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 ${\displaystyle \mathbb {Z} ^{d},d\geq 0}$.^[1] Affine monoids are closely connected to convex polyhedra, and their associated algebras are of much use in the algebraic study of these geometric objects.

• Affine monoids are finitely generated. This means for a monoid ${\displaystyle M}$, there exists ${\displaystyle m_{1},\dots ,m_{n}\in M}$ such that

${\displaystyle M=m_{1}\mathbb {Z_{+}} +\dots +m_{n}\mathbb {Z_{+}} }$.

• Affine monoids are cancellative. In other words,

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

• Affine monoids are also torsion free. For an affine monoid ${\displaystyle M}$, ${\displaystyle nx=ny}$ implies that ${\displaystyle x=y}$ for ${\displaystyle n\in \mathbb {N} }$, and ${\displaystyle x,y\in M}$.

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

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

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

${\displaystyle (x-y)+(u-v)=(x+u)-(y+v)}$ defines the addition.^[1]

• The rank of an affine monoid ${\displaystyle M}$ is the rank of a group of ${\displaystyle gp(M)}$.^[1]
• If an affine monoid ${\displaystyle M}$ is given as a submonoid of ${\displaystyle \mathbb {Z} ^{r}}$, then ${\displaystyle gp(M)\cong \mathbb {Z} M}$, where ${\displaystyle \mathbb {Z} M}$ is the subgroup of ${\displaystyle \mathbb {Z} ^{r}}$.^[1]

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

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

• If ${\displaystyle M}$ is a submonoid of an affine monoid ${\displaystyle N}$, then the submonoid

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

is the integral closure of ${\displaystyle M}$ in ${\displaystyle N}$. If ${\displaystyle M={\hat {M_{N}}}}$, then ${\displaystyle M}$ is integrally closed.

• The normalization of an affine monoid ${\displaystyle M}$ is the integral closure of ${\displaystyle M}$ in ${\displaystyle gp(M)}$. If the normalization of ${\displaystyle M}$, is ${\displaystyle M}$ itself, then ${\displaystyle M}$ is a normal affine monoid.^[1]
• A monoid ${\displaystyle M}$ is a normal affine monoid if and only if ${\displaystyle \mathbb {R} _{+}M}$ is finitely generated and ${\displaystyle M=\mathbb {Z} ^{r}\cap \mathbb {R} _{+}M}$ .

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

${\displaystyle f=\sum _{i=1}^{n}f_{i}x_{i}}$, where ${\displaystyle f_{i}\in R,x_{i}\in M}$, and ${\displaystyle n\in \mathbb {N} }$.

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

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

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

• Let ${\displaystyle C}$ be a rational convex cone in ${\displaystyle \mathbb {R} ^{n}}$, and let ${\displaystyle L}$ be a lattice in ${\displaystyle \mathbb {Q} ^{n}}$. Then ${\displaystyle C\cap L}$ is an affine monoid.^[1] (Lemma 2.9, Gordan's lemma)
• If ${\displaystyle M}$ is a submonoid of ${\displaystyle \mathbb {R} ^{n}}$, then ${\displaystyle \mathbb {R} _{+}M}$ is a cone if and only if ${\displaystyle M}$ is an affine monoid.
• If ${\displaystyle M}$ is a submonoid of ${\displaystyle \mathbb {R} ^{n}}$, and ${\displaystyle C}$ is a cone generated by the elements of ${\displaystyle gp(M)}$, then ${\displaystyle M\cap C}$ is an affine monoid.
• Let ${\displaystyle P}$ in ${\displaystyle \mathbb {R} ^{n}}$ be a rational polyhedron, ${\displaystyle C}$ the recession cone of ${\displaystyle P}$, and ${\displaystyle L}$ a lattice in ${\displaystyle \mathbb {Q} ^{n}}$. Then ${\displaystyle P\cap L}$ is a finitely generated module over the affine monoid ${\displaystyle C\cap L}$.^[1] (Theorem 2.12)

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