affine monoid

• Affine monoids are finitely generated. This means for a monoid $M$, there exists $m\_1,\; \backslash dots\; ,\; m\_n\; \backslash in\; M$ such that

:$M\; =\; m\_1\backslash mathbb\{Z\_+\}+\backslash dots\; +\; m\_n\backslash mathbb\{Z\_+\}$.

• Affine monoids are cancellative. In other words,

:$x\; +\; y\; =\; x\; +\; z$ implies that $y\; =\; z$ for all $x,y,z\; \backslash in\; M$, 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 $n\; \backslash in\; \backslash mathbb\{N\}$, and $x,\; y\; \backslash in\; M$.
• A subset $N$ of a monoid $M$ that is itself a monoid with respect to the operation on $M$ is a submonoid of $M$.

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

{{See also|Grothendieck group}}

: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.

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

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

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

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

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

is the integral closure of $M$ in $N$. If $M\; =\; \backslash hat\{M\_N\}$, 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 $\backslash mathbb\{R\}\_+M$ is finitely generated and $M\; =\; \backslash mathbb\{Z\}^r\; \backslash cap\; \backslash mathbb\{R\}\_+M$ .

: see also: Group ring

• 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 $f\; \backslash in\; R[M]$, then

: $f\; =\; \backslash sum\_\{i=1\}^\{n\}f\_\{i\}x\_i$, where $f\_i\; \backslash in\; R,\; x\_i\; \backslash in\; M$, and $n\; \backslash in\; \backslash mathbb\{N\}$.

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

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

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

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

Category:Algebraic structures