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 ℤd, d ≥ 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, 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.
- A subset of a monoid that is itself a monoid with respect to the operation on is a submonoid of.
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
- can be viewed as the set of equivalences classes, where if and only if, for, and
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
- If is an affine monoid, then the monoid homomorphism defined by satisfies the following 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
Definition
