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.

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.