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 $\mathbb {Z} ^{d},d\geq 0$ . 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 $x,y\in M$ .

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.

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 $\mathbb {Z} ^{r}$ , 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.

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 {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

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 $R$ .

$R[M]$ is a domain since, for some $d\geq 0$ , it embeds in $R[\mathbb {Z} ^{d}]$ which is a domain.

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)