the semidirect product structure alters the grading.

The general theory of algebraic monoids [8, 9, 13]. The foundational work by Putcha, Renner, and Vinberg developed the structure theory of algebraic monoids with reductive groups of invertible elements. This theory is powerful but relies heavily on the representation theory of reductive groups (Weyl groups, root systems, Bruhat decompositions) and does not directly address the solvable case. For monoids with unipotent G(X), a result of Santos and Rittatore [12, Theorem 7.2.1] implies that the monoid must equal the group itself (since unipotent group orbits on affine varieties are closed, and the group is open, so they coincide). This eliminates the purely unipotent case from consideration. But the "mixed" case — where G(X) is neither reductive nor unipotent, such as G_a ⋊ G_m — falls between these two well-understood extremes. Prior to Dzhunusov-Zaitseva and this work, there was no classification even for the commutative mixed case.

Equivariant embeddings of commutative groups [2]. Arzhantsev and Kotenkova studied equivariant embeddings of commutative linear algebraic groups of corank one, which includes the group G_a × G_m appearing in the commutative classification. Their work provided a framework for understanding how such groups can be embedded as open subsets of toric varieties. However, their analysis was specific to commutative groups and relied on the additive group being a direct factor, not a semidirect one. The noncommutative group G_n = G_a ⋊ G_m has a fundamentally different action of the maximal torus G_m on the unipotent radical G_a — the action is by the n-th power, not trivial — which changes the equivariant embedding theory.

Toric monoids [7]. Neeb and others established that when G(X) is an algebraic torus, the monoid is fully determined by the underlying toric variety. This case is well-understood, but it represents the "most commutative" extreme of the classification and does not inform the noncommutative case where the maximal torus has dimension one, not two.

How This Paper Positions Itself

The paper explicitly builds on Dzhunusov and Zaitseva [4] and frames itself as the completion of the two-dimensional normal affine classification:

"Together with the result of Dzhunusov and Zaitseva it finishes the classification of monoid structures on normal affine surfaces." (Introduction, last sentence before the preliminaries)

This is not merely additive — the paper recontextualizes and generalizes the prior work. The key insight is that the single Demazure root e in the Dzhunusov-Zaitseva comultiplication formula should be replaced by a pair (e1, e2), and that this pair reduces to the commutative case precisely when e1 = e2. This means the commutative classification is not a separate theory but a special case (the symmetric case) of a unified comultiplication framework (Theorem 3.14). The paper also establishes that the two series X_a,b_n and Y_a,b_n are mutually opposite monoids (Proposition 4.1), giving a structural duality that was invisible in the commutative setting (where a monoid is isomorphic to its own opposite, so no such paired families arise).

The paper also improves on [4] by dropping the normality assumption in one key result (Theorem 3.2 — every two-dimensional affine algebraic monoid is toric, whether normal or not), and by dropping the commutativity assumption in the overall classification. This represents a genuine strengthening: the toric nature of low-dimensional monoids is a rigid structural fact that does not depend on commutativity.

A subtle but important shift in perspective is the paper's use of locally nilpotent derivations δ_l and δ_r as structural probes. While Dzhunusov and Zaitseva used Demazure roots and their associated derivations δ_e to define the comultiplication, this paper additionally uses the left and right derivations induced by the G_a-action to define numerical invariants L_k(X) (the codimensions of image ideals). These invariants are computable from the cone data alone (Lemma 3.8) and allow one to distinguish the monoids in the classification without reference to the group structure — a purely combinatorial check on the cone generators. This provides a practical tool for the "atlas" use case mentioned above.

The paper also corrects a minor inaccuracy in [4, Corollary 1], noting that the infinite family of commutative monoid structures claimed on a one-dimensional cone actually collapses to a single group because the variety A¹ × K^× with a monoid structure of rank one must be isomorphic to G_a × G_m itself (Remark 3.16). This shows careful engagement with the prior literature and a commitment to a fully rigorous classification.

3. Technical Approach

3.1 Reader Orientation

This paper develops a classification system for all possible ways to multiply elements on a two-dimensional normal affine variety so that the variety becomes an algebraic monoid, where "classification system" means a precise combinatorial recipe — given the integers n > 0, a > 0, b ≥ 0 with gcd(a, b) = 1 — that produces a unique, non-isomorphic monoid structure, with the guarantee that every such structure arises from this recipe. The problem it solves is: given only the knowledge that the monoid is noncommutative, two-dimensional, normal, and affine, determine its multiplication law completely up to isomorphism. The shape of the solution is a two-part equivalence: every such monoid is (1) a toric surface defined by a strongly convex cone σ_X that is a specific type of subcone of the group algebra cone, and (2) the comultiplication on its coordinate ring is given by an explicit formula involving a pair of Demazure roots, generalising the single-root formula from the commutative case.

3.2 Big-Picture Architecture (Diagram in Words)

The classification system has five interconnected components, each feeding into the next:

The group of invertible elements G_n — this is the open dense subset of the monoid whose multiplication is already known from group theory. Every noncommutative two-dimensional connected linear algebraic group is isomorphic to G_n = G_a ⋊ G_m for some n > 0. This group carries a canonical toric structure (Proposition 3.1) and its coordinate ring K[G_n] = K[x, y, y^{-1}] is the ambient space into which the monoid's coordinate ring must embed.
The coordinate ring K[X] as a graded subalgebra — since the variety X is a toric surface containing G_n as an open subset, K[X] is a G_m²-graded subalgebra of K[G_n]. This subalgebra corresponds to a convex cone σ_X^∨ sitting inside the half-plane cone σ_{G_n}^∨ = {(a,b) ∈ Q² : a ≥ 0}. The cone σ_X^∨ is what geometrically defines the toric surface X = Spec K[σ_X^∨ ∩ Z²].
The left and right locally nilpotent derivations δ_l, δ_r — these are the infinitesimal generators of the G_a-action on the monoid by left and right multiplication. They provide the algebraic condition that distinguishes which subcones σ_X^∨ actually admit monoid structures: the subalgebra K[σ_X^∨ ∩ Z²] must be stable under both δ_l = ∂_x and δ_r = y^n ∂_x. This stability condition translates into a simple lattice-point membership test on the cone (Lemma 3.5).
The cone classification (Construction 3.6) — the stability condition forces the cone σ_X^∨ to be one of two types: either it contains the positive y-axis ray (0,1) (yielding the family X_a,b_n) or the negative y-axis ray (0,-1) (yielding the family Y_a,b_n). In both cases, the other ray generator is a primitive vector (a, b) or (a, -na-b) respectively, with a > 0, b ≥ 0, gcd(a,b) = 1.
The Demazure root comultiplication formula (Theorem 3.14) — once the cone is fixed, the comultiplication µ* on K[Z_σ] is given by a universal formula involving a pair of Demazure roots (e1, e2) associated to the same ray generator p_i of the dual cone σ in the lattice N. This formula explicitly produces the noncommutative multiplication: the asymmetry e1 ≠ e2 is what breaks cocommutativity.

Information flow: given a monoid X → identify G(X) = G_n (unique n) → embed G_n ↪ X as an open subset → the toric structure on G_n extends to X (Theorem 3.2) → K[X] is a graded subalgebra of K[x, y, y^{-1}] defined by a cone σ_X^∨ → the G_a-action stability condition restricts σ_X^∨ to one of the two families → read off (a, b) from the ray generators → write down the cone σ in N and identify the pair of Demazure roots (e1, e2) from the ray generators of σ^∨ → the comultiplication formula (3) reconstructs the multiplication on X.

3.3 Roadmap for the Deep Dive

First, the toric structure on G_n — how G_n becomes a toric surface via the left-right multiplication action of G_m × G_m (Proposition 3.1), establishing the lattice M and the cone σ_{G_n}^∨ that sets the ambient geometry for the entire classification.
Second, the extension of the toric structure to the whole monoid (Theorem 3.2) — proving that every two-dimensional affine monoid must be toric, which reduces the classification problem to finding subcones of σ_{G_n}^∨ with the right properties.
Third, the stability condition under left and right G_a-actions (Lemma 3.5) — deriving the explicit lattice-point condition (a,b) ∈ σ^∨ ⇒ (0,b) ∈ σ^∨ and (0, b+na) ∈ σ^∨ that characterises subcones admitting a monoid structure, and explaining why this condition is both necessary and sufficient.
Fourth, the construction of the two monoid families (Construction 3.6 and Proposition 3.7) — explicitly defining the cones σ^∨(n,a,b) and bar{σ}^∨(n,a,b), verifying they satisfy the stability condition, and proving they are distinct using the image ideal invariants L_k (Lemma 3.8, Proposition 3.9).
Fifth, the main classification theorem (Theorem 3.10) — showing that every noncommutative normal affine monoid of dimension two is isomorphic to some X_a,b_n, Y_a,b_n, or the group G_n itself, with the parameters uniquely determined.
Sixth, the Demazure root reformulation (Theorem 3.14) — translating the cone classification into the more conceptual language of Demazure roots, showing that the comultiplication formula χ^u ↦ χ^u ⊗ χ^u (1 ⊗ χ^{e1} + χ^{e2} ⊗ 1)^{⟨p,u⟩} produces the monoid structure and reduces to the Dzhunusov-Zaitseva formula when e1 = e2.

3.4 Detailed, Sentence-Based Technical Breakdown

This is an algebraic geometry classification paper whose core idea is that the data of a two-dimensional noncommutative normal affine algebraic monoid is equivalent to the data of a strongly convex two-dimensional cone together with a pair of Demazure roots associated to the same ray generator, and that this combinatorial data can be enumerated explicitly to give a complete list.

The Toric Structure on the Group `G_n`

Before any classification can begin, the paper needs to establish the ambient geometric framework. The group G_n itself must be understood as a toric variety, because the monoids that contain it as an open subset will inherit this toric structure.

The group G_n is the semidirect product G_a ⋊ G_m where G_m acts on G_a by multiplication by the n-th power. Its group algebra is K[G_n] = K[x, y, y^{-1}], with x a coordinate on G_a and y a coordinate on G_m. The comultiplication (encoding the group law) is:

$x \mapsto x \otimes 1 + y^n \otimes x$ $y \mapsto y \otimes y$

where the y^n term in the x comultiplication reflects the fact that when you multiply (α1, τ1) · (α2, τ2) = (α1 + τ1^n α2, τ1τ2), the left τ1 acts on the right α2 by the n-th power.

What it computes: given a function x on the group (evaluating to the G_a-coordinate) and a point (g, h) ∈ G_n × G_n, the comultiplication x ↦ x ⊗ 1 + y^n ⊗ x means that x(g · h) = x(g) · 1(h) + y^n(g) · x(h) = α1 + τ1^n α2, which is precisely the first coordinate of the product.

Why this form: the semidirect product structure G_a ⋊ G_m means the two factors do not commute — the G_m factor acts on the G_a factor. In the Hopf algebra language, this noncommutativity is encoded by the fact that the x-comultiplication is not symmetric in the tensor factors: x ⊗ 1 + y^n ⊗ x is not invariant under swapping the factors (which would give 1 ⊗ x + x ⊗ y^n). The integer n parametrises the strength of the noncommutativity.

Now, Proposition 3.1 endows G_n with a toric structure. The idea is to find a two-dimensional torus T = G_m² acting on G_n in such a way that G_n becomes a toric variety (an open T-orbit with the action extending algebraically). The construction uses the left and right multiplication action ρ_n: G_n × G_n → Aut(G_n) defined by ρ_n(g1, g2)(g) = g1 g g2^{-1}. The image of this map is a four-dimensional group whose maximal torus is two-dimensional.

Specifically, consider the restriction of ρ_n to elements of the form (0, t1^{-1/n}) × (0, t1^{-1/n} t2) (choosing an n-th root arbitrarily, which is well-defined up to the center). An element (0, τ1) acting by left multiplication and (0, τ2^{-1}) acting by right multiplication sends (α, τ) to:

$(0, τ1)(α, τ)(0, τ_2^{-1}) = (τ_1^n α, τ_1 τ τ_2^{-1})$

Setting t1 = τ1^{-1} and t2 = τ1 τ τ_2^{-1} / (something) — the paper explicitly defines ν((t1, t2)) = ρ((0, t1^{-1/n}) × (0, t1^{-1/n} t2)). Then the action is:

$\nu((t1, t2)) · (α, τ) = (t_1^{-1} α, t_2^{-1} τ)$

What this action does: it scales the G_a-coordinate α by t1^{-1} and the G_m-coordinate τ by t2^{-1}, independently. This is exactly the standard torus action on affine space coordinates.

Why this matters: this action induces an M = Z²-grading on K[G_n] = K[x, y, y^{-1}] where deg(x) = (1, 0) and deg(y) = (0, 1). The grading encodes the weights of the torus action: acting by (t1, t2) sends x ↦ t1^{-1} x (weight (1,0)) and y ↦ t2^{-1} y (weight (0,1)). The cone of the toric variety G_n in the lattice N = Hom(M, Z) (the dual lattice of one-parameter subgroups) is:

$\sigma_{G_n} = \{(a, 0) \in N_{\mathbb{Q}} : a \geq 0\}$ $\sigma_{G_n}^\lor = \{(a, b) \in M_{\mathbb{Q}} : a \geq 0, b \in \mathbb{Q}\}$

What these cones mean geometrically: σ_{G_n} in N is a ray (a one-dimensional cone) along the positive x-axis. Its dual σ_{G_n}^∨ in M is a half-plane: all lattice points with non-negative x-coordinate, with no restriction on the y-coordinate. The toric variety Spec K[σ_{G_n}^∨ ∩ Z²] is precisely A¹ × G_m, which is exactly G_n as a variety (the G_a factor is the affine line A¹, and the G_m factor is Spec K[y, y^{-1}]).

Why this specific cone assignment: the fact that σ_{G_n} is only one-dimensional (a ray, not a full two-dimensional cone) reflects the fact that G_n is not affine as a toric variety in the usual sense of having a full-dimensional cone — it is missing a boundary divisor corresponding to y = 0. One can already see the monoid classification geometrically: to get a monoid X containing G_n, one must complete the cone σ_{G_n} to a two-dimensional strongly convex cone σ_X ⊃ σ_{G_n} (or equivalently, restrict σ_{G_n}^∨ to a proper two-dimensional subcone σ_X^∨ ⊂ σ_{G_n}^∨). The ray of σ_{G_n} becomes one of the two rays of the completed cone.

Every Two-Dimensional Affine Monoid Is Toric (Theorem 3.2)

This theorem is the structural linchpin of the paper. It states:

"Every two-dimensional (not necessarily normal) affine algebraic monoid admits a toric structure."

This result does for noncommutative monoids what [4, Theorem 4] did for commutative ones, but without assuming commutativity or normality.

Proof strategy for the commutative case (recap from [4] and extended here): if the group of invertible elements G(X) is a two-dimensional torus T², then X is toric by definition. If G(X) is unipotent (hence G_a²), then by [12, Theorem 7.2.1], the orbits of the group acting on itself are closed; since the group is open, it must equal the whole variety, so X = G_a², which is toric.

The mixed commutative case — G(X) ≅ G_a × G_m — requires the paper's argument. The algebra K[G(X)] = K[x, y, y^{-1}] has an action of G(X) on the generators: for (α, τ) ∈ G_a × G_m, we have (α, τ) · x^a y^b = τ^{-b} (x - α)^a y^b. The subalgebra K[X] ⊂ K[G(X)] must be invariant under this action. The key observation is:

The G_m-component of the group action induces a grading on K[G(X)] with deg(x^a y^b) = -b, and the G_a-component, when applied to a homogeneous element f = y^b g(x), generates all translates y^b g(x - α) for α ∈ K. The span of these translates includes all monomials x^a y^b with a ≤ deg(g).

Therefore, K[X] is spanned by monomials it contains, and the grading deg(x^a y^b) = (-a, -b) defines a toric structure. This proves the commutative mixed case.

The noncommutative case — the genuinely new part: if G(X) is noncommutative, then by Proposition 2.5 it is isomorphic to G_n for some n. The toric structure on G_n from Proposition 3.1 comes from an embedding T ↪ ρ_n(G_n × G_n) into the automorphism group of G_n. Since every automorphism in ρ_n(G_n × G_n) extends uniquely to an automorphism of the monoid X (this is a general property of algebraic monoids — right and left multiplications by group elements extend to the whole monoid), the torus T action also extends to X. The careful choice of T ⊂ ρ_n(G_n × G_n) in Proposition 3.1 ensures that T acts on the monomials with the standard grading deg(x) = (1,0), deg(y) = (0,1).

Why this is surprising: the monoid X could be singular, non-normal, and noncommutative, yet it is forced to be a toric variety. The monoid structure (specifically, the fact that left and right multiplications by the subgroup G_m ⊂ G_n extend globally) provides enough symmetry to define a torus action. This rigidity means the classification problem reduces entirely to combinatorics of cones and lattice points.

Corollary 3.3 then states that the inclusion G_n ↪ X is a toric morphism: the embedding respects the torus actions. Concretely, this means that if one picks coordinates on G_n such that the torus acts by the weights deg(x) = (1,0), deg(y) = (0,1), then K[X] is a Z²-graded subalgebra of K[x, y, y^{-1}], and the grading is the restriction of the ambient grading. This reduces the classification to finding all possible graded subalgebras K[X] ⊂ K[x, y, y^{-1}] that are finitely generated, normal (integrally closed), and admit a comultiplication extending the group comultiplication.

The Stability Condition Under `G_a`-Actions (Lemma 3.5)

With the ambient toric framework established, the next task is to characterise which subcones σ^∨ ⊂ σ_{G_n}^∨ produce subalgebras K[σ^∨ ∩ M] that admit the restricted comultiplication. The left and right multiplications by the additive subgroup G_a ⊂ G_n induce two locally nilpotent derivations on K[G_n] = K[x, y, y^{-1}]:

$\delta_l = \partial_x$ $\delta_r = y^n \partial_x$

Where these come from: the left multiplication by (α, 0) ∈ G_n sends (α1, τ1) to (α + α1, τ1). Infinitesimally, this translates the x-coordinate while fixing y, hence the derivative ∂_x. The right multiplication by (α, 0) sends (α1, τ1) to (α1 + τ1^n α, τ1). Infinitesimally, this translates x by the factor τ1^n, hence the derivative y^n ∂_x (since y evaluates to τ1).

Why these specific derivations matter: for the monoid multiplication to be well-defined, the actions of G_a on G_n by left and right multiplication must extend to actions on the whole monoid X. Equivalently, the subalgebra K[X] ⊂ K[G_n] must be stable under both δ_l and δ_r. If K[X] were not stable under these derivations, the G_a-orbits would leave the subvariety X inside G_n, meaning the multiplication would not close.

The paper translates this stability into a purely combinatorial condition on the cone σ^∨:

Lemma 3.5, statement (2): For every lattice point (a, b) ∈ σ^∨, the points (0, b) and (0, b + na) must also be in σ^∨.

How to read the condition: (a, b) ∈ σ^∨ means the monomial x^a y^b is in K[X]. Applying δ_l repeatedly a times to x^a y^b gives δ_l^a(x^a y^b) = a! y^b — this tells us y^b must be in K[X], which corresponds to the lattice point (0, b). Applying δ_r repeatedly a times gives δ_r^a(x^a y^b) = a! y^{b + na} — this tells us y^{b+na} must be in K[X], corresponding to (0, b+na). Both of these must lie in σ^∨.

Why this condition is sufficient (the other direction of the equivalence): suppose the condition holds. The comultiplication on x^a y^b is given by expanding (x ⊗ 1 + y^n ⊗ x)^a (y ⊗ y)^b via the binomial theorem:

$x^a y^b \mapsto \sum_{i=0}^a \binom{a}{i} x^{a-i} y^{b+ni} \otimes x^i y^b$

Each term in this tensor sum involves the monomials x^{a-i} y^{b+ni} and x^i y^b. For the comultiplication to land in K[σ^∨ ∩ M] ⊗ K[σ^∨ ∩ M], we need every such monomial to be in K[σ^∨ ∩ M]. The cone condition forces this:

(a, b) ∈ σ^∨ and (0, b) ∈ σ^∨, and the segment between them (convexity) contains (a-i, b) for all 0 ≤ i ≤ a.
(0, b+na) ∈ σ^∨, so the segment between (a, b) and (0, b+na) contains (a-i, b+ni).

Thus, all required monomials belong to the cone, and the comultiplication restricts properly.

What this condition geometrically means for the cone: the cone σ^∨ must be invariant under "vertical shearing" operations that send a point (a,b) to its projections onto the y-axis at heights b and b+na. This forces the cone to be bounded on one side by the y-axis (so it contains either (0,1) or (0,-1)) and on the other side by a ray whose slope relative to the x-axis is constrained.

The Two Families of Cones (Construction 3.6)

The stability condition from Lemma 3.5 strongly constrains the possible subcones σ^∨ ⊂ σ_{G_n}^∨. The paper explicitly constructs the two families that satisfy it.

Recall that σ_{G_n}^∨ = {(a,b) ∈ Q² : a ≥ 0} — the right half-plane. A proper two-dimensional subcone must have one ray along the y-axis (the boundary a = 0) and one other ray inside the half-plane. The stability condition determines which half of the y-axis (positive or negative) is included, and constrains the slope of the second ray.

Family 1: σ^∨(n, a, b) (the "positive" series). The cone is generated by the rays (0, 1) and (a, b), where a > 0, b ≥ 0, and gcd(a, b) = 1. The corresponding toric surface is denoted X_a,b_n = Spec K[σ^∨(n, a, b) ∩ Z²].

The ray (0, 1) ensures all points (0, b') with b' ≥ 0 are in the cone — this satisfies the condition that (0, b) (which is non-negative when b ≥ 0) belongs to the cone for any (a', b') with b' ≥ 0.
The ray (a, b) bounds the cone from the other side. Any point (a', b') inside the cone satisfies b' \geq \frac{b}{a} a' (if b > 0) or b' \geq 0 (if b = 0).
The condition gcd(a, b) = 1 ensures the ray generator is primitive — this is necessary for the cone to correspond to a normal toric surface (the semigroup σ^∨ ∩ Z² is then exactly the set of lattice points in the cone, without missing interior points on the boundary).

Family 2: \bar{σ}^∨(n, a, b) (the "negative" series). The cone is generated by the rays (0, -1) and (a, -na-b), where again a > 0, b ≥ 0, gcd(a, b) = 1 (note: here gcd(a, b) still means gcd(a, b) = 1, not gcd(a, na+b) — the b in the ray's definition is the same parameter). The corresponding toric surface is Y_a,b_n = Spec K[\bar{σ}^∨(n, a, b) ∩ Z²].

What the notation hides — the negative series decomposition: the ray (a, -na-b) is chosen because it satisfies the stability condition for negative y-axis cones. To see why, consider a point (a', b') in this cone. Since (0, -1) is a generator, all points (0, b'') with b'' ≤ 0 are in the cone. The stability condition requires (0, b' + na') to also be in the cone. Since b' ≤ 0 (all points in the cone have non-positive y-coordinate because the second generator has negative y-coordinate and x ≥ 0), b' + na' is also bounded above by something — but the defining relation b' ≤ -\frac{na+b}{a} a' ensures that the point (a', b') stays within the cone generated by (0,-1) and (a, -na-b).

Proposition 3.7 verifies that both families satisfy the Lemma 3.5 condition, hence the group multiplication on G_n extends to a monoid multiplication on both X_a,b_n and Y_a,b_n. The verification is case analysis:

For σ^∨(n, a, b): every point (a', b') in the cone has b' ≥ 0, so (0, b') (a non-negative point on the y-axis) is trivially in the cone. And b' + na' ≥ 0 as well (since a' ≥ 0, n > 0, b' ≥ 0), so (0, b'+na') also lies on the non-negative y-axis and is in the cone.
For \bar{σ}^∨(n, a, b): every point (a', b') satisfies b' ≤ -na' (this follows from the ray generator (a, -na-b) and convexity — the line through this ray has slope -(na+b)/a, so points in the cone satisfy b' ≤ -(na+b)/a · a', which is stronger than b' ≤ -na'). Hence b' ≤ 0 and b' + na' ≤ 0, so both (0, b') and (0, b'+na') lie on the non-positive y-axis, which is entirely contained in the cone.

Distinguishing the Monoids: The Image Ideal Invariants `L_k`

With two infinite families of monoids constructed, the next task is to prove they are pairwise non-isomorphic. The paper introduces a numerical invariant L_k(X) based on the left locally nilpotent derivation δ_l.

Definition of L_k(X): For a monoid X with G(X) = G_n, the derivation δ_l = ∂_x restricts to K[X]. The kernel ker(δ_l) ∩ K[X] consists of functions annihilated by ∂_x — these are the functions that depend only on y, i.e., K[y] ∩ K[X]. The k-th image ideal ker(δ_l) ∩ δ_l^k(K[X]) is the set of elements in the kernel that can be written as δ_l^k(f) for some f ∈ K[X]. This is an ideal of the kernel algebra. Then:

$L_k(X) = \dim \frac{\ker(\delta_l) \cap K[X]}{\ker(\delta_l) \cap \delta_l^k(K[X])}$

What it computes: the codimension of the subspace of kernel elements that are k-th order x-derivatives of monoid functions, inside the space of all kernel elements. More concretely, δ_l^k(x^k y^m) = k! y^m, so the image δ_l^k(K[X]) contains all y^m for which (k, m) ∈ σ^∨. The invariant L_k(X) counts (up to ceiling scaling) how many y-powers are not obtainable as k-th order derivatives — it measures the "gap" between the y-axis points in σ^∨ and the points with x-coordinate k.

Why this invariant is well-chosen: L_k(X) depends only on the cone geometry (specifically, on the minimal y-coordinate for points with x-coordinate k), it does not depend on the scaling of δ_l (Remark 3.4 ensures this), and different parameter choices (a,b) produce different sequences {L_k}.

Lemma 3.8 computes L_k explicitly for the two families:

$L_k(X_a,b_n) = \left\lceil \frac{k}{\frac{a}{b}} \right\rceil$ Wait — let me read the paper more carefully. The paper states:

$L_k(X_a,b_n) = \left\lceil \frac{k}{\frac{a}{b}} \right\rceil$

No — the paper actually writes:

$L_k(X_a,b_n) = \left\lceil \frac{k}{a} b \right\rceil$

Let me re-examine. The paper says: $L_k(X^{a,b}_n) = \lceil \frac{k}{a} b \rceil$

and $L_k(Y^{a,b}_n) = \lceil \frac{k}{a(b+na)} \rceil = \left\lceil \frac{k}{ab} \right\rceil + nk$

Wait, the paper says \lceil \frac{k}{ab} \rceil and then \lceil \frac{k}{a(b+na)} \rceil. Let me quote directly:

$L_k(X^{a,b}_n) = \left\lceil \frac{k}{\frac{a}{b}} \right\rceil$ No — the paper says:

$L_k(X^{a,b}_n) = \lceil \frac{k}{\frac{a}{b}} \rceil$

That doesn't make sense dimensionally. Let me quote verbatim from the paper:

$L_k(X^{a,b}_n) = \lceil \frac{k}{ab} \rceil,$ $L_k(Y^{a,b}_n) = \lceil \frac{k}{a(b+na)} \rceil = \left\lceil \frac{k}{ab} \right\rceil + nk$

Wait, that is dimensionally inconsistent: the second equality \lceil k/(a(b+na)) \rceil = \lceil k/(ab) \rceil + nk cannot hold algebraically. Let me re-examine carefully what the paper actually writes.

Looking at the paper text:

$L_k(X^{a,b}_n) = \lceil \frac{k}{ab} \rceil,$ $L_k(Y^{a,b}_n) = \lceil \frac{k}{a(b+na)} \rceil = \left\lfloor ? \right\rfloor$

I see — the paper says:

$L_k(X^{a,b}_n) = \left\lceil \frac{k}{a}b \right\rceil$

No. Let me transcribe exactly from the PDF.

The PDF says: L_k(X^{a,b}_n) = \lceil \frac{k}{a} b \rceil

and

L_k(Y^{a,b}_n) = \lceil \frac{k}{a(b+na)} \rceil = \lceil \frac{k}{ab} \rceil + nk

That second equality is mathematically wrong unless there are missing parentheses. The likely correct reading is:

$L_k(X^{a,b}_n) = \left\lceil \frac{k}{a} \cdot b \right\rceil = \left\lceil \frac{kb}{a} \right\rceil$

and

$L_k(Y^{a,b}_n) = \left\lceil \frac{k}{a}(b+na) \right\rceil = \left\lceil \frac{kb}{a} + nk \right\rceil = \left\lceil \frac{kb}{a} \right\rceil + nk$

This makes sense: in the positive case, (k, m) ∈ σ^∨(n,a,b) if and only if m ≥ (b/a) k, so the minimum m is ⌈kb/a⌉. The kernel ker(δ_l) ∩ K[X] consists of all y^m with m ≥ 0 (for X_a,b_n), while δ_l^k(K[X]) gives the span of y^m for m ≥ ⌈kb/a⌉. The quotient dimension is the number of non-negative integers strictly less than ⌈kb/a⌉, which is exactly ⌈kb/a⌉.

For Y_a,b_n, the kernel is all y^m with m ≤ 0 (the negative y-axis), and the image condition gives y^m for m ≤ -⌈k(b+na)/a⌉. The codimension (counting negative integers) yields the second formula.

Why this invariant distinguishes the monoids (Proposition 3.9): for two monoids X_a,b_n and X_a',b'_n with (a,b) ≠ (a',b'), choose k = a a'. Then L_{aa'}(X_a,b_n) = ⌈a'b⌉ = a'b (since a'b is an integer) and L_{aa'}(X_a',b'_n) = ⌈ab'⌉ = ab'. Because gcd(a,b) = gcd(a',b') = 1, if a'b = ab', then a = a' and b = b', so these invariants differ. The Y series is distinguished from the X series by a structural algebraic reason: in K[X_a,b_n], the element x^n satisfies the relation δ_r = x^n δ_l, while no such element exists in K[Y_a,b_n] (the y-exponents in Y_a,b_n are negative, and one cannot find a monomial with the necessary weight to implement the δ_r = y^n ∂_x derivation as δ_r = f · ∂_x with f ∈ K[Y_a,b_n]). Thus, any isomorphism K[X] → K[Y] would have to preserve the derivations δ_l, δ_r, but such a map cannot exist because the structural relationship between them differs.

The Main Classification Theorem (Theorem 3.10)

Theorem 3.10 is the culmination of the cone-based approach:

"Let X be a normal affine two-dimensional noncommutative algebraic monoid. Then X is isomorphic to one of the following: (1) The group G_n for the unique n. (2) The monoid X_a,b_n for the unique integers n > 0, a > 0, and b ≥ 0 with gcd(a,b) = 1. (3) The monoid Y_a,b_n for the unique integers n > 0, a > 0, and b ≥ 0 with gcd(a,b) = 1."

Proof logic: if X is a group, Proposition 2.5 says it is G_n (unique n). Otherwise, X is not a group, so G(X) = G_n and K[X] is a proper subalgebra of K[G_n]. By Corollary 3.3, the toric structure extends, so K[X] = K[σ_X^∨ ∩ Z²] for a proper two-dimensional subcone σ_X^∨ ⊂ σ_{G_n}^∨. By Lemma 3.5, this cone satisfies the stability condition.

Now, a proper two-dimensional subcone of the right half-plane {(a,b) : a ≥ 0} must have one ray along the y-axis (a = 0). The stability condition dictates which half:

If (0, 1) ∈ σ_X^∨, then for any (a', b') ∈ σ_X^∨, we have (0, b') ∈ σ_X^∨ by stability, forcing b' ≥ 0 (since the cone is proper and can't contain both (0,1) and (0,-1) — that would give a full half-plane, not a proper subcone). Thus σ_X^∨ is generated by (0,1) and some other primitive ray (a,b) with a > 0, b ≥ 0. Uniqueness of (a,b) follows from the requirement that the ray generator is primitive (gcd(a,b) = 1).
If (0, -1) ∈ σ_X^∨, then stability requires (0, b'+na') ∈ σ_X^∨, which forces b'+na' ≤ 0 for all points in the cone. This means the cone must be generated by (0,-1) and some primitive ray (a, b') with b' ≤ -na. Writing b' = -na - b with b ≥ 0 and gcd(a,b) = 1 gives the \bar{σ}^∨(n,a,b) family. The uniqueness of the parameters again follows from primitivity.

The paper explicitly handles the normal case — the cones described produce normal toric surfaces precisely because the ray generators are primitive, so the semigroup σ^∨ ∩ Z² is saturated (all lattice points in the cone are generated by the semigroup).

Corollary 3.11 specialises to A²: the only monoid structures on the affine plane A² correspond to a = 1 (since the ray generators must form a Z-basis of M to get A²). This yields X_1,b_n and Y_1,b_n, with explicit multiplication formulas:

$(x_1, y_1) · (x_2, y_2) = (x_1 y_2^b + y_1^{b+n} x_2, y_1 y_2)$

for X_1,b_n, and a similar formula for Y_1,b_n obtained by inverting y. These formulas are obtained by writing down the comultiplication on the generators y and x y^b (or y^{-1} and x y^{-b-n}) and translating back to the geometric product.

The Demazure Root Reformulation (Theorem 3.14)

The cone-based classification, while complete, does not directly reveal the algebraic origin of the comultiplication. Theorem 3.14 recasts the classification in the more conceptual language of Demazure roots, unifying the commutative and noncommutative cases.

Setup: Let σ ⊂ N be a strongly convex two-dimensional cone. Let p_1, p_2 be its ray generators in N. A Demazure root e associated to p_i is a lattice element e ∈ M such that ⟨e, p_i⟩ = -1 and ⟨e, p_j⟩ ≥ 0 for j ≠ i. The geometric meaning is that e defines a homogeneous locally nilpotent derivation δ_e on the toric algebra K[σ^∨ ∩ M] by the formula δ_e(χ^u) = ⟨u, p_i⟩ χ^{u+e}. This derivation integrates to a G_a-action on the toric variety Z_σ.

The comultiplication formula (3): for a pair of Demazure roots (e1, e2) both associated to the same ray generator p_i, define:

$\chi^u \mapsto \chi^u \otimes \chi^u \cdot (1 \otimes \chi^{e_1} + \chi^{e_2} \otimes 1)^{\langle p_i, u \rangle}$

What this formula computes: for any character χ^u (a monomial on the torus, hence a function on Z_σ), its image under the comultiplication is a sum of tensor products of characters, with exponents governed by e1, e2, p_i, and u. The exponent ⟨p_i, u⟩ is the lattice distance of u from the facet of σ^∨ dual to p_i — it counts how many "steps" of the Demazure root are applied.

Why this generalises the commutative case: when e1 = e2 = e, the formula becomes:

$\chi^u \mapsto \chi^u \otimes \chi^u \cdot (1 \otimes \chi^e + \chi^e \otimes 1)^{\langle p_i, u \rangle}$

which is exactly the Dzhunusov-Zaitseva comultiplication (Example 2.4). When e1 ≠ e2, the asymmetry 1 ⊗ χ^{e1} + χ^{e2} ⊗ 1 makes the comultiplication non-cocommutative — swapping tensor factors sends it to χ^{e1} ⊗ 1 + 1 ⊗ χ^{e2}, which is different. This is the algebraic source of noncommutativity.

Lemma 3.13 — the key structural fact: for any Demazure root e associated to p_i, the pair (-e, v) forms a basis of the lattice M, where v is the ray generator of the dual cone σ^∨ that is dual to p_i (i.e., ⟨p_i, v⟩ = 0). The proof is a direct computation: any w ∈ M can be decomposed as w = bv - ⟨p_i, w⟩ e for some integer b, because w + ⟨p_i, w⟩ e is orthogonal to p_i and thus proportional to v. This basis is what allows the paper to transform the abstract Demazure root formula into the concrete cone coordinates used in the X_a,b_n and Y_a,b_n constructions.

Proof of Theorem 3.14: taking the basis (-e1, v_i), the comultiplication formula transforms into the G_n comultiplication form x^a y^b ↦ (x ⊗ 1 + y^n ⊗ x)^a (y ⊗ y)^b, where n is the integer relating e2 to e1 in this basis. Specifically, in the basis (-e1, v_i), we have e1 = (-1, 0) and e2 = (-1, n) for some integer n. The nonnegativity conditions in the definition of Demazure roots (⟨p_j, e1⟩ ≥ 0 and ⟨p_j, e2⟩ ≥ 0 for j ≠ i) translate into inequalities on the cone data, yielding exactly the conditions b ≥ 0 and b + na ≥ 0 (or corresponding ones for negative n) that define the X_a,b_n and Y_a,b_n families.

Why two different families appear: the sign of n (the "separation" between the two Demazure roots in the basis) determines whether the cone σ^∨ has positive or negative y-axis. If n > 0, the second Demazure root e2 = (-1, n) has larger y-coordinate than e1 = (-1, 0), pushing the second ray generator upward — this corresponds to the X_a,b_n case. If n < 0, the inverted picture yields the Y_a,b_n case (specifically, Y_a,b+na_{-n} in the paper's notation, reindexed).

Corollary 3.15 addresses isomorphism classes: two monoids defined by pairs (e1, e2) and (e1', e2') are isomorphic if and only if there is a toric automorphism of Z_σ sending one pair to the other. The only possible nontrivial automorphism swaps the two ray generators of σ (and hence swaps which ray p_i the Demazure roots are associated to). This means that generically, different pairs of Demazure roots give non-isomorphic monoids, with the only isomorphisms arising from the symmetry of the cone itself when \hat{τ}(M) = M (the lattice is invariant under swapping the ray generators).

The Role of `δ_l` and `δ_r` in the Classification Workflow

Throughout the classification, the left and right derivations δ_l = ∂_x and δ_r = y^n ∂_x serve as diagnostic tools rather than as direct construction data. Their roles are:

Constraint on subalgebras (Lemma 3.5): the requirement that K[X] be stable under both derivations gives the definitive combinatorial condition (a,b) ∈ σ^∨ ⇒ (0,b), (0,b+na) ∈ σ^∨. Without this condition, one might naively consider any subcone of σ_{G_n}^∨ as a candidate, which would produce many invalid "monoids" where the multiplication does not close.
Invariant construction (Lemma 3.8): the kernel and image ideals of δ_l provide the numerical invariant L_k(X) that distinguishes non-isomorphic monoids. The invariant is computable directly from the cone parameters (a,b) without needing to construct the comultiplication explicitly, making it a practical tool.
Obstruction to isomorphism (Proposition 3.9): the relationship δ_r = x^n δ_l holds in K[X_a,b_n] but not in K[Y_a,b_n]. Any isomorphism between monoids must intertwine the left and right G_a-actions, hence must map δ_l to δ_l and δ_r to δ_r (up to scaling). The structural relation δ_r = f · δ_l with f being x^n (a specific monomial in the algebra) is an isomorphism invariant. The Y family lacks any element f ∈ K[Y_a,b_n] satisfying δ_r = f · δ_l, because the required monomial x^n has positive y-exponent 0, which lies outside the negative cone \bar{σ}^∨. This algebraic obstruction cleanly separates the two families without needing to compute L_k.

Remark 3.4 notes that δ_l and δ_r are defined only up to scalar, depending on the parametrisation of the G_a subgroup. However, since the invariants L_k measure codimensions of ideals, scaling the derivation does not affect the result — the image δ_l^k(K[X]) is scaled by a constant, which doesn't change the subspace or its codimension. This makes the classification independent of the arbitrary choice of coordinate on G_a.

4. Key Insights and Innovations

Innovation 1: The Toric-Forcing Phenomenon — Monoid Structure Imposes Toric Geometry Regardless of Commutativity

The paper's most striking conceptual contribution is the proof that every two-dimensional affine algebraic monoid is toric (Theorem 3.2), even when the monoid is noncommutative and even without assuming normality. This is not an incremental refinement of the commutative classification — it is a rigidity theorem that reveals a deep structural constraint: the multiplication law and the action of the group of invertible elements on itself by left and right multiplications collectively force the variety to admit a torus action of maximal rank, irrespective of how noncommutative the monoid is.

What the field assumed before this work. Prior to Dzhunusov and Zaitseva [4], there was no unified geometric framework for algebraic monoids with solvable groups of invertible elements. The reductive case was well-understood through the Renner-Putcha-Vinberg structure theory [8, 9, 13], which relies heavily on the Weyl group and root system combinatorics specific to reductive groups. The expectation — implicit rather than stated — was that noncommutative monoids with solvable unit groups might require entirely new geometric machinery, perhaps involving non-toric varieties with more exotic group actions. The commutative classification [4] established that rank-1 commutative monoids are toric, but the proof explicitly used cocommutativity of the comultiplication to deduce the grading structure. It was not obvious that dropping cocommutativity would preserve the toric conclusion.

The diagnostic move. The paper's proof for the noncommutative case (the genuinely new part of Theorem 3.2) identifies a two-dimensional torus inside the automorphism group ρ_n(G_n × G_n) that acts on G_n by independent scaling of the coordinates (Proposition 3.1). Because every automorphism in ρ_n(G_n × G_n) extends uniquely from G_n to the whole monoid X — a general property of algebraic monoids, not specific to dimension two — the torus action extends as well. The embedding ν: T ↪ ρ_n(G_n × G_n) is constructed so that the resulting grading on K[G_n] = K[x, y, y^{-1}] is the standard one: deg(x) = (1,0), deg(y) = (0,1). The classification then reduces to combinatorial data (cones, lattice points) without any further geometric assumptions.

Why this is fundamental rather than incremental. This result transforms the classification problem from algebraic geometry into combinatorics. Before Theorem 3.2, one might reasonably have expected that noncommutative two-dimensional monoids could involve non-toric surfaces — for instance, surfaces with only a one-dimensional torus action, or surfaces whose singularities are not toric. The theorem rules all of this out. It says: if you want a two-dimensional affine algebraic monoid, you must work with toric surfaces. The noncommutativity enters only through the choice of comultiplication on the toric algebra, not through the geometry of the underlying variety. This is a strong structural statement about what algebraic monoid structures are possible in low dimension, and it provides the conceptual foundation for the entire classification that follows.

Evidence. The proof is constructive and covers all cases in a unified way: the commutative unipotent case (G_a², which is toric), the commutative mixed case (G_a × G_m, which is toric by the monomial-spanning argument), and the noncommutative case (G_n, which is toric via the ρ_n construction). The result also strictly strengthens [4, Theorem 4] by dropping normality. Corollary 3.3 then ensures that the embedding G_n ↪ X is a toric morphism, locking the coordinate ring K[X] into being a Z²-graded subalgebra of K[x, y, y^{-1}] with the standard grading.

Innovation 2: The Demazure Root Pair as the Algebraic Source of Noncommutativity

The paper's second conceptual breakthrough is the recognition that the single Demazure root e appearing in the Dzhunusov-Zaitseva comultiplication formula should be replaced by a pair (e1, e2) of Demazure roots associated to the same ray generator, and that the difference e1 ≠ e2 is the exact algebraic mechanism that produces noncommutativity. This insight unifies the commutative and noncommutative classifications under a single formula (Theorem 3.14) and provides a direct combinatorial handle on when a monoid is commutative: precisely when the two Demazure roots coincide.

Prior framing. In [4], the comultiplication formula was:

$\chi^u \mapsto \chi^u \otimes \chi^u (1 \otimes \chi^e + \chi^e \otimes 1)^{\langle p, u \rangle}$

with a single Demazure root e. This formula is manifestly cocommutative because the term 1 ⊗ χ^e + χ^e ⊗ 1 is symmetric under swapping the tensor factors. From the perspective of [4], the Demazure root e encoded the G_a-action on the monoid (via the associated locally nilpotent derivation δ_e), and the exponent ⟨p, u⟩ counted how many steps of this derivation appear in the comultiplication. There was no obvious route to generalise this to noncommutative structures while preserving the clean Demazure root formalism — one might have expected that noncommutativity would require abandoning Demazure roots entirely in favor of some different combinatorial data.

The reframing. Theorem 3.14 shows that replacing the single e with two potentially distinct Demazure roots (e1, e2) — both still associated to the same ray generator p_i — yields:

$\chi^u \mapsto \chi^u \otimes \chi^u (1 \otimes \chi^{e_1} + \chi^{e_2} \otimes 1)^{\langle p_i, u \rangle}$

The binomial expansion now produces terms of the form χ^{e_1} on the right tensor factor and χ^{e_2} on the left tensor factor in asymmetric combinations. Swapping tensor factors sends 1 ⊗ χ^{e_1} + χ^{e_2} ⊗ 1 to χ^{e_1} ⊗ 1 + 1 ⊗ χ^{e_2}, which is different whenever e1 ≠ e2. The cocommutative condition e1 = e2 recovers the Dzhunusov-Zaitseva formula exactly.

Why this is conceptually powerful. The Demazure root data is purely combinatorial — it depends only on the cone σ in the lattice N and the choice of ray generator. The pair (e1, e2) captures all the information needed to reconstruct the monoid multiplication, including the parameter n that distinguishes the groups G_n (it appears as the difference between the two roots in the basis (-e1, v_i), Lemma 3.13). This means the entire noncommutative classification can be read off from the toric data: given a toric surface Z_σ, one enumerates the Demazure roots associated to each ray generator, picks a pair for one of the rays, and writes down the comultiplication formula. The only monoid structures of rank 1 on Z_σ arise this way, and they are commutative exactly when the pair has equal entries.

Comparison to the cone classification. The cone-based classification (Theorem 3.10, Construction 3.6) gives a complete list but does not reveal why the particular comultiplication takes the form it does. The Demazure root formulation (Theorem 3.14) provides the conceptual origin: the comultiplication is the unique way to extend the group multiplication on G_n to the toric variety while respecting the G_a-action encoded by the Demazure roots. This is analogous to how, in the theory of algebraic groups, the comultiplication on the coordinate ring of a group is essentially forced by the group law; here, the monoid comultiplication is forced by the group law on the open subset G_n together with the requirement that the G_a-orbits close in the monoid.

The correction to [4] reinforces the insight. Remark 3.16 notes an inaccuracy in [4, Corollary 1], which claimed infinitely many non-isomorphic commutative monoid structures on a surface with a one-dimensional cone (A¹ × K^×). In fact, all such structures are isomorphic to the group G_a × G_m itself — the boundary collapses because the cone is not full-dimensional. This correction highlights that the full-dimensional cone (and hence the full toric surface geometry) is essential for producing nontrivial monoid structures, and it underscores the tightness of the relationship between the Demazure root pair and the cone: you need two independent ray generators to have distinct e1, e2 or even meaningful monoid boundaries.

Evidence. Theorem 3.14 is proved by applying Lemma 3.13 (the basis (-e1, v_i)) to transform the abstract Demazure root formula into the explicit G_n comultiplication in coordinates, then reading off the cone data from the nonnegativity conditions on Demazure roots. The equivalence between the X_a,b_n, Y_a,b_n classification and the Demazure root pair classification is established in the proof by showing that the two cases n > 0 and n < 0 correspond exactly to the positive and negative cone families, and that the parameters (a,b) correspond to the ray generator data of the dual cone. Corollary 3.15 then characterises when two different Demazure root pairs produce isomorphic monoids — only when there is a toric automorphism (swapping the ray generators) that matches the pairs.

Innovation 3: The Image Ideal Invariants `L_k` as a Practical Distinguishing Tool

The paper introduces a new numerical invariant L_k(X) — the codimension of the k-th image ideal of the left locally nilpotent derivation δ_l — and demonstrates that this invariant cleanly separates all the non-isomorphic monoids in the classification without requiring any structural group theory. This is a methodological innovation rather than a conceptual one, but it is significant because it provides a practical, computable criterion for distinguishing monoids that is independent of the classification itself.

What was done before. In many classification problems in algebraic geometry, proving that two objects are non-isomorphic requires constructing ad hoc invariants — something about the singularity type, the automorphism group, the intersection theory — that are often difficult to compute or require deep structural knowledge. In the commutative monoid classification [4], the parameters were distinguished by the cone data itself (the ray generators), but the proof that different cones give non-isomorphic monoids relied on the fact that the cone determines the toric variety up to isomorphism. In the noncommutative case, an additional subtlety arises: two different cones σ^∨ might conceivably produce isomorphic monoids if there is an isomorphism that does not preserve the toric structure (i.e., a non-toric automorphism of the variety that intertwines the monoid multiplications). One needs an invariant that depends on the monoid structure, not just the variety.

The diagnostic concept. The left G_a-action on the monoid (by left multiplication by elements of the additive subgroup) induces the locally nilpotent derivation δ_l = ∂_x on K[X]. The kernel ker(δ_l) ∩ K[X] consists of functions invariant under this left G_a-action — these are the functions of y alone. The image δ_l^k(K[X]) consists of functions that can be obtained as k-th order x-derivatives of monoid functions. The quotient measures, roughly, how many independent functions of y are not obtainable as k-th derivatives — it captures the "gap" between the monoid algebra and the subalgebra generated by applying δ_l repeatedly.

Why this is a good invariant. First, it is computed directly from the cone geometry (Lemma 3.8 gives explicit formulas in terms of a and b), so it is practical to evaluate. Second, it depends only on the derivation δ_l, which is defined up to scalar (Remark 3.4), so the codimension is independent of arbitrary coordinate choices. Third, it is sensitive to the monoid structure: the derivation δ_l exists only because of the monoid's G_a-action, and two monoids that are isomorphic as varieties but not as monoids will generally have different L_k invariants because the isomorphism won't intertwine the derivations.

Distinguishing the two families. Beyond separating different (a,b) parameters within each family, the invariant L_k also provides a clean algebraic obstruction separating X_a,b_n from Y_a,b_n. In K[X_a,b_n], the element x^n satisfies δ_r = x^n δ_l (the right derivation is a multiple of the left derivation by a monomial in the algebra). In K[Y_a,b_n], no such element exists because the required monomial x^n has weight (n, 0), whose y-coordinate is zero — but the cone \bar{σ}^∨(n,a,b) only contains points with negative y-coordinate. This difference is not captured by L_k alone (Lemma 3.8 shows L_k different by the shift nk), but the structural relation δ_r = f · δ_l with f in the algebra provides an additional, independent invariant, which the paper invokes in Proposition 3.9.

Evidence. Lemma 3.8 computes L_k(X_a,b_n) = ⌈kb/a⌉ and L_k(Y_a,b_n) = ⌈k(b+na)/a⌉ = ⌈kb/a⌉ + nk. Proposition 3.9 then uses these formulas to show that if X_a,b_n ≅ X_a',b'_n, then plugging k = aa' gives different values unless (a,b) = (a',b'). The algebraic obstruction δ_r = x^n δ_l in K[X] but not in K[Y] separates the families. This double-layered invariant system makes the non-isomorphism proof robust and avoids reliance on the classification theorem itself — one can distinguish any two monoids in the list without knowing that the list is complete.

Innovation 4: The Boundary Divisor as a Structural Invariant — Irreducibility, Zero Element, and the `G_n`-Module Structure

The paper's analysis of the boundary divisor R(X) = X \ G(X) in Theorem 4.3 and Corollary 4.4 reveals that, despite the infinite family of distinct monoids parameterized by (n, a, b), the boundary divisor is always isomorphic to the affine line A¹, always contains a zero element, and the multiplication restricted to the boundary is null (x · y = 0 for boundary elements). Moreover, the G_n × G_n-orbit structure on X is constant: three orbits — the group itself, the punctured boundary, and the zero element.

Why this is non-obvious. In the theory of algebraic monoids, the boundary divisor encodes the "degeneration" of the group multiplication at infinity. For reductive monoids, the boundary has a rich structure — it typically decomposes into multiple orbits corresponding to the Bruhat decomposition, with the number and dimensions of boundary orbits reflecting the representation theory of the group. One might expect that the two-dimensional solvable case would exhibit similar complexity, with different parameter choices (a,b) producing boundary divisors with different geometry (different numbers of irreducible components, different singularity types, different orbit structures). The paper shows the opposite: the boundary is maximally simple and constant across the entire classification.

What this says structurally. The fact that R(X) ≅ A¹ regardless of the cone parameters suggests that the boundary is "universal" for these monoids — it does not depend on the specific embedding of G_n into the toric surface but only on the fact that G_n has a one-dimensional unipotent radical and a one-dimensional torus. The null multiplication on the boundary (x · y = 0 for all x, y ∈ R) is a hallmark of monoids with a zero element, and its presence in all examples (except the degenerate X_1,0_n case) indicates that the monoid structure forces the boundary to collapse multiplicatively. Theorem 4.3(3) gives the precise action of G_n on the boundary line R ≅ A¹: left multiplication scales the coordinate by τ^{b+an}, while right multiplication scales by τ^b. These weights depend on the parameters (a,b,n), providing a way to recover the parameters from the boundary action even though the boundary variety itself is always A¹.

Why the X_1,0_n exception is revealing. The monoid X_1,0_n has b = 0 and a = 1. Its multiplication formula is (x1, y1) · (x2, y2) = (x1 + y1^n x2, y1 y2), and the boundary multiplication is (x1, 0) · (x2, 0) = (x1, 0) — it is not null. This is exactly the case where R(X) fails to contain a zero element in the usual sense (the multiplication on the boundary is the projection onto the first coordinate, not the constant zero map). Geometrically, this corresponds to the cone σ^∨(n, 1, 0) having its second ray generator (1, 0) lying on the x-axis, which is the boundary of the half-plane σ_{G_n}^∨. This degeneracy case is precisely where the boundary does not collapse to a multiplicative zero, and it is exactly the case where b = 0. The paper's analysis thus gives a precise criterion: the zero element exists in R(X) if and only if b > 0.

Corollary 4.4 — the orbit structure. The G_n × G_n-action on X by left and right multiplication has exactly three orbits: the open orbit G_n, the one-dimensional orbit R(X) \ {0}, and the zero-dimensional orbit {0}. This is the simplest possible orbit structure for a non-group monoid and stands in stark contrast to reductive monoids (which typically have a much richer poset of orbits). The three-orbit structure reflects the fact that the boundary is irreducible and contains a fixed point of the G_n × G_n-action.

Significance beyond the classification. The constancy of the boundary geometry across the family has a practical implication: one cannot distinguish non-isomorphic monoids by looking only at the boundary as an abstract variety. The monoid structure is encoded in how G_n acts on the boundary (the weight parameters) and in the interior geometry (the cone), not in the boundary's intrinsic geometry. This is a cautionary tale for classification problems: two non-isomorphic objects can have isomorphic "boundaries at infinity" but differ in how the interior group approaches that boundary. The paper's explicit weight formulas make this precise.

Evidence. Theorem 4.3 is proved first for the A² case (a = 1) by direct computation with the explicit multiplication formula, then extended to general (a,b) via the quotient isomorphism X_a,b_n ≅ X_1,b_an / C_a from Proposition 4.2, using the fact that the quotient of A¹ by a finite cyclic group is again A¹. The orbit decomposition follows directly from the weight formulas in Theorem 4.3(3).

5. Experimental Analysis

Evaluation Methodology

Dataset and domain. The entire paper operates within the universe of two-dimensional normal affine algebraic varieties over an algebraically closed field K of characteristic zero, equipped with an algebraic monoid structure whose group of invertible elements is noncommutative. This is not an empirical dataset in the machine-learning sense — there is no train/test split, no collection of example monoids to classify. Rather, the "data" are the objects of a mathematical category (normal affine surfaces with algebraic monoid structures), and the task is to produce a complete, redundancy-free list of isomorphism classes together with explicit structural descriptions. The "test set" is the infinite family of all such monoids; the claimed result is that the classification captures every one.
Base objects and parameter space. The classification is parameterized by three integers: n > 0 (encoding the group of invertible elements G_n = G_a ⋊ G_m), a > 0, and b ≥ 0 with gcd(a, b) = 1. The two families X_a,b_n and Y_a,b_n, together with the group G_n itself, form the complete list. The paper also provides an equivalent parameterization via a pair of Demazure roots (e1, e2) associated to a ray generator of a strongly convex two-dimensional rational cone σ ⊂ N. The choice of this parameterization (rather than, say, a parameterization by singularity types or embedding dimensions) is motivated by the proof strategy: the toric structure (Theorem 3.2) reduces the problem to the combinatorics of cones and lattice points, where the parameters (a,b) emerge naturally as the coordinates of the second ray generator of the dual cone σ^∨ after normalizing the first ray generator to (0,1) or (0,-1).
Metrics for classification quality. A classification theorem in algebraic geometry is evaluated not by accuracy or F1 scores but by three criteria: completeness (does every object in the target class appear somewhere in the list?), non-redundancy (are the entries in the list pairwise non-isomorphic?), and constructivity (can the structure of each entry be explicitly described?). The paper claims completeness via Theorem 3.10, whose proof shows that any normal two-dimensional noncommutative affine monoid X must have K[X] = K[σ_X^∨ ∩ Z²] for a cone σ_X^∨ satisfying Lemma 3.5, which forces it to be one of the two constructed families or the group itself. Non-redundancy is established in Proposition 3.9 using the image ideal invariants L_k and the structural relation δ_r = x^n δ_l. Constructivity is provided by the explicit comultiplication formulas (Theorem 3.14, Corollary 3.11, Example 3.12) and the boundary divisor analysis (Theorem 4.3).
Baselines and prior classification. The commutative classification by Dzhunusov and Zaitseva [4] serves as the primary baseline. Their result classified all two-dimensional normal affine commutative monoids and established that rank-1 commutative monoids are toric with a comultiplication involving a single Demazure root. This paper's noncommutative classification must:
- reproduce the commutative case as a special sub-case (which it does when e1 = e2 in Theorem 3.14, recovering the Dzhunusov-Zaitseva formula exactly);
- extend the classification to cover all noncommutative monoids (which it does through the (n, a, b) parameterization);
- avoid introducing spurious additional structures that would make the commutative and noncommutative cases incommensurable (the unified Demazure root formulation achieves this).
The paper also acknowledges and corrects an inaccuracy in [4, Corollary 1] regarding monoids on one-dimensional cones (Remark 3.16), which serves as a consistency check: the new classification must not produce "extra" monoids in the degenerate case where the cone is not full-dimensional.
Invariants for distinguishing objects. The paper deploys a multi-layered system of isomorphism invariants rather than a single "compute budget":
- L_k(X) — the codimension of the k-th image ideal of δ_l, computed directly from the cone parameters (a,b,n) (Lemma 3.8). This separates different (a,b) pairs within each family.
- The algebraic relation δ_r = x^n δ_l holding in K[X_a,b_n] but not in K[Y_a,b_n] — a structural obstruction separating the two families (Proposition 3.9 end).
- The group of invertible elements G(X) ≅ G_n, whose parameter n is recovered from Z(G_n) (center is cyclic of order n, Proposition 2.6) or from the Demazure root pair data (the difference between e1 and e2 in the Lemma 3.13 basis).
- Toric automorphism equivalence: Corollary 3.15 checks when two Demazure root pairs (e1, e2) and (e1', e2') can be related by swapping the ray generators of σ, which is the only possible source of isomorphism between formally different parameter choices.
These invariants are applied in Proposition 3.9 to prove pairwise non-isomorphism, and their computability from the cone data without reference to the full monoid structure is a deliberate design choice — it makes the classification self-contained and verifiable.
Cross-validation / proof protocol. There is no statistical cross-validation; instead, the paper's proof architecture serves an analogous function:
- Two-path redundancy. The classification is established twice: once through the cone-based approach (Construction 3.6 → Theorem 3.10) and once through the Demazure root approach (Theorem 3.14). The equivalence of these two descriptions is proved in the body of Theorem 3.14 by explicit coordinate transformation (Lemma 3.13), providing an internal consistency check.
- Boundary case verification. Degenerate and edge cases are explicitly checked: the X_1,0_n case (where b = 0, producing non-null boundary multiplication, Theorem 4.3), the group case G_n itself (which is not covered by the a > 0, b ≥ 0 parameterization but appears as a separate entry in Theorem 3.10), and the one-dimensional cone case (Remark 3.16, correcting [4]).
- Completeness argument by exhaustion. The proof of Theorem 3.10 proceeds by case analysis on which half of the y-axis is contained in σ_X^∨, with Lemma 3.5 providing the necessary and sufficient condition. Every subcone satisfying the condition falls into one of the two families or the degenerate (non-proper subcone) group case. This leaves no unclassified territory.

Main Quantitative Results

This paper is a pure mathematics classification result — it does not report numerical measurements in the conventional sense. The "quantitative results" are the cardinalities and structural descriptions of the classified families, the explicit formulas for the comultiplication and boundary multiplication, and the invariant computations that separate the isomorphism classes. What follows is organized by the paper's logical groupings.

Theorem 3.10: The Complete List — How Many and Which Ones

The headline result is that the set of isomorphism classes of two-dimensional normal affine noncommutative algebraic monoids is in bijection with the disjoint union of:

One group family: G_n for n ∈ Z_{>0} — countably infinite, parameterized by a single positive integer.
The X series: X_a,b_n for n ∈ Z_{>0}, a ∈ Z_{>0}, b ∈ Z_{≥0} with gcd(a, b) = 1 — countably infinite, parameterized by three integers subject to one coprimality condition.
The Y series: Y_a,b_n for n ∈ Z_{>0}, a ∈ Z_{>0}, b ∈ Z_{≥0} with gcd(a, b) = 1 — same cardinality as the X series, but non-isomorphic to any X monoid (Proposition 3.9) and representing the opposite monoid structure (Proposition 4.1).

What this means concretely. For each choice of (n, a, b), there is exactly one monoid X_a,b_n and exactly one monoid Y_a,b_n, and every noncommutative normal affine monoid of dimension two appears exactly once in this list (except the groups G_n, which are the case where the monoid has no boundary). The parameter n encodes the "strength of noncommutativity" — it is the integer defining the semidirect product G_n = G_a ⋊ G_m and cannot be changed by isomorphism (Corollary 2.7). The parameters (a, b) encode the cone geometry of the toric surface, with gcd(a, b) = 1 ensuring the ray generator is primitive (necessary for normality of the toric variety, [3]).

Evidence for completeness. Theorem 3.10, proof. The argument shows that for any normal affine noncommutative monoid X, the cone σ_X^∨ must be a proper two-dimensional subcone of σ_{G_n}^∨ satisfying Lemma 3.5(2). The y-axis (the line a = 0 in M_Q) must intersect σ_X^∨ in a ray, and the half of the y-axis that lies in the cone determines which family we are in. The second ray generator, by primitivity and the stability condition, must take the form (a, b) or (a, -na-b) with the stated constraints. Since every such cone produces a valid monoid (Proposition 3.7), and the parameters are uniquely determined by the cone (up to the normalization of the y-axis ray to (0,1) or (0,-1)), the list is complete.

Evidence for non-redundancy. Proposition 3.9. The argument has three parts:

Within the X series: if X_a,b_n ≅ X_a',b'_n, then L_{aa'}(X_a,b_n) = a'b must equal L_{aa'}(X_a',b'_n) = ab'. The coprimality gcd(a,b) = gcd(a',b') = 1 forces (a,b) = (a',b').
Within the Y series: analogous computation with the Y-specific L_k formula (Lemma 3.8).
Between X and Y: the algebraic relation δ_r = x^n δ_l holds in K[X] (since x^n is a monomial in the algebra) but cannot hold in K[Y] because the required monomial x^n has weight (n, 0), whose y-coordinate 0 lies outside the negative cone \bar{σ}^∨(n,a,b).

Theorem 3.14: Unified Comultiplication Formula — The Single Algebraic Expression That Generates All Monoids

The Demazure root formulation provides a single comultiplication formula that, for appropriate choices of the pair (e1, e2), produces every monoid in the classification:

$\chi^u \mapsto \chi^u \otimes \chi^u (1 \otimes \chi^{e_1} + \chi^{e_2} \otimes 1)^{\langle p_i, u \rangle}$

What this formula unifies. When e1 = e2, it reduces to the Dzhunusov-Zaitseva commutative formula [4, Theorem 3]. When e1 ≠ e2, the asymmetry produces noncommutativity. The exponent ⟨p_i, u⟩ is the pairing between the lattice point u ∈ σ^∨ ∩ M and the ray generator p_i of the cone σ in the dual lattice N. This pairing measures the "distance" of u from the facet of σ^∨ dual to p_i.

How the parameter n emerges. In the basis (-e1, v_i) provided by Lemma 3.13, the two Demazure roots take the form e1 = (-1, 0) and e2 = (-1, n) for some integer n. The sign of n determines whether we are in the X family (n > 0) or Y family (converted via n < 0 to a positive index). The magnitude |n| is the n parameter in G_n.

How the parameters (a, b) emerge. The second ray generator of the cone σ^∨ in the basis (-e1, v_i) has coordinates (a, b) with a > 0. The Demazure root condition ⟨p_j, e2⟩ ≥ 0 (for j ≠ i) translates to b + na ≥ 0 when n > 0, or more generally to the inequality that distinguishes which y-axis direction is included in the cone. The coprimality condition gcd(a, b) = 1 corresponds to the ray generator being primitive in M.

Corollary 3.15 — when do different root pairs give the same monoid? The only possible isomorphism between monoids defined by different Demazure root pairs arises when there is a toric automorphism of Z_σ that swaps the two ray generators of σ. This happens if and only if \hat{τ}(M) = M, where \hat{τ} is the linear map swapping the two ray generators of σ^∨. When this lattice condition fails, every distinct pair (e1, e2) produces a distinct monoid. This result provides a precise count of isomorphism classes on a given toric surface Z_σ.

Proposition 3.9 and Lemma 3.8: Invariant Computations That Separate the Families

The L_k invariant provides explicit numerical separation:

$L_k(X_a,b_n) = \left\lceil \frac{kb}{a} \right\rceil, \qquad L_k(Y_a,b_n) = \left\lceil \frac{k(b+na)}{a} \right\rceil = \left\lceil \frac{kb}{a} \right\rceil + nk$

What these formulas compute. For X_a,b_n, the integer L_k(X) counts the number of nonnegative integers m such that m < kb/a — i.e., y^m is in the kernel of δ_l but cannot be expressed as δ_l^k(f) for any f ∈ K[X]. Equivalently, it is the minimal nonnegative integer m such that the lattice point (k, m) lies in the cone σ^∨(n, a, b).

The Y formula and the shift nk. The +nk term in L_k(Y_a,b_n) reflects the fact that the Y cone is "tilted" downward by the semidirect product structure. The point (k, m) lies in \bar{σ}^∨(n, a, b) if and only if m ≤ -k(b+na)/a. The codimension counts negative integers, and the shift nk appears because the G_a-action encoded by δ_l interacts with the G_m-action in the noncommutative group G_n — the right multiplication by G_a picks up a factor of y^n, which shifts the relevant exponents.

Diagnostic use in Proposition 3.9. To distinguish X_a,b_n from X_a',b'_n, evaluate L_{aa'} on both. The values are a'b and ab' respectively. If (a,b) ≠ (a',b'), these differ because if a'b = ab', then since gcd(a,b) = gcd(a',b') = 1, we would have a = a' and b = b'. This is a clean arithmetic criterion with no geometric reasoning required.

Why the derivation scaling doesn't matter. Remark 3.4 notes that δ_l is only defined up to a nonzero scalar multiple (depending on the choice of coordinate on G_a). However, L_k(X) = dim(ker ∩ K[X]) / (ker ∩ δ_l^k(K[X])) is invariant under scaling δ_l by c ∈ K^×: scaling δ_l scales δ_l^k by c^k, which does not change the subspace δ_l^k(K[X]) and hence leaves the codimension unchanged.

Proposition 4.1: Opposite Monoids and the X-Y Duality

The monoid Y_a,b_n is isomorphic to the opposite monoid (X_a,b_n)^{op}. This is proved by constructing an explicit lattice automorphism \hat{ϕ}: M → M sending (0,1) ↦ (0,-1) and (1,0) ↦ (1,-n), which maps the cone σ^∨(n, a, b) onto \bar{σ}^∨(n, a, b). Under this isomorphism, the comultiplication on K[Y_a,b_n] is the tensor-swapped version of the comultiplication on K[X_a,b_n] — exactly what it means for the monoids to be opposite.

What this means classificationally. The Y family is not an independent second series — it is the "mirror image" of the X family under the operation of reversing the multiplication order. In the commutative case (e1 = e2), every monoid is isomorphic to its own opposite, so no separate Y family appears. The fact that X and Y are generally not isomorphic (Proposition 3.9) is a direct reflection of noncommutativity: a monoid is commutative if and only if it is isomorphic to its opposite via the identity map.

Theorem 4.3: Boundary Divisor Structure — Universality Across the Family

The boundary divisor R(X_a,b_n) = X_a,b_n \ G_n is always isomorphic to the affine line A¹, and the multiplication restricted to R × R satisfies x · y = 0 for all x, y ∈ R when b > 0. The action of G_n on R by left and right multiplication is:

$(\alpha, \tau) · x = \tau^{b+an} x, \qquad x · (\alpha, \tau) = \tau^b x$

for x ∈ R ≅ A¹ (identifying the zero element with 0 ∈ A¹).

The b = 0 exception. When b = 0, the monoid X_1,0_n (the only case where the cone ray (a,b) lies on the x-axis) has boundary multiplication (x_1, 0) · (x_2, 0) = (x_1, 0), which is not null. This is exactly the degenerate case where the cone σ^∨(n, 1, 0) has its second ray generator (1,0) on the boundary of the ambient half-plane — the boundary divisor does not collapse to a zero element. The general formula τ^{b+an} and τ^b still holds with b = 0, giving the weights τ^{an} and τ^0 = 1.

Proof structure. The theorem is first proved for a = 1 (the affine plane case, Corollary 3.11) by direct coordinate computation: the monoid is A² with explicit multiplication, the non-invertible locus is {y = 0} ≅ A¹, and the boundary multiplication (x_1, 0) · (x_2, 0) = (x_1·0^b + 0^{b+n}·x_2, 0·0) = (0, 0) for b > 0. The general case then follows from the quotient isomorphism X_a,b_n ≅ X_1,b_an / C_a (Proposition 4.2) and the fact that the quotient of A¹ by a finite cyclic group is again A¹.

Corollary 4.4: Orbit Structure — Exactly Three Orbits

The action of G_n × G_n on X_a,b_n by left and right multiplication has exactly three orbits:

The open orbit: G_n itself (dimension 2).
The punctured boundary: R \ {0} (dimension 1).
The zero element: {0} (dimension 0).

Why this is the "quantitative" orbit count. This is the simplest possible orbit structure for a non-group algebraic monoid, contrasting sharply with reductive monoids where the Bruhat decomposition produces a rich graded poset of orbits. The three-orbit structure is a direct consequence of Theorem 4.3(3): the G_n × G_n-action on R ≅ A¹ is given by the character (τ_1, τ_2) ↦ τ_1^{b+an} τ_2^{-b}, which has a dense orbit (the nonzero elements) and a fixed point (zero). There are no intermediate orbit closures because A¹ with this torus action has exactly two orbits ({0} and its complement).

Ablation Studies and Robustness Checks

In a pure mathematics classification paper, the analogs of "ablation studies" are edge case verifications, consistency checks against prior work, and robustness of the classification under weakening of hypotheses. The paper provides several of these.

Dropping the normality assumption (Theorem 3.2). The proof that every two-dimensional affine algebraic monoid is toric does not assume normality. This is a robustness check on the geometric framework: if the classification depended sensitively on integral closedness of the coordinate ring, non-normal monoids would require a separate treatment. Theorem 3.2 shows they do not — the toric structure is forced at the level of the underlying variety regardless of singularities. However, the subsequent cone-based classification (Construction 3.6, Theorem 3.10) does assume normality (to identify the coordinate ring with the semigroup algebra K[σ^∨ ∩ Z²] rather than a general subalgebra with the same cone). This means the paper's main classification theorem is stated for normal surfaces only, but the toric-forcing result holds more broadly. The paper acknowledges this distinction implicitly by stating "normal affine" in Theorem 3.10 while proving Theorem 3.2 without it.

Recovery of the commutative classification as e1 = e2. Theorem 3.14 states that the Demazure root comultiplication formula with a pair (e1, e2) defines a monoid, and that every rank-1 monoid on Z_σ arises this way. When e1 = e2 = e, the formula reduces to the Dzhunusov-Zaitseva formula χ^u ↦ χ^u ⊗ χ^u (1 ⊗ χ^e + χ^e ⊗ 1)^{⟨p,u⟩} [4, Theorem 3]. This is an explicit consistency check: the noncommutative classification contains the commutative one as a sub-case. The paper also verifies that the isomorphism conditions (Corollary 3.15) reduce correctly in the commutative case, noting that the commutative version in [4, Corollary 1] correctly identifies the role of \hat{τ} but incorrectly claims infinitely many structures on one-dimensional cones (corrected in Remark 3.16).

The X_1,0_n degenerate case (Theorem 4.3 boundary). The boundary divisor analysis reveals a qualitative difference when b = 0: the boundary multiplication is not null, and there is no zero element in the monoid. This is the only case in the X family where the boundary fails to collapse multiplicatively. The paper explicitly computes this case (Corollary 3.11 for a = 1, b = 0 gives the group-like formula (x1, y1) · (x2, y2) = (x1 + y1^n x2, y1 y2)), confirming that the b > 0 and b = 0 cases are genuinely distinct in their algebraic structure, not just in parameter values. This is analogous to checking that a machine learning result does not collapse at extreme hyperparameter values.

The one-dimensional cone correction (Remark 3.16). The paper identifies and corrects an inaccuracy in [4, Corollary 1], which claimed infinitely many non-isomorphic commutative monoid structures on the surface corresponding to a one-dimensional cone (A¹ × K^×). The paper explains that all such structures are actually isomorphic to the group G_a × G_m itself — the boundary collapses because the cone is not full-dimensional, and the variety is just the group. This correction is a robustness check on the necessity of the full-dimensional cone hypothesis in Theorem 3.14. It confirms that nontrivial monoid structures exist only when the cone is two-dimensional, which is exactly the setting of the main classification.

The quotient consistency check (Proposition 4.2). The proof that X_a,b_mn / C_m ≅ X_a',b'_n (where a' = a · m/gcd(m,b) and b' = b/gcd(m,b)) provides an internal consistency check on the parameterization. If one takes a monoid with parameters (a, b, mn) and quotients by the central subgroup of order m, one lands back in the same classification scheme with renormalized parameters (a', b', n). This shows that the parameterization is closed under taking quotients by central subgroups, and that the formulas for a' and b' are consistent with the requirement that gcd(a', b') = 1. This is the algebraic geometry analog of checking that a learned representation is equivariant under a group action — the classification should commute with natural operations on the objects being classified.

Invariant independence from coordinate choices (Remark 3.4). The derivation δ_l is only defined up to a nonzero scalar multiple (depending on the parametrization of the G_a subgroup in G_n). The invariant L_k(X), being a codimension of subspaces, is unaffected by this scaling ambiguity. Similarly, the structural relation δ_r = x^n δ_l in K[X_a,b_n] is invariant under scaling both derivations by the same constant. This is a robustness check on the well-definedness of the invariants used to separate isomorphism classes: the classification does not depend on arbitrary choices made in the proof.

Critical Assessment

Claim 1: Every two-dimensional affine algebraic monoid is toric (Theorem 3.2).

What the proof actually shows. The theorem is proved by case analysis on the group of invertible elements G(X). For G(X) a torus or unipotent, prior results or simple arguments apply. For G(X) = G_a × G_m, the proof uses the G_a-action to show that K[X] is spanned by monomials in K[G(X)], which gives the toric grading. For G(X) = G_n (noncommutative), the proof constructs a torus T ⊂ ρ_n(G_n × G_n) inside the automorphism group and argues by extension to the monoid that T acts on X, giving a toric structure.

Strengths. The proof is complete across all cases and drops both the normality and commutativity assumptions that prior work required. The construction of the torus T via ρ_n (Proposition 3.1) is explicit and verifiable: one can check that the action ν((t1, t2)) · (α, τ) = (t1^{-1} α, t2^{-1} τ) integrates to the standard grading deg(x) = (1,0), deg(y) = (0,1).

Latent assumptions. The proof for the G_n case relies on the fact that automorphisms in ρ_n(G_n × G_n) extend uniquely to the monoid X. This is stated without proof (the paper says it follows from the monoid axioms — right and left multiplications by fixed group elements are morphisms X → X that restrict to automorphisms on the open subset G_n, and since G_n is dense in X, the extension is unique). This is a standard argument in algebraic geometry (two morphisms agreeing on a dense open subset are equal), but the paper does not elaborate on why the extension of the G_n × G_n action to X is well-defined for non-group elements (i.e., why g x h^{-1} makes sense when x is a boundary element and g, h ∈ G_n). This is a standard property of algebraic monoids — the multiplication µ: X × X → X is defined everywhere, so the left and right multiplication maps L_g(x) = g·x and R_h(x) = x·h are morphisms defined on all of X for g, h ∈ G_n. The combined map (g, h) ↦ L_g ∘ R_{h^{-1}} then defines the G_n × G_n action. This reasoning is implicit but not spelled out in the proof of Theorem 3.2.

What would strengthen it. The theorem states it holds for "not necessarily normal" monoids, but the rest of the classification (Theorem 3.10) assumes normality. The paper does not explore whether the X_a,b_n and Y_a,b_n construction produces non-normal monoids for non-primitive ray generators (i.e., when gcd(a,b) > 1). A comment on what happens in the non-normal case — whether the same cone parameters with gcd(a,b) > 1 classify non-normal monoids, or whether non-normal monoids can arise from non-saturated semigroups that are not of the form σ^∨ ∩ Z² — would clarify the scope of Theorem 3.2 relative to the main classification.

Claim 2: The complete list of noncommutative normal affine two-dimensional monoids is `G_n`, `X_a,b_n`, and `Y_a,b_n` (Theorem 3.10).

What the proof actually shows. The proof takes a monoid X with G(X) = G_n, embeds it as a toric surface via Corollary 3.3, and argues that σ_X^∨ must be a proper two-dimensional subcone of the half-plane σ_{G_n}^∨ satisfying Lemma 3.5(2). It then exhausts the possibilities based on which y-axis ray (0,1) or (0,-1) lies in the cone, deriving the two families.

Strengths. The exhaustion argument is logically tight: Lemma 3.5 provides a necessary and sufficient condition for a subcone to admit the restricted comultiplication; the condition forces the cone to contain exactly one of (0,1) or (0,-1) (for proper subcones); and the second ray generator is determined up to the normalization gcd(a,b) = 1 for normality. The parameters are uniquely determined because the cone uniquely determines the toric surface up to toric isomorphism, and the toric structure is fixed by Corollary 3.3.

One implicit step worth flagging. The proof of Theorem 3.10 assumes that the cone σ_X^∨ is two-dimensional. It rules out the one-dimensional case implicitly: if σ_X^∨ were one-dimensional, the corresponding variety would be A¹ × K^× (as noted in Remark 3.16), and Remark 3.16 argues that the only monoid structure on this variety is the group G_n itself. However, this conclusion is stated in a remark about [4], not in the proof of Theorem 3.10. A reader skimming the proof might wonder whether one-dimensional subcones could produce non-group monoids not covered by the two families. The paper would be stronger if it explicitly addressed this case in the theorem proof rather than delegating it to a remark about prior work.

What is not tested. The paper claims that G_n, X_a,b_n, and Y_a,b_n are pairwise non-isomorphic (Proposition 3.9). The proof that X ≅ Y is impossible uses the structural relation δ_r = x^n δ_l. This argument assumes that the derivation δ_r is exactly y^n ∂_x and δ_l is exactly ∂_x — i.e., that the isomorphism would preserve not just the abstract derivations but their specific expressions in the standard coordinates. Since an isomorphism of monoids induces an isomorphism of the corresponding G_a-actions (left and right multiplications by the additive subgroup), the induced map on coordinate rings must intertwine δ_l with δ_l and δ_r with δ_r (up to the scalar ambiguity of Remark 3.4). The paper's argument that K[Y] lacks an element f satisfying δ_r = f · δ_l requires checking that no element of K[Y] can serve this role, not just that the specific monomial x^n is absent. The paper states "there is no such element in K[Y]" — this is true because any such element would have to be homogeneous of degree (n, 0) (to convert ∂_x to y^n ∂_x), and points with y-coordinate zero are not in \bar{σ}^∨(n,a,b). This reasoning is sound but compressed; a reader unfamiliar with graded algebras might need to fill in the step that the derivation-twisting element must be homogeneous.

Claim 3: Every monoid structure of rank 1 on a toric surface `Z_σ` is given by the Demazure root pair comultiplication formula (Theorem 3.14).

What the proof actually shows. The proof transforms the Demazure root formula into the G_n comultiplication via the basis (-e1, v_i) from Lemma 3.13, then identifies the resulting monoid as one of the X_a,b_n or Y_a,b_n families (or their commutative e1 = e2 limit). The converse — that every monoid in the X and Y families can be written this way — follows from the explicit parameter translation: given (n, a, b), the cone σ (in N) is the dual of σ^∨(n, a, b) or \bar{σ}^∨(n, a, b), and the Demazure roots are e1 = (-1, 0) and e2 = (-1, n) in the basis (-e1, v_i).

Strengths. The Demazure root formulation is genuinely more conceptual than the cone parameterization. It reveals that the comultiplication is the "binomial expansion" of the G_a-action encoded by the Demazure roots, and it makes the commutative/noncommutative dichotomy transparent (e1 = e2 vs. e1 ≠ e2). The equivalence between the two parameterizations is proved constructively rather than by abstract existence arguments.

What the proof does not address. Theorem 3.14 states that the formula "defines a comultiplication on K[Z_σ]." The proof of coassociativity — that the comultiplication satisfies the bialgebra axiom (µ* ⊗ id) ∘ µ* = (id ⊗ µ*) ∘ µ* — is not provided. The paper appears to rely on the fact that the formula comes from restricting the group comultiplication on G_n to the subalgebra K[Z_σ], and coassociativity of the group comultiplication is known. The verification in Lemma 3.5 shows that the comultiplication restricts to the subalgebra, but does not re-verify coassociativity for the restricted structure. This is a standard argument — coassociativity is inherited by sub-bialgebras — but the paper does not flag this explicitly.

The parameter shift for negative n. The proof of Theorem 3.14 handles n < 0 by reindexing: it obtains a monoid isomorphic to Y_a,b+na_{-n}. A reader needs to track the parameter shift carefully: the a parameter remains the same, the b parameter changes from b to b+na (which is nonnegative by the Demazure root condition ⟨p_j, e2⟩ ≥ 0), and the n parameter changes sign. The paper asserts this without walking through the index transformation, which might cause confusion when comparing the Demazure root classification to the cone classification.

Claim 4: The boundary divisor is always `A¹` with null multiplication and a zero element (Theorem 4.3).

What the proof actually shows. For a = 1, the proof is a direct coordinate computation. For general a, it uses the quotient X_a,b_n ≅ X_1,b_an / C_a (Proposition 4.2) and the fact that A¹ / C_a ≅ A¹ (since the quotient of the affine line by a finite group acting linearly is always an affine line). The formulas for the G_n-action weights are inherited from the quotient.

Strengths. This is a complete structural description of the boundary. The weights τ^{b+an} (left action) and τ^b (right action) provide explicit, parameter-dependent invariants that distinguish the G_n × G_n-module structure on the boundary. The three-orbit decomposition (Corollary 4.4) is a clean consequence.

A subtle point. The proof in Theorem 4.3 first handles X_1,b_n (the A² case) and then extends to general X_a,b_n via the quotient. For this to work, one needs to know that the boundary divisor of X_a,b_n is the quotient of the boundary divisor of X_1,b_an by C_a. This requires checking that the quotient map X_1,b_an → X_a,b_n restricts to a map on the boundaries (the non-invertible loci), which follows from the fact that the quotient map is a monoid homomorphism (being the quotient by a central subgroup) and therefore maps non-invertible elements to non-invertible elements. The paper does not elaborate on this, leaving it to the reader.

Overall Assessment

The paper's classification is exhaustive and rigorous within its stated scope (normal affine surfaces, characteristic zero, algebraically closed field). The proofs are constructive: given a monoid, one can extract the parameters (n, a, b) from the group of invertible elements and the cone of the toric structure, and given the parameters, one can write down the explicit multiplication formula. The internal consistency checks (commutative limit, opposite monoid duality, quotient closure, boundary behavior) confirm that the classification is not ad hoc but fits into a coherent structural framework.

Genuine limitations:

Characteristic dependence. The paper works over an algebraically closed field of characteristic zero. The structure of G_n (Proposition 2.5) and the classification of solvable groups may differ in positive characteristic, where the group G_n could have different center structure or the Levi-Malcev decomposition may fail. The paper does not address whether the classification extends to positive characteristic.
The non-normal case is not classified. Theorem 3.2 establishes that non-normal two-dimensional affine monoids are toric, but Theorem 3.10 classifies only the normal ones. Non-normal toric surfaces correspond to non-saturated semigroups — sub-semigroups of σ^∨ ∩ Z² that generate the same cone but miss some lattice points on the boundary. Which of these admit monoid structures extending the G_n multiplication? The paper does not explore this. The stability condition (Lemma 3.5) would still apply, but the classification would be finer (more monoids, some of which might be non-isomorphic to the normal ones even with the same parameters).
Dimension restriction. The paper explicitly restricts to dimension two. The Demazure root formulation (Theorem 3.14) and the stability condition (Lemma 3.5) use the fact that the cone is two-dimensional in essential ways (e.g., the y-axis is a one-dimensional boundary of the half-plane, Demazure roots are associated to specific ray generators). Extending to higher dimensions would require a more sophisticated combinatorial framework. The paper does not discuss which aspects might generalize (the Demazure root pair idea likely extends, but the classification of subcones satisfying the analog of Lemma 3.5 for higher-dimensional cones would be substantially more complex).
No explicit verification of coassociativity. As noted above, the paper relies on the fact that the comultiplication restricts from the group comultiplication to prove that it defines a monoid, but the restriction argument (Lemma 3.5) only checks that the comultiplication lands in K[X] ⊗ K[X]. Coassociativity, the unit property, and compatibility with the algebra structure are inherited from the ambient group bialgebra but this inheritance is not explicitly verified or cited. In a rigorous classification paper, this is a minor gap that most algebraic geometers would fill in, but it is a gap nonetheless.

6. Limitations and Trade-offs

Limitation 1: The Classification Assumes Normality of the Underlying Variety

The assumption or constraint. Theorem 3.10, the main classification result, is stated explicitly for "normal affine two-dimensional noncommutative algebraic monoids." Normality — the condition that the coordinate ring K[X] is integrally closed — corresponds geometrically to the variety having no singularities in codimension one (roughly, the surface is smooth except possibly at isolated points, and those singularities are "mild" enough that the algebraic functions behave well). Concretely, normality is what allows the coordinate ring to be identified with the semigroup algebra K[σ^∨ ∩ Z²] of all lattice points in the dual cone — a sublattice missing some boundary points would also produce a toric surface invariant under the G_a-actions and satisfying Lemma 3.5, but it would not correspond to a saturated semigroup and would therefore be non-normal.

The consequence. The classification does not cover non-normal two-dimensional affine algebraic monoids, even though Theorem 3.2 proves that such monoids must also be toric. Non-normal toric surfaces correspond to subsemigroups of σ^∨ ∩ Z² that generate the same cone but lack some lattice points (typically on the facets). Such subsemigroups, provided they satisfy the Lemma 3.5 stability condition (a,b) ∈ S ⇒ (0,b), (0,b+na) ∈ S, would yield monoid structures not isomorphic to those in the X_a,b_n and Y_a,b_n families, because the normalization map (adding in the missing lattice points) would change the variety but not the group G_n. The paper provides no enumeration of which such non-normal subsemigroups actually admit a comultiplication restricting from K[G_n], nor any criterion for when two such non-normal monoids are isomorphic. An investigator encountering a non-normal monoid in the wild (e.g., as a degeneration or a quotient of a higher-dimensional monoid) has no classification to consult.

What evidence exists in the paper. Theorem 3.2 explicitly states it applies to "not necessarily normal" monoids, acknowledging the generality gap. Theorem 3.10 then restricts to "normal affine" without further discussion. Construction 3.6 defines σ^∨(n,a,b) with the primitivity condition gcd(a,b) = 1, which is precisely the condition for the semigroup σ^∨ ∩ Z² to be saturated (normal). The paper does not include an experiment, example, or even a remark about what happens when gcd(a,b) > 1 — whether the corresponding non-normal monoids are valid, how many there are, or whether they can be reduced to the normal ones by some quotient or extension procedure.

Mitigation status. Not addressed. The paper does not flag the non-normal case as a limitation, does not provide a single example of a non-normal monoid, and does not discuss whether the classification strategy would extend. A reader interested in the non-normal setting would need to work this out from scratch, using Lemma 3.5 as a starting point but without guidance on the isomorphism problem or the comultiplication formulas.

Limitation 2: The Entire Classification Is Confined to Dimension Two

The assumption or constraint. Every theorem in the paper — the toric-forcing result (Theorem 3.2), the cone construction (Construction 3.6), the Demazure root formula (Theorem 3.14), the boundary divisor analysis (Theorem 4.3), the invariant L_k computations (Lemma 3.8) — is proved under the explicit hypothesis that the variety X has dimension two. The ambient group algebra K[G_n] = K[x, y, y^{-1}] has two variables, the torus action grading is by Z², the cone σ_{G_n}^∨ is a half-plane in Q², and the classification of possible subcones σ_X^∨ depends on the geometry of two-dimensional convex cones (which have exactly two ray generators, making the case analysis in the proof of Theorem 3.10 exhaustive).

The consequence. The paper provides no evidence and no conjectural framework for how the classification might extend to dimension three or higher. Several structural elements that simplify the two-dimensional case fail in higher dimensions:

Cones in higher dimensions. In dimension d, a strongly convex cone has d extremal rays. The "stability condition" under δ_l and δ_r will constrain which subcones of the ambient group cone admit monoid structures, but the combinatorics of such subcones is far richer — there are continuous families of cones in dimension ≥ 3, not the discrete enumeration by two integer parameters (a,b) that occurs in dimension two.
Demazure roots and toric geometry. In higher dimensions, Demazure roots are associated to facets (codimension-one faces) of the cone rather than to ray generators. The decomposition in Lemma 3.13 (w = bv - ⟨p_i, w⟩e making (-e, v) a basis of M) uses the fact that the orthogonal complement of a ray generator is one-dimensional — in higher dimensions it is (d-1)-dimensional, and the basis argument collapses. The Demazure root pair formulation (Theorem 3.14) has no obvious generalisation to cones of dimension ≥ 3.
The group of invertible elements. In dimension two, the only noncommutative connected linear algebraic groups are the semidirect products G_n = G_a ⋊ G_m (Proposition 2.5). In higher dimensions, there are many more such groups (iterated semidirect products, groups with higher-dimensional unipotent radicals, groups with more complicated torus actions, etc.). Each would require its own analysis of the toric structure on the group and the extension to boundary divisors. The paper's strategy of reducing to G_n and then classifying subcones of a specific half-plane does not obviously generalise.

What evidence exists in the paper. None. The paper neither discusses the higher-dimensional problem nor claims that its methods extend. The introduction frames the work as completing the two-dimensional classification, which is a self-contained goal. But the absence of even a remark about higher dimensions — whether the Demazure root pair idea suggests a natural generalisation, or whether the two-dimensional case is expected to be profoundly simpler — leaves a reader wondering about the scope of the techniques.

Mitigation status. Not addressed. The paper is strictly a dimension-two classification and does not claim otherwise. A practitioner working with higher-dimensional monoids (e.g., closures of solvable groups in endomorphism varieties, or monoids arising in spherical geometry) will find no guidance here, apart from the general principle that monoid structures force toric geometry — a result the paper proves only for dimension two and which may or may not hold in higher dimensions.

Limitation 3: The Classification Depends on the Base Field Being Algebraically Closed of Characteristic Zero

The assumption or constraint. The paper states in Section 1: "Let K be an algebraically closed field of characteristic zero. We assume all varieties to be defined over K." This is a standard setup in algebraic geometry over algebraically closed fields, but it rules out both positive characteristic (where the group theory and the structure of locally nilpotent derivations can differ substantially) and non-algebraically closed fields (where rational points, Galois actions, and forms of algebraic groups enter the picture).

The consequence. If the base field has positive characteristic p > 0, several steps in the paper break or require significant modification:

The group G_n may not be the only two-dimensional noncommutative connected group. Proposition 2.5 uses the Levi-Malcev decomposition and structure theory of solvable groups, which may fail or take a different form in positive characteristic. In characteristic p, there exist infinitesimal group schemes (Frobenius kernels, Witt vectors) and unipotent groups with more complicated structure (e.g., the group law (α1, τ1) · (α2, τ2) = (α1 + τ1^p α2, τ1τ2) might not be the only option). The center of G_n (Proposition 2.6) might have additional p-power torsion.
Locally nilpotent derivations behave differently. The paper's invariant L_k(X) uses that the kernel of δ_l = ∂_x consists of functions of y alone, and that δ_l^k(x^k) = k! is a nonzero constant. In characteristic p, k! vanishes whenever k ≥ p, which would collapse the image ideal invariants for large k and potentially merge distinct monoids into the same isomorphism class. The entire calculus of derivations and their kernels (the image ideal theory from Freudenburg [5]) is more subtle in positive characteristic, where derivations can have p-powers that vanish identically.
The toric structure argument (Theorem 3.2) uses the fact that the torus T = G_m² acts with weights (1,0) and (0,1), and that the grading on K[G_n] decomposes the algebra into one-dimensional weight spaces. In positive characteristic, the representation theory of tori is still well-behaved (tori are linearly reductive in all characteristics), but the extension of the G_n × G_n-action from G_n to X may be affected if the stabilizers of boundary points involve inseparable morphisms.
For non-algebraically closed fields, the classification would need to account for Galois twists: a monoid over Q might become isomorphic to X_a,b_n after base change to \bar{Q}, but there may be multiple non-isomorphic Q-forms corresponding to the same geometric monoid. The paper's classification is strictly geometric (over the algebraic closure), and the descent problem is not addressed.

What evidence exists in the paper. The paper provides no characteristic-p examples, no discussion of the breakdown of the k! factor in δ_l^k, and no mention of forms over non-algebraically closed fields. The statement "characteristic zero" appears in the preliminaries and is not revisited.

Mitigation status. Not addressed. This is a standard scope limitation in algebraic geometry classification papers — one first solves the problem over an algebraically closed field of characteristic zero (the "geometric" case) and leaves the arithmetic and characteristic-p refinements to future work. The paper does not flag this as a limitation or suggest what would be needed to extend the classification. For a practitioner working over F_p or over Q, the results are suggestive but not directly applicable without revisiting the proofs.

Limitation 4: The Classification Is Specific to the Affine Case and Relies on the Algebraic Structure of the Coordinate Ring

The assumption or constraint. The entire paper operates in the category of affine algebraic monoids — monoids whose underlying variety is affine, meaning the coordinate ring K[X] is a finitely generated K-algebra and the maximal spectrum recovers the variety. This is not an incidental choice: the paper's methods (analysing the comultiplication on K[X], using cones to describe subalgebras of K[G_n], invoking locally nilpotent derivations on the coordinate ring) are fundamentally algebraic and rely on the global coordinate ring as the main object of study.

The consequence. The classification says nothing about non-affine (e.g., projective or quasi-projective) algebraic monoids, even though such objects exist and arise naturally. For example, a toric variety defined by a fan (not just a single cone) is typically not affine unless the fan consists of a single cone and its faces. The paper's construction produces only affine monoids (spectra of semigroup algebras for a single cone). If one wanted to classify all two-dimensional algebraic monoids — including, say, complete (compact) monoids that are closures of G_n in some projective embedding — the paper offers no method and no classification.

Furthermore, even within the affine case, the paper's technique depends heavily on the fact that the coordinate ring of G_n is a Laurent polynomial ring K[x, y, y^{-1}], which has a simple monomial basis and a manifestly computable comultiplication. For non-affine monoids, the algebra of global functions is not a finitely generated K-algebra (or loses information by not separating points), and one must work with local charts, line bundles, or projective embeddings — techniques this paper does not develop.

What evidence exists in the paper. The paper never claims to address non-affine monoids. The title specifies "normal affine surfaces." All constructions produce affine varieties (spectra of finitely generated algebras). There is no discussion of whether the X_a,b_n and Y_a,b_n monoids admit natural completions to projective monoids, or whether additional monoid structures exist on two-dimensional projective varieties.

Mitigation status. Not addressed. This is a deliberate scope restriction, and the paper is transparent about it in the title and introduction. However, a practitioner who encounters a two-dimensional monoid that is quasi-projective but not affine (e.g., a toric surface that is not defined by a single strongly convex cone) will need a separate theory. The paper provides no roadmap for extending the classification beyond the affine case, even for the natural next step of "toric monoids defined by fans."

Limitation 5: The Demazure Root Comultiplication Formula Is Presented Without Verifying the Bialgebra Axioms for the Restricted Structure

The assumption or constraint. Theorem 3.14 claims that the formula

$\chi^u \mapsto \chi^u \otimes \chi^u (1 \otimes \chi^{e_1} + \chi^{e_2} \otimes 1)^{\langle p_i, u \rangle}$

"defines a comultiplication on K[Z_σ]." For this to be a genuine comultiplication in the sense of algebraic monoids, the map must satisfy:

Coassociativity: (µ* ⊗ id) ∘ µ* = (id ⊗ µ*) ∘ µ*
Counit property: The counit η* (evaluation at the identity of G_n) must satisfy (η* ⊗ id) ∘ µ* = id = (id ⊗ η*) ∘ µ*
Algebra homomorphism: µ*(fg) = µ*(f) µ*(g)

The paper's proof of Theorem 3.14 shows that in the appropriate basis, the formula transforms into the G_n comultiplication x^a y^b ↦ (x⊗1 + y^n⊗x)^a (y⊗y)^b. Lemma 3.5 then checks that this comultiplication maps K[σ_X^∨ ∩ Z²] into K[σ_X^∨ ∩ Z²] ⊗ K[σ_X^∨ ∩ Z²] — i.e., that the image lands in the correct subalgebra. This is the restriction step: the comultiplication on the ambient group algebra K[G_n] restricts to the subalgebra K[X].

The consequence. The paper implicitly relies on the fact that coassociativity, the counit property, and the algebra homomorphism property are inherited by the restricted comultiplication from the ambient group bialgebra. This is true: if µ* : A → A ⊗ A is a coassociative comultiplication on an algebra A, and B ⊂ A is a subalgebra such that µ*(B) ⊂ B ⊗ B, then µ*|_B is automatically coassociative (coassociativity is an equality of maps B → B ⊗ B ⊗ B, and this equality already holds in A, so it holds in the subspace B). Similarly for the counit and algebra homomorphism properties.

However, the paper never states this inheritance argument. The verification in Lemma 3.5 checks only that the image condition holds — it does not mention coassociativity, the counit, or the algebra homomorphism condition. A reader unfamiliar with bialgebra restriction arguments might wonder whether additional checks are needed, whether the restricted structure could fail to be coassociative because K[X] ⊗ K[X] is not a sub-coalgebra in the appropriate sense, or whether the image condition µ*(K[X]) ⊂ K[X] ⊗ K[X] is sufficient by itself.

What evidence exists in the paper. The proof of Lemma 3.5(2) verifies only the image condition for monomials. The proof of Proposition 3.7 says "It is enough to prove that the cones... satisfy the second condition of Lemma 3.5" — i.e., the image condition is treated as sufficient for the existence of a monoid structure. The paper never writes down the coassociativity check explicitly, never discusses the counit η* beyond its definition in Section 2.4, and never verifies that µ* is an algebra homomorphism.

Mitigation status. Implicitly mitigated by the ambient bialgebra structure of K[G_n], but not explicitly addressed. In a rigorous algebraic geometry paper, this is a standard argument that experienced readers fill in automatically. However, given that the paper is a classification that must ensure every listed object is genuinely a monoid, the omission of an explicit statement like "coassociativity is inherited from K[G_n]" is a minor gap. The paper would be stronger if it stated this inheritance principle explicitly, particularly since Theorem 3.14 is presented as generating a comultiplication from Demazure roots without reference to the ambient group, making the inheritance argument less obvious.

Limitation 6: The Difficulty of Recognising a Given Monoid in the Classification Without Explicit Coordinates

The assumption or constraint. The classification is constructive — given parameters (n, a, b), one can write down the monoid explicitly (its toric surface as a cone, its comultiplication formula). However, the inverse problem — given a monoid presented in some other form (e.g., as a subvariety of A³ defined by equations, with a multiplication given by polynomial formulas in ambient coordinates), determine its (n, a, b) parameters — requires nontrivial computation that the paper does not algorithmize.

The consequence. A practitioner who encounters a two-dimensional normal affine noncommutative monoid in the wild — say, arising as the closure of a G_n-orbit in some representation, or as the spectrum of an invariant ring — cannot simply "look up" its place in the classification without performing the following steps:

Identify the group of invertible elements and determine n (which may require analyzing the center or the commutator subgroup).
Find the toric structure and compute the cone σ_X^∨ (which requires identifying a G_m²-grading on the coordinate ring and computing the weights of generators).
Determine the ray generators of σ_X^∨ and normalize them to extract (a,b), checking the stability condition from Lemma 3.5 to confirm that the monoid is indeed one of the classified ones.
Distinguish between X_a,b_n and Y_a,b_n using the L_k invariant or the δ_r = x^n δ_l structural relation. None of these steps is automated or even given as an algorithm in the paper — they are distributed across the proofs of Lemma 3.8, Proposition 3.9, and Theorem 3.10.

What evidence exists in the paper. The paper provides explicit formulas for the A² case (Corollary 3.11) and for one singular example (Example 3.12, X_2,2k+1_n), which demonstrate the process of reading off the parameters from the multiplication formulas. However, these examples start from the (n,a,b) parameters and derive the multiplication, rather than the reverse. The invariant L_k (Lemma 3.8) is computable from the cone data but requires knowing the cone — which itself requires knowing the toric structure on the monoid. If the monoid is presented without an obvious torus action (e.g., the multiplication is given by some ad hoc polynomials), finding the toric structure is a nontrivial first step that the paper does not address.

Mitigation status. Not addressed. This is more of a practical limitation than a theoretical one — the classification is mathematically complete, but using it as a "lookup table" requires computational work that the paper does not provide tools for. There is no discussion of algorithms for extracting the parameters from a given monoid, no mention of computational complexity (e.g., whether the L_k invariant can be computed without first finding a basis of the coordinate ring), and no reference to computational algebra systems that might automate any of these steps. For a practitioner with an explicit monoid in hand, the paper functions as a theoretical guarantee that the monoid is in the list, but not as a practical guide to finding which entry it is.

7. Implications and Future Directions

How This Work Changes the Landscape

This paper completes a program that has been running since the 1980s: classify all algebraic monoid structures on low-dimensional affine varieties. By providing the first full classification of noncommutative two-dimensional normal affine algebraic monoids, it closes the last open case in dimension two — the commutative classification having been settled by Dzhunusov and Zaitseva in [4]. This is not merely additive; the paper fundamentally reframes how one should think about the relationship between commutativity, toric geometry, and monoid structures.

The paradigm shift: from "commutative plus exceptions" to "Demazure root pairs as the primitive data." Prior to this work, the commutative Dzhunusov-Zaitseva comultiplication formula — involving a single Demazure root e — stood as the canonical way to construct rank-1 monoids on toric surfaces. One might reasonably have assumed that noncommutative monoids would require an entirely different construction, perhaps abandoning Demazure roots altogether in favor of some representation-theoretic data tied to the noncommutative group G_n. Theorem 3.14 shows the opposite: the pair (e1, e2) of Demazure roots is the correct primitive, and the single-root commutative case is simply the specialization e1 = e2. This is a genuine reframing — it says that noncommutativity is not a departure from the Demazure root framework but is already encoded in it, through the asymmetry between two roots associated to the same ray generator. The paper thus provides a unified algebraic origin for both commutative and noncommutative monoid multiplications on toric surfaces, collapsing what appeared to be two separate theories into one.

Reconciliation of conflicting intuitions. The structure theory of algebraic monoids developed by Putcha, Renner, and Vinberg [8, 9, 13] focused heavily on the reductive case, where the Bruhat decomposition and Weyl group combinatorics govern the orbit structure. Solvable monoids — particularly noncommutative ones with unipotent radicals — were largely uncharted territory. The implicit expectation was that solvable monoids might exhibit fundamentally different geometric behavior from reductive ones, perhaps involving non-toric varieties or more exotic singularities. Theorem 3.2 demolishes this expectation for dimension two: every two-dimensional affine algebraic monoid is toric, regardless of commutativity or normality. This is a rigidity theorem of the strongest kind — the monoid structure forces toric geometry, full stop. The paper thus reconciles the reductive and solvable pictures in low dimension by showing that the underlying geometry is, in both cases, combinatorial (cones and fans), with the group-theoretic complexity entering only through the choice of Demazure roots and the comultiplication formula, not through the variety itself.

A new diagnostic tool: the image ideal invariants L_k. The paper introduces the codimensions L_k(X) = dim(ker(δ_l) ∩ K[X]) / (ker(δ_l) ∩ δ_l^k(K[X])) as practical isomorphism invariants computable directly from the cone parameters (Lemma 3.8: L_k(X_a,b_n) = ⌈kb/a⌉, with the Y series shifted by nk). This provides a purely combinatorial method for distinguishing non-isomorphic monoids that neither requires analyzing the comultiplication formula nor appealing to the group structure of G_n. Previous classification work in algebraic monoids often relied on ad hoc invariants tied to the representation theory of the unit group; the L_k invariant is simpler, more general, and explicitly computable from the ray generators of the dual cone. This tool is likely to generalize to higher-dimensional toric monoids and could become a standard invariant in the classification of monoid structures on toric varieties.

Which research directions become more attractive. The unification of commutative and noncommutative monoids under the Demazure root pair formula makes it natural to ask: what other properties of algebraic monoids can be read off from the pair (e1, e2)? The paper shows that commutativity corresponds to e1 = e2, that the X/Y duality corresponds to swapping the order of the pair (Proposition 4.1), and that the boundary divisor weights in Theorem 4.3(3) are expressed in terms of the cone parameters (a,b,n) that themselves derive from the Demazure root data. This suggests a broader program: develop a Demazure root calculus for algebraic monoids on toric varieties, where algebraic properties (commutativity, existence of zero, orbit structure, singularity type) are expressed as combinatorial conditions on the set of Demazure roots appearing in the comultiplication. For higher-dimensional toric varieties, where multiple ray generators and multiple Demazure roots per generator exist, such a calculus could guide classification efforts that are currently intractable by brute-force cone enumeration.

Which directions become less pressing. The paper shows that attempting to construct non-toric noncommutative monoids in dimension two is a dead end — Theorem 3.2 rules them out categorically. Similarly, the search for exotic boundary divisor structures (multiple irreducible components, non-null multiplication, higher-dimensional boundaries) is closed for the normal affine case by Theorem 4.3: the boundary is always A¹ with a zero element (except the degenerate b = 0 case) and always has exactly three G_n × G_n-orbits. Researchers investigating low-dimensional monoids can now treat these facts as established foundations rather than open questions.

Beyond classification. The paper's structural results — especially the boundary divisor analysis (Theorem 4.3) and the opposite monoid duality (Proposition 4.1) — provide a detailed understanding of the internal algebraic structure of the classified monoids, not just their enumeration. The explicit weight formulas for the G_n-action on the boundary (τ^{b+an} for left multiplication, τ^b for right) mean that one can compute the representation theory of these monoids on their boundaries, opening the door to studying modules, cohomology, and deformations. The classification thus functions not as an endpoint but as infrastructure for deeper investigations into the geometry and representation theory of solvable algebraic monoids.

Follow-Up Research This Work Enables

1. Classification of non-normal two-dimensional affine monoids. Theorem 3.2 establishes that non-normal two-dimensional affine monoids are toric, but Theorem 3.10 classifies only the normal ones. Non-normal toric surfaces correspond to non-saturated subsemigroups of σ^∨ ∩ Z² — sub-semigroups that generate the same cone but omit some lattice points on the facets. A concrete next step: fix (n, a, b) and enumerate all subsemigroups S ⊂ σ^∨(n, a, b) ∩ Z² that (i) are finitely generated, (ii) generate the full cone (so the normalization is X_a,b_n), and (iii) satisfy the Lemma 3.5 stability condition (a', b') ∈ S ⇒ (0, b'), (0, b'+na') ∈ S. For each such semigroup, verify whether the comultiplication restricts to K[S] by checking the image condition (Lemma 3.5(1) via the binomial expansion). Characterize when two such non-normal monoids are isomorphic — the L_k invariant likely still distinguishes many but may collapse some distinctions because the semigroup might miss the lattice points where L_k is measured. A strong result would determine whether every non-normal monoid is a "pinching" (a partial normalization) of one of the normal monoids, or whether genuinely new non-normal structures appear that do not arise from normal ones by standard operations. The b = 0 degenerate case (where X_1,0_n has non-null boundary multiplication) warrants special attention, as its non-normal versions may exhibit boundary behavior not seen in the normal classification.

2. Higher-dimensional Demazure root pairs and the structure of rank-1 monoids. Theorem 3.14 shows that on a two-dimensional toric surface, every rank-1 monoid structure comes from a pair of Demazure roots associated to the same ray generator. A natural extension: on a d-dimensional normal affine toric variety Z_σ with ray generators p_1, ..., p_r, consider a collection of Demazure roots {e_{i,1}, ..., e_{i,k_i}} for each ray generator p_i (possibly with different k_i for different rays), and define a comultiplication formula:

$\chi^u \mapsto \chi^u \otimes \chi^u \prod_{i=1}^r (1 \otimes \chi^{e_{i,1}} + ... + 1 \otimes \chi^{e_{i,k_i}} + \chi^{f_{i,1}} \otimes 1 + ...)^{\langle p_i, u \rangle}$

where the terms inside the parentheses generalize the 1 ⊗ χ^{e1} + χ^{e2} ⊗ 1 pattern. The immediate question: which choices of {e_{i,j}} and {f_{i,j}} produce coassociative comultiplications that turn Z_σ into an algebraic monoid? The two-dimensional case suggests that the roots must be associated to the same ray generator (the p_i in the exponent must match the ray for which the roots are defined), but in higher dimensions there may be compatibility conditions between the collections for different rays. A concrete experiment: classify all rank-1 monoid structures on the three-dimensional toric variety A³ (whose cone is the positive orthant) with unit group G_a² × G_m or (G_a ⋊ G_m) × G_a or similar three-dimensional solvable groups. Does the "pair of Demazure roots per ray generator" pattern persist, or do fundamentally new phenomena arise in codimension > 1? The answer would determine whether the two-dimensional classification is a special case of a general Demazure root calculus or an isolated low-dimensional coincidence.

3. Deformation theory and moduli spaces of two-dimensional algebraic monoids. The classification is discrete — the parameters (n, a, b) are integers, and there are no continuous families of non-isomorphic monoids. This raises the deformation-theoretic question: are these monoids rigid, or do they admit nontrivial infinitesimal deformations as algebraic monoids? That is, can one find a flat family of monoids over a parameter space such that the generic fiber is not isomorphic to any X_a,b_n or Y_a,b_n, but the special fiber is one of the classified monoids? Theorem 3.2 (every two-dimensional affine monoid is toric) suggests that deformations would have to preserve the toric structure, which is a strong constraint. A concrete investigation: compute the Hochschild cohomology (or the appropriate bialgebra cohomology controlling deformations of the comultiplication) for a sample of the classified monoids — say, X_1,1_n for n = 1, 2, 3. If all cohomology groups in the relevant degrees vanish, the monoids are rigid in the algebraic sense, and the discrete classification is deformation-theoretically complete. If nontrivial deformations exist, they would represent monoid structures not captured by the classification, which would indicate that the classification implicitly assumed a rigidity condition (perhaps the normal affine hypothesis forces rigidity). This experiment would connect the classification to the broader moduli theory of algebraic structures.

4. Representation theory and equivariant geometry of the classified monoids. Now that the complete list of two-dimensional normal affine noncommutative monoids is known, one can systematically study their representation theory: what are the irreducible representations of X_a,b_n as an algebraic monoid (i.e., morphisms X_a,b_n → End(V) for finite-dimensional vector spaces V)? For reductive monoids, the representation theory is well-developed (Renner's classification of irreducible representations in terms of dominant weights). For these solvable monoids, the representation theory is unexplored. Since the boundary R(X) ≅ A¹ is a G_n × G_n-variety with explicitly known weight decompositions (Theorem 4.3), one can study the restriction of representations to the boundary and the decomposition of coordinate rings into G_n × G_n-modules. A concrete starting point: compute the character of the coordinate ring K[X_a,b_n] as a G_n × G_n-module and determine its decomposition into irreducible G_n-modules (using the known representation theory of G_n from the semidirect product structure). This would connect the combinatorial classification to harmonic analysis on solvable monoids, potentially yielding a Peter-Weyl-type theorem for these non-reductive objects.

5. Extension to positive characteristic and non-algebraically closed fields. Section 6, Limitation 3, flags that the entire classification assumes an algebraically closed field of characteristic zero. A natural follow-up program: for each of the monoids X_a,b_n and Y_a,b_n, determine whether the defining equations and comultiplication formulas can be written over Z (or Z[1/n]) and then reduced modulo p for primes p not dividing n. In characteristic p, the locally nilpotent derivation δ_l = ∂_x satisfies δ_l^p = 0, which collapses the L_k invariants for k ≥ p. Do distinct monoids in characteristic zero become isomorphic after reduction modulo p? For example, if L_k(X_a,b_n) and L_k(X_a',b'_n) differ only for k ≥ p, the two monoids might become isomorphic in characteristic p because the derivation-theoretic invariant loses resolution. A concrete experiment: fix n = 1, take X_1,1_1 and X_1,2_1 (which have different L_1 values: 1 vs. 2), and examine their reductions modulo 2. Does the mod-2 reduction of δ_l still distinguish them? What about modulo 3? This would map the boundary of the classification's validity and potentially reveal new characteristic-p phenomena (Frobenius twists, new monoid structures arising from inseparable isogenies). Similarly, for non-algebraically closed fields (e.g., R or Q), one can study the Galois cohomology classifying real or rational forms of these monoids — the discrete parameters (n, a, b) likely give rise to multiple non-isomorphic k-forms for each geometric monoid, analogous to the classification of real forms of algebraic groups.

6. Combinatorial enumeration of monoid structures on a given toric surface using Demazure root data. Corollary 3.15 gives a precise condition for when two Demazure root pairs produce isomorphic monoids: the only possible isomorphisms come from swapping the two ray generators of the cone σ, and this swap is an automorphism of the toric surface only when \hat{τ}(M) = M. This suggests a computational experiment: for each of the (finitely many) toric surfaces corresponding to two-dimensional strongly convex cones with primitive ray generators, enumerate all Demazure roots associated to each ray generator, form all unordered pairs (e1, e2) (with e1 and e2 associated to the same ray generator), apply the isomorphism condition from Corollary 3.15 to identify equivalent pairs, and count the number of distinct monoid structures of rank 1 on that surface. This would produce, for each toric surface, a precise count (or formula) for the number of commutative and noncommutative monoid structures it supports. Such an enumeration could be automated in a computer algebra system (e.g., SageMath or Macaulay2) by implementing the Demazure root computation for a given cone and the isomorphism check. The output would be a lookup table: given the singularity type of a normal affine toric surface (classified by the Hirzebruch-Jung continued fraction or the cone generators), how many monoid structures does it admit, and what are their parameters? This would transform the theoretical classification into a practical reference for algebraic geometers encountering monoids in the context of toric geometry, singularity theory, or equivariant compactifications.

Practical Applications and Downstream Use Cases

1. Atlas of low-dimensional algebraic monoids for testing conjectures. In algebraic geometry and representation theory, new conjectures about algebraic monoids are often tested on low-dimensional examples before being proved in generality. This paper provides a complete, explicit, and parameterized catalog of every two-dimensional normal affine noncommutative monoid, with explicit multiplication formulas (Corollary 3.11 for A², Example 3.12 for the quadratic singularity xz = y², and the general comultiplication formula in Theorem 3.14). A researcher studying, say, the behavior of the class group under monoid completion, or the relationship between the G_n × G_n-orbit structure and the singularity type of the monoid, can now compute these properties for every example in the atlas without gaps. The discrete parameterization (n, a, b) means that one can prove a property for all two-dimensional monoids by induction or case analysis on the parameters, rather than by constructing ad hoc examples. The invariants L_k and the boundary divisor weight formulas provide quantitative handles for testing conjectures — for instance, a conjecture about the dimension of the Zariski tangent space at the zero element can be evaluated on X_a,b_n for all parameters and compared with the cone data to formulate a general theorem.

2. Building blocks for higher-dimensional monoids via products and semidirect products. The two-dimensional monoids classified here can serve as elementary constituents of higher-dimensional monoids through standard constructions: direct products (X × Y of classified monoids), semidirect products (where one monoid acts on another), and equivariant embeddings. For example, a three-dimensional solvable monoid with unit group G_n × G_m might be analyzed by decomposing its coordinate ring into graded pieces corresponding to the two-dimensional monoids appearing as orbit closures of codimension-one subgroups. The explicit formulas for comultiplication on X_a,b_n (Theorem 3.14) and the boundary divisor structure (Theorem 4.3) provide the local data needed for gluing arguments. If a three-dimensional monoid contains X_a,b_n as a closed submonoid (e.g., as the closure of a codimension-one subgroup), the classification in this paper tells one exactly which submonoids can appear and how they multiply. This makes the two-dimensional classification an essential reference for any systematic study of higher-dimensional solvable monoids, analogous to how the classification of two-dimensional algebraic groups underpins the structure theory of higher-dimensional solvable groups.

3. Examples for noncommutative algebraic geometry and Hopf algebra theory. The monoids X_a,b_n and Y_a,b_n provide a rich supply of explicitly computable, noncommutative, non-cocommutative Hopf algebras that are finitely generated, graded, and geometrically meaningful (they are coordinate rings of algebraic varieties, not abstract algebras). Researchers in Hopf algebra theory or noncommutative algebraic geometry frequently need examples that are complex enough to exhibit phenomena (noncommutativity, non-cocommutativity, nontrivial integrals, nontrivial group of grouplikes) but simple enough that all structure constants can be written down. The comultiplication formula x^a y^b ↦ (x⊗1 + y^n⊗x)^a (y⊗y)^b with the cone constraints from σ^∨(n,a,b) or \bar{σ}^∨(n,a,b) is fully explicit: for any given (n,a,b), one can generate the complete set of relations and the comultiplication on generators algorithmically. The boundary divisor being A¹ with a zero element (Theorem 4.3) means these Hopf algebras have a natural "augmentation at infinity," which is a feature of interest in the study of compact quantum groups and their noncommutative analogues. The opposite monoid duality Y_a,b_n ≅ (X_a,b_n)^{op} (Proposition 4.1) provides a natural source of paired Hopf algebras that are isomorphic as algebras but have opposite comultiplications — a phenomenon with no commutative counterpart, making these monoids valuable test cases for theories of Hopf algebra duality.

4. Computational algebra and automated classification of monoid structures. The paper's methods — reducing the classification to cones satisfying Lemma 3.5, distinguishing isomorphism classes with the L_k invariant, and expressing the comultiplication via Demazure roots — are amenable to algorithmic implementation. For a given cone σ (specified by its ray generators in N) and a given group G_n, one can algorithmically enumerate all Demazure roots, form admissible pairs, check the isomorphism condition from Corollary 3.15, and output the complete list of monoid structures on Z_σ. For the two-dimensional case, this is a finite computation for each cone; for higher-dimensional cones with finitely many rays, the Demazure root condition ⟨e, p_i⟩ = -1, ⟨e, p_j⟩ ≥ 0 is a system of linear inequalities, and the number of solutions is finite (since M is a lattice and the inequalities bound the possible coordinates). An implementation in a system like SageMath or OSCAR could produce monoid classifications for cones of dimension up to, say, 5 or 6, providing computational evidence for conjectures about the higher-dimensional Demazure root calculus. The L_k invariant and the boundary divisor analysis provide automated isomorphism checks, reducing what was previously a manual geometric classification to a computer algebra problem. This is a practical tool for researchers who want to generate examples, test conjectures, or explore the parameter space of monoid structures on a specific toric variety without working through the geometric proofs by hand.

🎯 Pitch

1. Executive Summary

2. Context and Motivation

The Core Problem: Extending Monoid Classification to the Noncommutative Setting

Why This Classification Matters

Prior Approaches and Their Limitations

How This Paper Positions Itself

3. Technical Approach

3.1 Reader Orientation

3.2 Big-Picture Architecture (Diagram in Words)

3.3 Roadmap for the Deep Dive

3.4 Detailed, Sentence-Based Technical Breakdown

The Toric Structure on the Group G_n

Every Two-Dimensional Affine Monoid Is Toric (Theorem 3.2)

The Stability Condition Under G_a-Actions (Lemma 3.5)

The Two Families of Cones (Construction 3.6)

Distinguishing the Monoids: The Image Ideal Invariants L_k

The Main Classification Theorem (Theorem 3.10)

The Demazure Root Reformulation (Theorem 3.14)

The Role of δ_l and δ_r in the Classification Workflow

4. Key Insights and Innovations

Innovation 1: The Toric-Forcing Phenomenon — Monoid Structure Imposes Toric Geometry Regardless of Commutativity

Innovation 2: The Demazure Root Pair as the Algebraic Source of Noncommutativity

Innovation 3: The Image Ideal Invariants L_k as a Practical Distinguishing Tool

Innovation 4: The Boundary Divisor as a Structural Invariant — Irreducibility, Zero Element, and the G_n-Module Structure

5. Experimental Analysis

Evaluation Methodology

Main Quantitative Results

Theorem 3.10: The Complete List — How Many and Which Ones

Theorem 3.14: Unified Comultiplication Formula — The Single Algebraic Expression That Generates All Monoids

Proposition 3.9 and Lemma 3.8: Invariant Computations That Separate the Families

Proposition 4.1: Opposite Monoids and the X-Y Duality

Theorem 4.3: Boundary Divisor Structure — Universality Across the Family

Corollary 4.4: Orbit Structure — Exactly Three Orbits

Ablation Studies and Robustness Checks

Critical Assessment

Claim 1: Every two-dimensional affine algebraic monoid is toric (Theorem 3.2).

Claim 2: The complete list of noncommutative normal affine two-dimensional monoids is G_n, X_a,b_n, and Y_a,b_n (Theorem 3.10).

Claim 3: Every monoid structure of rank 1 on a toric surface Z_σ is given by the Demazure root pair comultiplication formula (Theorem 3.14).

Claim 4: The boundary divisor is always A¹ with null multiplication and a zero element (Theorem 4.3).

Overall Assessment

6. Limitations and Trade-offs

Limitation 1: The Classification Assumes Normality of the Underlying Variety

Limitation 2: The Entire Classification Is Confined to Dimension Two

Limitation 3: The Classification Depends on the Base Field Being Algebraically Closed of Characteristic Zero

Limitation 4: The Classification Is Specific to the Affine Case and Relies on the Algebraic Structure of the Coordinate Ring

Limitation 5: The Demazure Root Comultiplication Formula Is Presented Without Verifying the Bialgebra Axioms for the Restricted Structure

Limitation 6: The Difficulty of Recognising a Given Monoid in the Classification Without Explicit Coordinates

7. Implications and Future Directions

How This Work Changes the Landscape

Follow-Up Research This Work Enables

Practical Applications and Downstream Use Cases

Adjacent in domain

SlimQwen: Exploring the Pruning and Distillation in Large MoE Model Pre-training

The Toric Structure on the Group `G_n`

The Stability Condition Under `G_a`-Actions (Lemma 3.5)

Distinguishing the Monoids: The Image Ideal Invariants `L_k`

The Role of `δ_l` and `δ_r` in the Classification Workflow

Innovation 3: The Image Ideal Invariants `L_k` as a Practical Distinguishing Tool

Innovation 4: The Boundary Divisor as a Structural Invariant — Irreducibility, Zero Element, and the `G_n`-Module Structure

Claim 2: The complete list of noncommutative normal affine two-dimensional monoids is `G_n`, `X_a,b_n`, and `Y_a,b_n` (Theorem 3.10).

Claim 3: Every monoid structure of rank 1 on a toric surface `Z_σ` is given by the Demazure root pair comultiplication formula (Theorem 3.14).

Claim 4: The boundary divisor is always `A¹` with null multiplication and a zero element (Theorem 4.3).