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 ${\mathbb{\mathbb{Z}}}^d,d\ge 0$. 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

$M=m_1{\mathbb{Z}}_{\mathbb{+}}+\dots +m_n{\mathbb{Z}}_{\mathbb{+}}$.

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

Affine monoids

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

• $gp(M)$ 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

$(x-y)+(u-v)=(x+u)-(y+v)$ defines the addition.

• 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 ${\mathbb{\mathbb{Z}}}^r$, then $gp(M)\cong \mathbb{\mathbb{Z}}M$, where $\mathbb{\mathbb{Z}}M$ is the subgroup of ${\mathbb{\mathbb{Z}}}^r$.

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 $\phi :M\to G$, where $G$ is a group, there is a unique group homomorphism $\psi :gp(M)\to G$, such that $\phi =\psi \circ \iota$, 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.

Normal affine monoids

Definition

• If $M$ is a submonoid of an affine monoid $N$, then the submonoid

${\hat{M}}_N=\{x\in N\mid mx\in M,m\in \mathbb{\mathbb{N}}\}$

is the integral closure of $M$ in $N$. If $M=\widehat{M_N}$, 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 $M={\mathbb{\mathbb{Z}}}^r\cap {\mathbb{ℝ}}_{+}M$ .

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 $n\in \mathbb{\mathbb{N}}$.

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 ${\mathbb{\mathbb{Q}}}^n$. 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 ${\mathbb{\mathbb{Q}}}^n$. Then $P\cap L$ is a finitely generated module over the affine monoid $C\cap L$. (Theorem 2.12)

See also

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