User:MCPOliseno /Math735 AffineMonoids

Michelle Poliseno

Affine Monoids
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, usually denoted by 0). 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.

Information down to here has been extracted into the page Definition:Affine Monoid.

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 the map $ \phi: M \to \ $ gp($ M \ $), $ \phi \ $ (x) = x - 0, is a monoid homomorphism which satisfies the universality condition. It is obvious that in this map, when $ M \ $ is cancellative, $ \phi \ $ is injective.

A monoid is finitely generated if there exists generators, $ a_1, \dots, a_n \ $, such that and element $ m \in M \ $ can be written as $ m = \lambda_1a_1 + \dots + \lambda_na_n \ $, for $ \lambda_i \in \Z_{\ge 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. This definition of rank, however, is not restricted to finitely generated monoids.

Every submonoid of $ \Z \ $ is finitely generated and is isomorphic to a submonoid of $ \Z_+ \ $, unless, however it is a subgroup of $ \Z \ $. These submonoids of $ \Z_+ \ $ are called numerical semigroups.

If $ C \ne 0 $ is a subcone of $ \R^d \ $ it is an example of a continuous monoid. If $ C \ $ = 0, then the monoid is not finitely generated. Note, $ \C \cap \Q^d \ $ is not finitely generated if it contains the nonzero vector.

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 not finitely generated.

A monoid algebra, or monoid ring, $ R \ $[$ M \ $], is constructed by an arbitrary monoid $ M \ $ and every commutative ring of coefficients $ R \ $. $ R \ $[$ M \ $] is free with a basis that consists of symbols $ X^a \ $, called a monomial of $ R \ $[$ M \ $], such that $ a \in M \ $. The operation of multiplication is denoted $ X^aX^b = X^{a+b} \ $. Note that if $ M \ $ is a monoid, $ N \ $ is an $ M \ $-module, and $ R \ $ is a ring, then $ M \ $ is finitely generated if and only if $ R \ $[$ M \ $] is a finitely generated $ R \ $-algebra, and $ N \ $ is a finitely generated $ M \ $-module if and only if $ RN \ $ is a finitely generated $ R \ $[$ M \ $]-module. Also, if $ M \ $ is a finitely generated monoid and $ N \ $ is a finitely generated $ M \ $-module, then every $ M \ $-submodule of $ N \ $ 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. This result can provide an array of examples of affine monoids.

If $ P \subset \R^d \ $ is a rational polyhedron, $C \ $ is the recession cone of $ P \ $, and $ L \subset \Q^d\ $ is a lattice, then $ P \cap L \ $ is a finitely generated module over the affine monoid $ C \cap L \ $.

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, meaning the only invertible element, in monoid, $ M \ $, then $ M \ $ is called positive. Following from above, if 0 is the only unit in $ M \ $, then $ M \ $ has only finitely many irreducible elements. $ M \ $ is positive if and only if $ C \ $ ($ M  \ $) is pointed. There exists a unique minimal system of generators of positive affine monoid $ M \ $, given by its irreducible elements. This system is called the Hilbert basis of $ M \ $ and denoted by Hilb($ M \ $).

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. Note that the total degree $ \tau \ $ when on a positive affine monoid is a grading which only 0 has a degree of 0. This is due to the injectivity of $ \sigma \ $.

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 \ $). This can be shown by letting $ L \ $ = gp($ M \ $). Then claim that $ U \ $($ M \ $) is the kernel of $ \sigma \ $. Then clearly, $ U \ $($ M \ $) $ \subset \ $ Ker($ \sigma \ $). Conversely, let $ x \in \ $ Ker($ \sigma \ $), then $ x \in C \cap L \ $, where $ C \ $ is the cone generated by $ M \ $. The normality of $ M \ $, $ \overline{M} \ $, then shows that $ x \in M \ $ and thus $ x \in U \ $($ M \ $). Since $ U \ $($ M \ $) is a direct summand of $ L \ $, there exists a projection $ \pi: L \to U \ $($ M \ $), which is a surjective $ \Z \ $-linear map, such that $ \pi^2 = \pi \ $. Then since $ x \in M \ $, $ (\pi (x), \sigma(x),) \in U \ $($ M \ $) ⨁ $ \sigma \ $($ M\ $). Conversely, given ($x_0, y^' \ $) $ \in U \ $($ M\ $)⨁ $ \sigma \ $($ M\ $), we choose $ y \in M \ $ with $ y' = \sigma \ $($ y \ $). Then $ x_0 + y - \pi \ $($ y \ $) $ \in M, \pi \ $($ x_0 + y - \pi \ $($ y \ $)) = $ y' \ $.

$ 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 \cap 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. $ I \ $ is a radical ideal in a monoid $ M \ $ if $ I \ $ = Rad($ I \ $) and $ I \ $ is a prime ideal if $ I \ne M \ $. Also, m + n $ \in I \ $, for m, n $ \in M \ $, only if m $ \in I \ $, or n $ \in I \ $. There are only finitely many radical ideal in an affine monoid, and they are determined by the geometry of $ \R_+ M \ $. Thus, if $ M \ $ is an affine monoid and $ I \subset M \ $ is an ideal, then (1) $ I \ $ is a radical ideal if and only if $ I \ $ is the intersection of the sets $ M \ $ \ $ F \ $, where $ F \ $ is a face of $ \R_+ M \ $ with $ F \cap I = \varnothing \ $, and (2) $ I \ $ is a prime ideal if and only if there exists a face $ F \ $ with $ I = M \ $ \ $ F \ $.

A seminormal monoid is a monoid in which every element x $ \in \ $ gp($ M \ $) with 2x, 3x $ \in M \ $ (and therefore mx $ \in M \ $ for m $ \in \Z_+ \ $, m $ \ge \ $ 2) is itself in $ M \ $. Then the subnormalizaion sn($ M \ $) of $ M \ $ is the intersection of all seminormal submonoid of gp($ M \ $) containing $ M \ $. Note that the subnormalization sn($ M \ $) of an affine monoid, $ M \ $, is affine itself. A normal monoid is seminormal, but a seminormal monoid is not necessarily normal. An affine monoid $ M \ $ is seminormal if and only if $ (M \cap F)_* \ $ is a normal monoid for every face $ F \ $ of $ \R_+ M \ $ and thus $ M_* = \overline{M_*} \ $ if $ M \ $ is seminormal.

$ M_* \ $ is the filtered union of affine submonoids, and if $ M \ $ is seminormal, then the submonoids can be chosen to be normal. Thus, if $ M \ $ is a positive affine monoid, $ M_* \ $ is the filtered union of affine submonoids, and if $ M_* \ $ is normal, then these submonoids can be chosen to be normal. There exists a family of affine monoids, $ M_i \ $, where i are elements of a set $ I \ $ such that $ M \ $ = $ \cup_{i \in I} M_i \ $ and where for all i, j $ \in I \ $, there exists a k $ \in I \ $ such that $ M_i, M_j \subset M_k \ $.

If we have $ M \subset \Z^r \ $ be a monoid with gp($ M \ $) = $ \Z^r \ $, then the following are equivalent, (1) $ M \ $ is seminormal and affine, and (2) there exists finitely many rational halfspaces, $ H_{i}^{+} \ $ and subgroups $ U_i \subset H_i \cap \Z^r \ $ such that rank $ U_i \ $ = r-1 and $ M \ $ = $ \bigcap(U_i \cup (H_{i}^{>} \cap \Z^r)) \ $.

If $ M \ $ is a reductive monoid, with zero, then $ M \ $ is affine. We can prove this by supposing that $ M' \ $ is irreducible. Then, if $ M \ $ is not normal, we can just take the normalization $ \sigma: \overline{M} \to M \ $. Then $ \overline{M} \ $ is a normal monoid with zero, and so $ \overline{M} \ $ is affine and therefore it follows that $ M \ $ is affine. Note that since the morphism of the normal $ \sigma \ $ is a finite surjective morphism, then $ \overline{M} \ $ is affine if and only if $ M \ $ is affine.