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 $$ \Q \ $$ ⨂ 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, $$ \tau = \sigma_1 + \dots + \sigma_s \ $$, 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 $$ M \ $$.

When 0 is the only unit (invertible element) in monoid, $$ M \ $$, then $$ M \ $$ is called positive. $$ M \ $$ is positive if and only if $$ C \ $$ ($$ M  \ $$) is pointed. When looking at positive affine monoids there is a standard embedding, meaning that the standard map is injective. Thus, consider affine monoid $$ M \ $$ with gp($$ M \ $$) = $$ \Z^r \ $$ and $$ C = \R_+M \subset \R_+ \ $$. Then first, $$ M \ $$ is positive, secondly, the standard map $$ \sigma \ $$ is injective on $$ M \ $$, thirdly, $$ \sigma: \R^r \to \R^s \ $$ is injective and lastly, $$ C \ $$ is pointed.

Furtherly, if $$ M \ $$ is an affine monoid of rank r with s support forms, then the following are equivalent: (1) $$ M \ $$ is positive; (2) $$ M \ $$ is isomorphic to a submonoid of $$ \Z_{+}^{d} \ $$ for some $$ d \ $$; (3) $$ M \ $$ is isomorphic to a submonoid $$ M' \ $$ of $$ \Z_{+}^{s} \ $$ such that the intersections $$ H_i \cap \R M' \ $$ of the coordinate hyperplanes $$ H_1, \dots, H_s \ $$ are exactly the support hyperplanes of $$ M' \ $$; (4) $$ M \ $$ is isomorphic to a submonid $$ M' \ $$ of $$ \Z_{+}^{r} \ $$ such that the intersections $$ H_i  \cap \R M'  \ $$  of the coordinate hyperplanes $$ H_1, \dots, H_r \ $$ are among the support hyperplanes of $$ M' \ $$; (5) $$ M \ $$ is isomorphic to a submonoid of $$ M' \ $$ of $$ \Z_{+}^{r}  \ $$ with gp($$ M \ $$) = $$ \Z^r \ $$; and (6) $$ M \ $$ has a positive grading, meaning $$ \sigma: M \to \Z_+ \ $$ such that $$ \sigma(x) = 0 \implies x = 0 \ $$.

$$ M \ $$ ($$ P \ $$) is called the polytopal affine monoid, where $$ L \ $$ is an affine lattice in $$ \R^d \ $$ and $$ P \ $$ is an $$ L \ $$-polytope associated with the monoid $$ \Z_+ \ $${(x, 1) : x $$ \in \ $$ lat($$ P \ $$)} in $$ \R^{d+1} \ $$. Note that the set {(x, 1) : x $$ \in \ $$ lat($$ P \ $$)} generating $$ M \ $$ ($$ P \ $$) is denoted $$ E \ $$ ($$ P \ $$). The set lat($$ P \ $$) is finite and thus $$ M \ $$ ($$ P \ $$) is an affine monoid and evidently positive. Polytopal monoids are special instances of homogeneous affine monoid, such that a monoid $$ M \ $$ is positive and admit a positive grading in which all irreducible elements have a degree of 1.

In order to discuss normalization, we must consider $$ M \ $$ to be a submonoid of commutative monoid $$ N \ $$, where the integral closure of $$ M \ $$ in $$ N \ $$ is the submonoid $$ \overbrace{M_N} \ $$ = {x $$ \in N : mx \in M \ $$ for some $$ m \in \N \ $$}. $$ M \ $$ is integrally closed in  $$ N \ $$ if $$ M =  \overbrace{M_N} \ $$. Then the $$ \overline{M} \ $$ is the normalization of a cancellative monoid $$ M \ $$ is the integral closure of $$ M \ $$ in gp($$ M \ $$). Therefore if $$ \overline{M} \ $$ = $$ M \ $$, then M is considered "normal".

Observe that when $$ M \ $$ is an affine monoid, then $$ M \ $$ is polytopal and $$ M \ $$ is homogeneous and coincides with $$ \overline{M} \ $$ in degree 1. The definition of polytopal basically implies that $$ M \ $$ is homogeneous and coincides with $$ \overline{M} \ $$ in degree 1. The height 1 lattice points of $$ \R_+ M \ $$ ($$ P \ $$) are exactly the generators of $$ M \ $$($$ P \ $$), and thus are contained in $$ M \ $$($$ P \ $$). Then to show the converse, let gp($$ M \ $$) = $$ \Z^d \ $$. By the hypothesis, $$ M \ $$ has a grading $$ y \ $$. We can extend it to $$ \Z \ $$-linear form on $$ \Z^d \ $$ and then to a linear form on $$ \R^d \ $$. Then since $$ M \ $$ is generated by elements of degree 1, $$ \R_+ M \ $$ is generated by integral vectors of degree 1. Their convex hull is the lattice polytope $$ P \ $$ = {x $$ \in \R_+ M \ $$ : $$ y \ $$(x) = 1}. Since all of the lattice points of $$ P \ $$ correspond to elements of $$ M \ $$, by hypothesis, $$ M \cong M \ $$($$ P \ $$).

Assume $$ M \ $$ is an integrally closed monoid of affine monoid $$ N \ $$ and the rank $$ M \ $$ = rank $$ N \ $$, then gp($$ M \ $$) = gp($$ N \ $$). (gp($$ N_* \ $$) = gp($$ N \ $$). Assume that gp($$ N \ $$) = $$ \Z^r \ $$ and since rank$$ M \ $$ = r, $$ \R M = \R N = \R^r \ $$. If r = 0, then there is nothing to show. Then suppose that r > 0. Choose elements $$ x_1, \dots, x_r \in M \ $$ generating $$ \R^r \ $$ as a vector space. Now, let $$ M' \ $$ = $$ \Z_+x_1 + \dots + \Z_+x_r \ $$, and the set $$ M = \R_+ M' \cap N \ $$. Then $$ M \ $$ is the integral closure of $$ M' \ $$ in $$ N \ $$ and an affine monoid itself. Then, since $$ M' \subset M \ $$, we can say $$ M \subset M \ $$, by hypothesis on $$ M \ $$. Then we can replace $$ M \ $$ by $$ M \ $$ and assume that $$ M \ $$ is affine. Now, choose $$ x \in \ $$ int($$ M \ $$). Then all support forms of $$ M \ $$ must have a positive value on $$ x \ $$, and they are linear forms on $$ \R N = \R M \ $$. Then for $$ y \in N \ $$ it follows that y + kx $$ \in \R_+ M \ $$ for k >> 0. Then y + kx is integral over $$ M \ $$, and thus y +kx $$ \in M \ $$ by hypothesis and therefore y $$ \in \ $$ gp($$ M \ $$).

When $$ M \subset \Z^r \ $$ is a monoid such that gp($$ M \ $$), then $$ \Z^r \cap \R_+ M \ $$ is the normalization of $$ M \ $$. In this case, $$ M \ $$ is normal and affine, $$ \R_+ M \ $$ is finitely generated and $$ M = \Z^r \cap \R_+ M \ $$, and there exists finitely many rational halfspaces $$ H_{i}^{+} \subset \R^r \ $$ such that $$ M = \cap_{i} H_{i}^{+} \cap \Z^r \ $$.

Then if $$ M \ $$ is a normal affine monoid, its subgroup of units, $$ U \ $$($$ M \ $$) and $$ \sigma: \ $$ gp($$ M \ $$) $$ \to \Z^s \ $$ is the standard map on $$ M \ $$, then $$ M \ $$ is isomorphic to $$ U \ $$($$ M \ $$) ⊕ $$ \sigma \ $$ ($$ M \ $$).

$$ M \ $$ is called "pure" in $$ N \ $$ if $$ M \ $$ is a submonoid of $$ N \ $$ and $$ M \ $$ = $$ N \cap \ $$ gp($$ M \ $$). By this we can state that $$ N \ $$\$$ M \ $$ is an $$ M \ $$-submodule of $$ N \ $$. The purity of $$ M \ $$ in $$ N \ $$ means that R[$$ M \ $$] is the direct summand of R[$$ N \ $$] as an R[$$ M \ $$]-module. A normal affine monoid is not only a pure submonoid of $$ \Z_{+}^{n} \ $$ for suitable $$ n \ $$, but is also a pure, integrally closed submonoid for a free monoid. If $$ M \ $$ is a pure submonoid of an affine monoid $$ N \ $$, then $$ M \ $$ is also affine.

The smallest submonoid of group, $$ G \ $$ containing monoid, $$ M \ $$, where $$ N \subset M \ $$, is denoted as $$ M \ $$[-$$ N \ $$] with all the elements $$ -x, x \in N \ $$. $$ M \ $$[-$$ N \ $$] is also considered the localization with respect to $$ N \ $$.

If $$ M \ $$ is a normal affine monoid with gp($$ M \ $$) = $$ \Z^d \ $$, $$ x \in M \ $$, and $$ H_1, \dots, H_s \ $$ its support hyperplanes, and $$ F \ $$ is the face of $$ \R_+ M \ $$, with $$ x \in \ $$ int($$ F \ $$), then $$ M \ $$[-$$ x \ $$] = $$ M \ $$[-($$ F \cap M \ $$)] = $$ \bigcap {H_{i}^{+} : x \in H_i } \cap \Z^d \ $$. Moreover, $$ M \ $$[-$$ x \ $$] splits into a direct sum $$ L \ $$ ⊕ $$ M' \ $$, where $$ L \cong \Z^e \ $$, e = dim($$ F \ $$). Note that if $$ M \ $$ is positive, then $$ M' \ $$ is positive.

For an ideal $$ I \ $$ in a monoid $$ M \ $$, we call the radical of $$ I \ $$, Rad($$ I \ $$) = {x $$ \in M \ $$ : ax $$ \in I \ $$ for some a $$ \in  \N \ $$}. Then if $$ M \ $$ is an affine monoid, (1) Rad( c \ ($$ \overline{M} \ $$ / $$ M \ $$)) is the set of all $$ x \in M \ $$ such that $$ M \ $$[-$$ x \$$] is normal and (2) ($$ \overline{M} \ $$ / $$ M \ $$) is the union of a finite family of sets $$ x \ $$ + ($$ F \cap M \ $$) where $$ x \in M \ $$ and $$ F \ $$ is a face such that $$ F \caps c( \overline{M} \ $$ \ $$ M) \ $$ = $$ \varnothing \ $$. Moreover, if $$ F \ $$ is maximal among these faces, then at least one set of type $$ x \ $$ + $$ F \cap M \ $$ must appear.