User:MCPOliseno /Math735 AffineMonoids

Affine Monoid

A monoid, $$ M \ $$ is a set together with an operation $$ M \ $$ x $$ M \to M \ $$, that is associative and has a neutral element. An affine monoid is a monoid that is finitely generated and is isomorphic to a submonoid of a free abelian group $$ \Z^d \ $$, for some d $$ \ge \ $$ 0. Affine monoids are characterized by being (1) finitely generated, (2) cancellative, and (3) torsionfree. within the class of commutative monoids.

The operation in $$ M \ $$ uses additive notaion and thus makes the condition that they are finitely generated imply that there exists $$ x^1, x_2, \dots, x_n \in M \ $$ such that

$$ M = \Z_+x_1 + \dots \Z_+x_n \ $$ = {$$ a_1x_1 + \dots + a_nx_n : a_i \in \Z_+ \ $$}.

Since additive notation is used, cancellativity implies that an equation x + y = x + z for x, y, z $$ \in M \ $$ implies that y = z. Torsionfree implies that if ax = ay for a $$ \in \N \ $$ and x, y $$ \in M \ $$ implies that x = y.

For every commutative monoid, $$ M \ $$, there exists a group of differences, gp($$ M \ $$), which is unique up to isomorphism. There also exists a monoid homomorphism $$ \phi: M \to \ $$ gp($$ M \ $$) such that for each monoid homomorphism $$ \psi: M \to H \ $$, where H is a group which factors in a unique way as $$ \psi = \pi \circ \phi \ $$ with unique group homomorphism $$ \pi: \ $$ gp($$ M \ $$) $$ \to H \ $$.

gp($$ M \ $$) is a set that consists of the equivalence classes x-y of pairs (x, y) $$ \in M^2 \ $$. x-y = u-v if and only if x+v+z = u+y+z for some z $$ \in M \ $$. The operation of this group is addition defined as (x-y) + (u-v) = (x+u) -(y+v). Then in the map $$ \phi: M \to \ $$ gp($$ M \ $$), $$ \phi \ $$ (x) = x - 0.

Any finitely generated monoid, $$ M \ $$ can be embedded into a finitely generated group that is torsionfree. In other words, it is isomorphic to a free abelian group $$ \Z^r \ $$. Looking at the rank of a monoid, M, which is the vector space dimension of $$ \Q \ $$ x gp($$ M \ $$) over $$ \Q \ $$, we can determine that if M is affine and gp($$ M \ $$) is isomorphic to $$ \Z^r \ $$, then the rank of $$ M \ $$ is r.

An $$ M \ $$-module is when in a set $$ N \ $$, with additive operation $$ M \ $$ x $$ N \to N \ $$, (a + b) + x = a + (b + x) and 0 + x = x for all a, b $$ \in M \ $$ and x $$ \in N \ $$.