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 . 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, there exists such that
  • Affine monoids are cancellative. In other words,
  • Affine monoids are also torsion free. For an affine monoid, implies that for, and.

Properties and examples

  • Every submonoid of is finitely generated. Hence, every submonoid of is affine.
  • The submonoid of is not finitely generated, and therefore not affine.
  • The intersection of two affine monoids is an affine monoid.

Affine monoids

Group of differences

Definition

defines the addition.
  • The rank of an affine monoid is the rank of a group of.
  • If an affine monoid is given as a submonoid of, then, where is the subgroup of.

Universal property

Normal affine monoids

Definition

  • If is a submonoid of an affine monoid, then the submonoid
is the integral closure of in. If, then is integrally closed.
  • The normalization of an affine monoid is the integral closure of in. If the normalization of, is itself, then is a normal affine monoid.
  • A monoid is a normal affine monoid if and only if is finitely generated and .

Affine monoid rings

Let be an affine monoid, and a commutative ring. Then one can form the affine monoid ring. This is an -module with a free basis, so if, then
In other words, is the set of finite sums of elements of with coefficients in.
is a domain since, for some, it embeds in which is a domain.

Connection to [convex geometry]

  • Let be a rational convex cone in, and let be a lattice in. Then is an affine monoid.
  • If is a submonoid of, then is a cone if and only if is an affine monoid.
  • If is a submonoid of, and is a cone generated by the elements of, then is an affine monoid.
  • Let in be a rational polyhedron, the recession cone of, and a lattice in. Then is a finitely generated module over the affine monoid.