Affine monoid explained

Z ^{d,} d \ge 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

M , there exists m _{1,} ... , m _{n} \in M such that M = m _{1 Z _{+} +...} + m _{n Z _{+}} .

• Affine monoids are cancellative. In other words,

x + y = x + z implies that y = z for all x,y,z \in M , where + denotes the binary operation on the affine monoid M .

• Affine monoids are also torsion free. For an affine monoid

M , nx = ny implies that x = y for n \in N , and x, y \in M .

• A subset

N of a monoid M that is itself a monoid with respect to the operation on M is a submonoid of M .

Properties and examples

• Every submonoid of

Z is finitely generated. Hence, every submonoid of Z is affine.

• The submonoid

\{(x,y)\in Z x Z \mid y > 0 \} \cup \{( 0 , 0 )\} of Z x Z is not finitely generated, and therefore not affine.

• The intersection of two affine monoids is an affine monoid.

Affine monoids

Group of differences

See also: Grothendieck group.

If

M is an affine monoid, it can be embedded into a group. More specifically, there is a unique group gp(M) , called the group of differences, in which M can be embedded.

Definition

gp(M) can be viewed as the set of equivalences classes x - y , where x - y = u - v if and only if x + v + z = u + y + z , for z \in M , and (x - y) + (u - v) = (x + u) - (y + v) defines the addition.

• The rank of an affine monoid

M is the rank of a group of gp(M) .

• If an affine monoid

M is given as a submonoid of Z ^{r} , then gp(M) \cong Z M , where Z M is the subgroup of Z ^{r} .

Universal property

• If

M is an affine monoid, then the monoid homomorphism \iota : M \to gp(M) defined by \iota(x) = x + 0 satisfies the following universal property:

for any monoid homomorphism

\varphi: M \to G , where G is a group, there is a unique group homomorphism \psi : gp(M) \to G , such that \varphi = \psi \circ \iota , and since affine monoids are cancellative, it follows that \iota is an embedding. In other words, every affine monoid can be embedded into a group.

Normal affine monoids

Definition

• If

M is a submonoid of an affine monoid N , then the submonoid\hat{M} _{N} = \{x\in N \mid mx \in M, m \in N \} is the integral closure of M in N . If M = \hat{M _{N}} , then M is integrally closed.

• The normalization of an affine monoid

M is the integral closure of M in gp(M) . If the normalization of M , is M itself, then M is a normal affine monoid.

• A monoid

M is a normal affine monoid if and only if R _{+M} is finitely generated and M = Z ^{r} \cap R _{+M} .

Affine monoid rings

see also: Group ring

Definition

• Let

M be an affine monoid, and R a commutative ring. Then one can form the affine monoid ring R[M] . This is an R -module with a free basis M , so if f \in R[M] , then f = \sum ^{n} _{i =1} f _{i} x _{i} , where f _{i} \in R, x _{i} \in M , and n \in N .

In other words,

R[M] is the set of finite sums of elements of M with coefficients in R .

Affine monoids arise naturally from convex polyhedra, convex cones, and their associated discrete structures.

• Let

C be a rational convex cone in R ^{n} , and let L be a lattice in Q ^{n} . Then C \cap L is an affine monoid. (Lemma 2.9, Gordan's lemma)

• If

M is a submonoid of R ^{n} , then R _{+M} is a cone if and only if M is an affine monoid.

• If

M is a submonoid of R ^{n} , and C is a cone generated by the elements of gp(M) , then M \cap C is an affine monoid.

• Let

P in R ^{n} be a rational polyhedron, C the recession cone of P , and L a lattice in Q ^{n} . Then P \cap L is a finitely generated module over the affine monoid C \cap L . (Theorem 2.12)

See also

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

References

1. Book: Bruns, Winfried . Joseph . Gubeladze . Polytopes, Rings, and K-Theory . Springer . Monographs in Mathematics . 2009 . 0-387-76356-2 .