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, implies that for all , where denotes the binary operation on the affine monoid .
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
If is an affine monoid, it can be embedded into a group. More specifically, there is a unique group , called the group of differences, in which can be embedded.
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:
for any monoid homomorphism , where is a group, there is a unique group homomorphism , such that , and since affine monoids are cancellative, it follows that is an embedding. In other words, every affine monoid can be embedded into a group.
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 , where , and . 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 Affine monoids arise naturally from convex polyhedra, convex cones, and their associated discrete structures.
Let be a rational convex cone in , and let be a lattice in . Then is an affine monoid. (Lemma 2.9, Gordan's lemma) 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 . (Theorem 2.12)
See also Monoid Convex cone Convex polytope Lattice (group) K-theory
References