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 [math]\displaystyle{ \mathbb{Z}^d, d \ge 0 }[/math]. 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 [math]\displaystyle{ M }[/math], there exists [math]\displaystyle{ m_1, \dots , m_n \in M }[/math] such that

[math]\displaystyle{ M = m_1\mathbb{Z_+}+\dots + m_n\mathbb{Z_+} }[/math].

• Affine monoids are cancellative. In other words,

[math]\displaystyle{ x + y = x + z }[/math] implies that [math]\displaystyle{ y = z }[/math] for all [math]\displaystyle{ x,y,z \in M }[/math], where [math]\displaystyle{ + }[/math] denotes the binary operation on the affine monoid [math]\displaystyle{ M }[/math].

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

Properties and examples

• Every submonoid of [math]\displaystyle{ \mathbb{Z} }[/math] is finitely generated. Hence, every submonoid of [math]\displaystyle{ \mathbb{Z} }[/math] is affine.
• The submonoid [math]\displaystyle{ \{(x,y)\in \mathbb{Z} \times \mathbb{Z} \mid y \gt 0\} \cup \{(0,0)\} }[/math] of [math]\displaystyle{ \mathbb{Z} \times \mathbb{Z} }[/math] is not finitely generated, and therefore not affine.
• The intersection of two affine monoids is an affine monoid.

Affine monoids

Group of differences

If [math]\displaystyle{ M }[/math] is an affine monoid, it can be embedded into a group. More specifically, there is a unique group [math]\displaystyle{ gp(M) }[/math], called the group of differences, in which [math]\displaystyle{ M }[/math] can be embedded.

Definition

• [math]\displaystyle{ gp(M) }[/math] can be viewed as the set of equivalences classes [math]\displaystyle{ x - y }[/math], where [math]\displaystyle{ x - y = u - v }[/math] if and only if [math]\displaystyle{ x + v + z = u + y + z }[/math], for [math]\displaystyle{ z \in M }[/math], and

[math]\displaystyle{ (x-y) + (u-v) = (x+u) - (y+v) }[/math] defines the addition.

• The rank of an affine monoid [math]\displaystyle{ M }[/math] is the rank of a group of [math]\displaystyle{ gp(M) }[/math].
• If an affine monoid [math]\displaystyle{ M }[/math] is given as a submonoid of [math]\displaystyle{ \mathbb{Z}^r }[/math], then [math]\displaystyle{ gp(M) \cong \mathbb{Z}M }[/math], where [math]\displaystyle{ \mathbb{Z}M }[/math] is the subgroup of [math]\displaystyle{ \mathbb{Z}^r }[/math].

Universal property

• If [math]\displaystyle{ M }[/math] is an affine monoid, then the monoid homomorphism [math]\displaystyle{ \iota : M \to gp(M) }[/math] defined by [math]\displaystyle{ \iota(x) = x + 0 }[/math] satisfies the following universal property:

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

Normal affine monoids

Definition

• If [math]\displaystyle{ M }[/math] is a submonoid of an affine monoid [math]\displaystyle{ N }[/math], then the submonoid

[math]\displaystyle{ \hat{M}_N = \{x\in N \mid mx \in M, m \in \mathbb{N}\} }[/math]

is the integral closure of [math]\displaystyle{ M }[/math] in [math]\displaystyle{ N }[/math]. If [math]\displaystyle{ M = \hat{M_N} }[/math], then [math]\displaystyle{ M }[/math] is integrally closed.

• The normalization of an affine monoid [math]\displaystyle{ M }[/math] is the integral closure of [math]\displaystyle{ M }[/math] in [math]\displaystyle{ gp(M) }[/math]. If the normalization of [math]\displaystyle{ M }[/math], is [math]\displaystyle{ M }[/math] itself, then [math]\displaystyle{ M }[/math] is a normal affine monoid.
• A monoid [math]\displaystyle{ M }[/math] is a normal affine monoid if and only if [math]\displaystyle{ \mathbb{R}_+M }[/math] is finitely generated and [math]\displaystyle{ M = \mathbb{Z}^r \cap \mathbb{R}_+M }[/math] .

Affine monoid rings

see also: Group ring

Definition

• Let [math]\displaystyle{ M }[/math] be an affine monoid, and [math]\displaystyle{ R }[/math] a commutative ring. Then one can form the affine monoid ring [math]\displaystyle{ R[M] }[/math]. This is an [math]\displaystyle{ R }[/math]-module with a free basis [math]\displaystyle{ M }[/math], so if [math]\displaystyle{ f \in R[M] }[/math], then

[math]\displaystyle{ f = \sum_{i=1}^{n}f_{i}x_i }[/math], where [math]\displaystyle{ f_i \in R, x_i \in M }[/math], and [math]\displaystyle{ n \in \mathbb{N} }[/math].

In other words, [math]\displaystyle{ R[M] }[/math] is the set of finite sums of elements of [math]\displaystyle{ M }[/math] with coefficients in [math]\displaystyle{ R }[/math].

Connection to convex geometry

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

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

See also

• Monoid
• Convex cone
• Convex polytope
• Lattice (group)
• K-theory