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

The interior of $$ M \ $$ can be denoted as $$ int(M) \ $$ = $$ M \cap int(\R_+M) \ $$, when $$ M \subset \Z^d \ $$ is an affine monoid. Since x+y $$ \in int(\R_+M) \ $$, for x $$ \in int(\R_+M) \ $$ and y $$ \in \R_+M \ $$, then it follows that int($$ M \ $$) is an ideal. Consider 0 $$ \in \ $$ int($$ M \ $$). This occurs if and only if M is a group, which implies that int($$ M \ $$) = $$ M \ $$. If 0 $$ \notin \ $$ int($$ M \ $$), then int($$ M \ $$) is $$ not \ $$ a monoid.

Int($$ M \ $$) $$ \cup \ $$ {0} is equal to $$ M_* \ $$, which is a submonoid of $$ M \ $$, where $$ M_* \ $$ = $$ M \iff \ $$ rank $$ M \le \ $$ 1 or $$ M \ $$ = int($$ M \ $$). Otherwise, $$ M_* \ $$ is finitely generated.

Suppose $$ C \ $$ is a rational cone in $$ \R^d \ $$ and $$ L \subset \Q^d \ $$ is a lattice. Then $$ C \cap L \ $$ is an affine monoid. This is known as Gordan's Lemma. To prove this lemma set $$ C' \ $$ = $$ C \cap \R L \ $$. Then $$ C' \ $$ is a rational cone as well and every element of $$ x \in \Q^d \cap \R L \ $$ is a rational linear combination of elements of $$ L \ $$ and so $$ \exists \ $$ an $$ a>0 \in \Z \ $$ with $$ ax \in L \ $$. Choose a finite system of generators $$ x_1, x_2, \dots, x_n \ $$ of $$ C' \ $$. Assume that $$ x_1, x_2, \dots, x_n \in L \ $$ and let $$ M' \ $$ be the affine monoid generated by $$ x_1, x_2, \dots, x_n \ $$. Then every element $$ x \in C' \cap L \ $$ has a representation $$ x = a_1x_1 + \dots + a_nx_n \ $$ for all $$ a_i \in \R_+ \ $$. Then $$ x = (\left \lfloor {a_i} \right \rfloor x_i + \dots + \left \lfloor {a_i} \right \rfloor x_n) \ $$ + $$ (q_1x_1 + \dots + q_nx_n) \ $$, where $$ \left \lfloor {x} \right \rfloor \ $$ = max{$$ z \in \Z : z \le x \ $$}, for $$ x \in \R \ $$. Then 0 $$ \le q_i = a_i - \left \lfloor {a_i} \right \rfloor < 1, i = 1, \dots, n \ $$. The first summand on the right hand side is in $$ M' \ $$ and the second is an element of $$ C' \cap L \ $$ that belongs to a bounded subset $$ B \ $$ of $$ \R^n \ $$. Then it follows that $$ C' \cap L \ $$ is generated as an $$ M' \ $$ - module by the finite set $$ B \cap C' \cap L \ $$ Being a finitely generated module over an affine monoid, the monoid $$ C \cap L \ $$ is itself finitely generated.

Given $$ M \ $$ as a submonoid of $$ \R^d, L \ $$ a lattice in $$ \R^d \ $$ containing $$ M \ $$ and $$ C = \R_+M \ $$, then $$ M \ $$ is an affine monoid, $$ \overbrace{M_L} = C \cap L \ $$ is also an affine monoid- where $$ \overbrace{M_L} \ $$ is also a finitely generated $$ M \ $$-module - and $$ C \ $$ is a cone.

It is also true that if $$ M \ $$ and $$ N \ $$ are affine submonoids of $$ \R^d \ $$ and $$ C \ $$ is a cone generated by the elements of gp($$ M \ $$), then (1) $$ M \cap N \ $$ is an affine monoid, (2) $$ M \cap C \ $$ is an affine monoid and (3) the extreme submonoids of $$ M \ $$ are affine.

When looking at an integral domain, the nonzero elements form a commutative cancellative monoid with respect to multiplication.

The standard map on an affine monoid is defined as the group gp($$ M \ $$) which is isomorphic to $$ \Z^r \ $$, where r = rank $$ M \ $$. The cone $$ C = \R_+M \subset \R^r \ $$ is generated by $$ M \ $$ and has a representation $$ C = H_{\sigma_1}^{+} \cap \dots \cap H_{\sigma_s}^{+} \ $$, which is an irredundant intersection of halfspaces defined by linear forms on $$ \R^+ \ $$. $$ H_{\sigma_s} \ $$ is a hyperplane generated as a vector space by integral vectors. Thus we can assume that $$ \sigma_i \ $$ is the \Z^r \ height above $$ H_{\sigma_i} \ $$. Then $$ \sigma_i \ $$ are called the support forms of $$ M \ $$ and $$ \sigma_i: M \to \Z_{+}^{s},  \sigma(x) = (\sigma_1(x), \dots, \sigma_s(x)) \ $$ is considered the standard map on $$ M \ $$.

The standard map has a natural extension to $$ \R^r \ $$ with values, $$ \sigma \in \R^s \ $$ and $$ \sigma_1, \dots, \sigma_s \ $$ is the minimal set of generators of the dual cone $$ C* \ $$. The standard map depends on the order of $$ \sigma_1, \dots, \sigma_s \ $$.

Note that $$ x \ $$ is a unit of a monoid, $$ M \ $$ if $$ x \ $$ has an inverse in $$ M \ $$. The units of $$ M \ $$ form a group, U($$ M \ $$). Now, let $$ M \ $$ be an affine monoid with the standard map $$ \sigma \ $$. Then (1), the units of $$ M \ $$ are precisely the elements $$ x \ $$ with $$ \sigma(x) \ $$ = 0, or equivalently, the total degree, $$ t = \sigma_1 + \dots + \sigma_s \ $$, on M is equal to zero. Also, (2), every element $$ x \in M \ $$ has a presentation $$ x = u + y_1 + \dots + y_m \ $$ in which $$ u \ $$ is a unit and $$ y_1, \dots, y_m \ $$ are irreducible, meaning if $$ y_i \ $$ = p + q, then one of the summands, p, q, must be a unit. Thirdly, up to two differences by unit, there exists only finitely many irreducible elements in $$ M \ $$.