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Affine monoid

From Wikipedia, the free encyclopedia

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 .[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

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  • Affine monoids are finitely generated. This means for a monoid , there exists such that
.
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 .
  • A subset of a monoid that is itself a monoid with respect to the operation on is a submonoid of .

Properties and examples

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  • 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

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Group of differences

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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

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  • can be viewed as the set of equivalences classes , where if and only if , for , and

defines the addition.[1]

  • The rank of an affine monoid is the rank of a group of .[1]
  • If an affine monoid is given as a submonoid of , then , where is the subgroup of .[1]

Universal property

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  • 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

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Definition

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  • 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.[1]
  • A monoid is a normal affine monoid if and only if is finitely generated and .

Affine monoid rings

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see also: Group ring

Definition

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  • 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 .

Connection to convex geometry

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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.[1] (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 .[1] (Theorem 2.12)

See also

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References

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  1. ^ a b c d e f g Bruns, Winfried; Gubeladze, Joseph (2009). Polytopes, Rings, and K-Theory. Monographs in Mathematics. Springer. ISBN 0-387-76356-2.