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.

Characterization

• 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 ${\displaystyle x,y\in M}$.
• A subset ${\displaystyle N}$ of a monoid ${\displaystyle M}$ that is itself a monoid with respect to the operation on ${\displaystyle M}$ is a submonoid of ${\displaystyle M}$.

Affine monoids

Affine monoid rings

see also: Group ring

Connection to convex geometry

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)

See also

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

References