Affine monoid

In abstract algebra, a branch of mathematics, an affine monoid is a commutative monoid that is finitely generated, and is isomorphic to a submonoid of a free abelian group $$\mathbb{Z}^d, d \ge 0$$. Affine monoids are closely connected to convex polyhedra, and their associated algebras are of much use in the algebraic study of these geometric objects.

Characterization

 * Affine monoids are finitely generated. This means for a monoid $$ M $$, there exists $$ m_1, \dots, m_n \in M $$ such that
 * $$ M = m_1\mathbb{Z_+}+\dots + m_n\mathbb{Z_+} $$.


 * Affine monoids are cancellative. In other words,
 * $$x + y = x + z$$ implies that $$y = z$$ for all $$x,y,z \in M$$, where $$+$$ denotes the binary operation on the affine monoid $$M$$.


 * Affine monoids are also torsion free. For an affine monoid $$M$$, $$nx = ny$$ implies that $$x = y$$ for $$ n \in \mathbb{N}$$, and $$ x, y \in M$$.
 * A subset $$N$$ of a monoid $$M$$ that is itself a monoid with respect to the operation on $$M$$ is a submonoid of $$M$$.

Properties and examples

 * Every submonoid of $$\mathbb{Z}$$ is finitely generated. Hence, every submonoid of $$\mathbb{Z}$$ is affine.
 * The submonoid $$\{(x,y)\in \mathbb{Z} \times \mathbb{Z} \mid y > 0\} \cup \{(0,0)\}$$ of $$\mathbb{Z} \times \mathbb{Z}$$ is not finitely generated, and therefore not affine.
 * The intersection of two affine monoids is an affine monoid.

Group of differences

 * If $$M$$ is an affine monoid, it can be embedded into a group. More specifically, there is a unique group $$gp(M)$$, called the group of differences, in which $$M$$ can be embedded.

Definition
$$(x-y) + (u-v) = (x+u) - (y+v)$$ defines the addition.
 * $$gp(M)$$ can be viewed as the set of equivalences classes $$x - y$$, where $$x - y = u - v$$ if and only if $$x + v + z = u + y + z$$, for $$z \in M$$, and


 * The rank of an affine monoid $$M$$ is the rank of a group of $$gp(M)$$.
 * If an affine monoid $$M$$ is given as a submonoid of $$\mathbb{Z}^r$$, then $$gp(M) \cong \mathbb{Z}M$$, where $$\mathbb{Z}M$$ is the subgroup of $$\mathbb{Z}^r$$.

Universal property

 * If $$M$$ is an affine monoid, then the monoid homomorphism $$\iota : M \to gp(M)$$ defined by $$\iota(x) = x + 0$$ satisfies the following universal property:


 * for any monoid homomorphism $$\varphi: M \to G$$, where $$G$$ is a group, there is a unique group homomorphism $$\psi : gp(M) \to G$$, such that $$\varphi = \psi \circ \iota$$, and since affine monoids are cancellative, it follows that $$\iota$$ is an embedding. In other words, every affine monoid can be embedded into a group.

Definition

 * If $$M$$ is a submonoid of an affine monoid $$N$$, then the submonoid
 * $$ \hat{M}_N = \{x\in N \mid mx \in M, m \in \mathbb{N}\}$$

is the integral closure of $$M$$ in $$N$$. If $$M = \hat{M_N}$$, then $$M$$ is integrally closed.
 * The normalization of an affine monoid $$M$$ is the integral closure of $$M$$ in $$gp(M)$$. If the normalization of $$M$$, is $$M$$ itself, then $$M$$ is a normal affine monoid.
 * A monoid $$M$$ is a normal affine monoid if and only if $$\mathbb{R}_+M$$ is finitely generated and $$M = \mathbb{Z}^r \cap \mathbb{R}_+M$$.

Affine monoid rings

 * see also: Group ring

Definition

 * Let $$M$$ be an affine monoid, and $$R$$ a commutative ring. Then one can form the affine monoid ring $$R[M]$$. This is an $$R$$-module with a free basis $$M$$, so if $$f \in R[M]$$, then
 * $$f = \sum_{i=1}^{n}f_{i}x_i$$, where $$f_i \in R, x_i \in M$$, and $$n \in \mathbb{N}$$.
 * In other words, $$R[M]$$ is the set of finite sums of elements of $$M$$ with coefficients in $$R$$.

Connection to convex geometry

 * Affine monoids arise naturally from convex polyhedra, convex cones, and their associated discrete structures.


 * Let $$C$$ be a rational convex cone in $$\mathbb{R}^n$$, and let $$L$$ be a lattice in $$\mathbb{Q}^n$$. Then $$C \cap L$$ is an affine monoid. (Lemma 2.9, Gordan's lemma)
 * If $$M$$ is a submonoid of $$\mathbb{R}^n$$, then $$\mathbb{R}_+M$$ is a cone if and only if $$M$$ is an affine monoid.
 * If $$M$$ is a submonoid of $$\mathbb{R}^n$$, and $$C$$ is a cone generated by the elements of $$gp(M)$$, then $$M \cap C$$ is an affine monoid.
 * Let $$P$$ in $$\mathbb{R}^n$$ be a rational polyhedron, $$C$$ the recession cone of $$P$$, and $$L$$ a lattice in $$\mathbb{Q}^n$$. Then $$P \cap L$$ is a finitely generated module over the affine monoid $$C \cap L$$. (Theorem 2.12)