Combinatorial properties and characterization of glued semigroups

This work focuses on the combinatorial properties of glued semigroups and provides its combinatorial characterization. Some classical results for affine glued semigroups are generalized and some methods to obtain glued semigroups are developed.


Introduction
Let S = n 1 , . . . , n l be a finitely generated commutative semigroup with zero element such that it is reduced (i.e. S ∩ (−S) = (0)). We suppose that S is cancellative, that is to say, if m+n = m+n , with m, n, n ∈ S, then n = n . With these conditions, we may assume that S is a subsemigroup of a non necessarily torsion-free group. If S is torsion-free, then S is an affine semigroup.
From now on, we assume that all the semigroups appearing in this work are finitely generated, commutative and reduced, thus in the sequel we omit these adjectives.
Let k be a field and k[X 1 , . . . , X l ] the polynomial ring in l indeterminates. This polynomial ring is obviously an S−graded ring (by assigning the S-degree n i to the indeterminate X i , the S-degree of X α = X α1 1 · · · X α l l is It is well known that the ideal (denoted by I S ) generated by is an S−homogeneous binomial ideal called semigroup ideal (see [6] for details). If S is torsion-free, the ideal obtained defines a toric variety (see [12] and the references therein). By Nakayama's lemma, all minimal generating sets of I S have same cardinality and the S−degrees of its elements are determinated.
In [1], [4] and [7] the authors study the minimal generating sets of semigroup ideals by means of the homology of different simplicial complexes (with isomorphic homologies) associated to the semigroup. For any m ∈ S, set we consider the abstract simplicial complex (used in [4] and [7]) on the vertex set C m , where gcd(F ) is the greatest common divisor of the monomials in F. The main aim of this work is to study the semigroups which result from the gluing of other two. This concept was introduced by Rosales in [10], and it is closely related to the ideals that are complete intersections (see [13] and the references therein).
A semigroup S minimally generated by A 1 A 2 (with A 1 = {n 1 , . . . , n r } and A 2 = {n r+1 , . . . , n l }) is the gluing of S 1 = A 1 and S 2 = A 2 , if there exists a set of generators, ρ, of I S of the form where ρ 1 , ρ 2 are generating sets of I S1 and I S2 respectively, and X γ − X γ ∈ I S such that the support of γ (supp (γ)) is included in {1, . . . , r} and supp (γ ) ⊂ {r + 1, . . . , l}. Equivalently, S is the gluing of S 1 and S 2 if I S = I S1 + I S2 + X γ − X γ . We call glued semigroups to this kind of semigroups.
In Section 1, we define the required mathematical elements in order to generalize to non torsion-free semigroups a classical result concerning affine semigroups (Proposition 2).
In Section 2, we examine the non-connected simplicial complexes ∇ m associated to the glued semigroups. By understanding the vertices of the connected components of these complexes, we give a combinatorial characterization of the glued semigroups as well as their glued degrees (Theorem 6). Besides, in Corollary 7 we deduce the conditions under which the ideal of a glued semigroup is uniquely generated. Despite the fact that Theorem 6 and Corollary 7 provide the basis to implement algorithms, they may be, however, no efficient. In this sense, the goal of this section is to provide further knowledge about (glued) semigroups employing combinatorial theory no matter the efficiency of the obtained algorithms.
We devote the last part of this work, Section 3, to construct glued semigroups (Corollary 10), complete intersection glued semigroups and affine glued semigroups (Subsection 3.1). We create the affine glued semigroups by solving an integer programming problem.

Preliminaries and generalizations about glued semigroups
In this section we summarize some notations and definitions, and give a generalization to non-torsion free semigroups of [10,Theorem 1.4].
We say that a binomial in I S is indispensable if it is in all the system of generators of I S (up to a scalar multiple). This kind of binomials were introduced in [9]. This notion comes from the Algebraic Statistics. In [8] the authors characterize the indispensable binomials by using the simplicial complexes ∇ m . Note that if I S is generated by its indispensable binomials, I S is uniquely generated (up to a scalar multiple).
Being the notation set as in the introduction, we associate a lattice to the semigroup S: ker S ⊂ Z l , α = (α 1 , . . . , α l ) ∈ ker S if l i=1 α i n i = 0. The property "S is reduced" is equivalent to ker S ∩ N l = (0). Given a system of binomial generators of I S , ker S is generated by a set whose elements are α − β with X α − X β being in the system of binomial generators.
We call M(I S ) to a minimal generating set of I S , and M(I S ) m ⊂ M(I S ) to the set of their elements whose S−degree are equal to m ∈ S. Betti(S) is the set of the S−degrees of the elements in M(I S ).
S is called a complete intersection semigroup if I S is minimally generated by rank(ker S) elements.
Let C(∇ m ) be the number of connected components of a non-connected ∇ m , this means that the cardinality of M(I S ) m is C(∇ m ) − 1 (see Remark 2.6 in [1] and Theorem 3 and Corollary 4 in [7]). Note that the complexes associated to the elements in Betti(S) are non-connected. The relation between M(I S ) and Betti(S) is studied next.

Construction 1. ([4, Proposition 1]).
For each m ∈ Betti(S), one can construct M(I S ) m by taking C(∇ m ) −1 binomials whose monomials are in different connected components of ∇ m and satisfying that two different binomials have not their corresponding monomials in the same components. This let us construct a minimal generating set of I S in a combinatorial way. Now, we are going to introduce the notations that we use to work with glued semigroups.
Let S be minimally 1 generated by A 1 A 2 with A 1 = {a 1 , . . . , a r } and A 2 = {b 1 , . . . , b t }. From now on, we identify the sets A 1 and A 2 with the matrixes We denote by k[A 1 ] and k[A 2 ] to the polinomial rings k[X 1 , . . . , X r ] and k[Y 1 , . . . , Y t ], respectively. We call pure monomials to the monomials with indeterminates only in X 1 , . . . , X r or Y 1 , . . . , Y t . Conversely, we call mixed monomials to the monomials with indeterminates in Xs and Y s. Given S, the gluing of S 1 = A 1 and S 2 = A 2 , we say that In this way, it is clear that if S is a glued semigroup, the lattice ker S has a basis such as where the supports of the elements in L 1 are in {1, . . . , r}, the supports of the elements in L 2 are in {r + 1, . . . , r + t}, Moreover, since S is reduced, one has that L 1 Z ∩ N r+t = L 2 Z ∩ N r+t = (0). We will denote by {ρ 1i } i to the elements in L 1 and by {ρ 2i } i to the elements in L 2 .
and dZ are the associated commutative groups of S 1 , S 2 and {d}.
Proof. Let's assume that S is the gluing of S 1 and S 2 . In this case, ker S is generated by the set (3).
Conversely, we suppose that there exists d ∈ (S 1 ∩S 2 )\{0} such that G(S 1 )∩ G(S 2 ) = dZ. Assuming this, we will prove that I S = I S1 This last polynomial is in I S1 • The case λ < 0 can be solved likewise.
Therefore, we conclude that It follows that, given the partition of the system of generators of S, the glued degree is unique.

Glued semigroups and combinatorics
In this section, we approach the study of simplicial complexes ∇ m associated with glued semigroups. We characterize the glued semigroups by means of the non-connected simplicial complexes.
For any m ∈ S, we redefine C m from (1), as and consider the vertex sets and the simplicial complexes where A 1 = {a 1 , . . . , a r } and A 2 = {b 1 , . . . , b t } as in Section 1. Trivially, the relations between ∇ A1 m , ∇ A2 m and ∇ m are The following result shows a relevant property of the simplicial complexes associated to glued semigroups.
Lemma 3. Let S be the gluing of S 1 and S 2 , and m ∈ Betti(S). Then all the connected components of ∇ m have at least a pure monomial. In addition, all mixed monomials of ∇ m are in the same connected component.
Proof. Supposed that there exists C, a connected component of ∇ m only with mixed monomials. In this case, in any generating set of I S there is, at least, a binomial with a mixed monomial (by Construction 1). But there is not this binomial. This is not possible because S is the gluing of S 1 and S 2 .
Since S is a glued semigroup, ker S has a system of generators as the intro- • The case λ < 0 can be solved likewise.
In any case, X α Y β and X γ Y δ are in the same connected component of ∇ m .
The following Lemma describes the simplicial complexes that correspond to the S−degrees that are multiples of the glued degree. Proof The following Lemma is a combinatorial version of [5,Lemma 9] and it is a necessary condition for our combinatorial characterization theorem (Theorem 6).
Lemma 5. Let S be the gluing of S 1 and S 2 , and d ∈ S the glued degree. Then the elements in C d are pure monomials and d ∈ Betti(S).
Proof. The reader can check that m S m if m − m ∈ S, is a well defined partial order on S.
Let's assume that there exists a mixed monomial T ∈ C d . By Lemma 3, there exists a pure monomial in But this is not possible due to the fact that d ≺ S d. Consequently, one can consider that T 1 is a mixed monomial and C A1 d = ∅, but C A2 d is not empty. If there exists a pure monomial in C A2 d connected to a mixed monomial in C d , the above process can be repeated until T 2 , Y b2 ∈ C d are obtained, with T 2 a mixed monomial. This process is finite by degree reasons.
So, after some steps, one find a d (i) ∈ S such that ∇ d (i) is not connected (i.e. d (i) ∈ Betti(S)) and it has a connected component whose vertices are only mixed monomials. This contradicts Lemma 3.
After examining the structure of the simplicial complexes associated to the glued semigroups, we enunciate a combinatorial characterization theorem by means of the non-connected simplicial complexes ∇ m . This result helps to understand the nature of glued semigroups and increase our knowledge on them.

For all
Besides, the above d ∈ Betti(S) is the glued degree.
Proof. If S is the gluing of S 1 and S 2 , we obtain immediately the theorem from Lemmas 3, 4 and 5.
Conversely, given d ∈ Betti(S) \ {d}, by the hypothesis 1 and 3, we can construct the sets, M(I S1 ) d and M(I S2 ) d , in a similar way to Construction 1, but only taking binomials whose monomials are in C A1 d or C A2 d . Analogously, if we consider d ∈ Betti(S), we construct the set M(I S ) d with C(∇ d ) − 1 binomials as the union of: We conclude that m∈Betti(S) generating set of I S . So S is the gluing of S 1 and S 2 .
From Theorem 6 we obtain an equivalent property to that in [5, Theorem 12] by using the language of monomials and binomials.
Corollary 7. Let S be the gluing of S 1 and S 2 , and X γ X − Y γ Y ∈ I S a glued binomial with S−degree d. Then, I S is (minimally) generated by its indispensable binomials if and only if: • I S1 and I S2 are (minimally) generated by their indispensable binomials.
• For all d ∈ Betti(S), the elements of C d are pure monomials.
Proof. Suppose that I S is generated by its indispensable binomials. By [8,Corollary 6], ∀m ∈ Betti(S), ∇ m has only two vertices. In particular, by d or ∇ A2 d (by Lemma 1). In any case, X γ X − Y γ Y ∈ I S is an indispensable binomial, and I S1 , I S2 are generated by their indispensable binomials.
Conversely, we suppose that I S is not generated by its indispensable binomials. So, ∃d ∈ Betti(S) \ {d} such that ∇ d has more than two vertices in at least two different connected components. Taking into account our hypothesis, there are not mixed monomials in ∇ d and so: , then I S1 (or I S2 ) is not generated by its indispensable binomials.
• In other case, C A1 d = ∅ = C A2 d , by Lemma 4, d = jd, with j ∈ N, and so X (j−1)γ X Y γ Y ∈ C d , which contradicts our hypothesis.
Thus, we conclude that I S is generated by its indispensable binomials.
We illustrate the above results with the following example taken from [13].
From Corollary 7, I S is not generated by its indispensable binomials (I S has only four indispensable binomials).

Generating glued semigroups
In this section, we give an algorithm with the aim of producing many examples of glued semigroups. Furthermore, we construct affine glued semigroups by means of solving an integer programming problem.
First of all, we consider two semigroups T 1 and T 2 . Keeping the same notation we have followed throughout the whole article, let A 1 = {a 1 , . . . , a r } and A 2 = {b 1 , . . . , b t } be two minimal generator sets of the semigroups T 1 = A 1 and T 2 = A 2 , and L j = {ρ ji } i be a basis of ker T j with j = 1, 2.
Let γ X and γ Y be two nonzero elements in N r and N t respectively 2 , and consider the integer matrix Let S be a semigroup such that ker S is the lattice generated by the rows of the matrix A. This semigroup can be computed by using the Smith Normal Form (see [11,Chapter 2]). Denote by B 1 , B 2 to two sets of cardinality r and t respectively, satisfying S = B 1 , B 2 and ker( B 1 , B 2 ) is generated by the rows of A.
The following Proposition shows that the semigroup S satisfies one of the conditions to be a glued semigroup.
Proof. Likewise in the proof of the necessary condition of Proposition 2, since we only used that ker S has a basis as (3).
This condition is not enough for S to be a glued semigroup, because the generating set B 1 ∪ B 2 could be non-minimal. For example, if one get the numerical semigroups T 1 = 3, 5 , T 2 = 2, 7 and (γ X , γ Y ) = (1, 0, 2, 0), one have the matrix as (5)  Next corollary is devoted to solve this issue.
2 Note that γ X / ∈ ker T 1 and γ Y / ∈ ker T 2 because these semigroups are reduced.
• If λγ X1 = 0, T 1 is not minimally generated, but it is not possible by hypothesis.
We have just proved that γ X = (1, 0, . . . , 0). In the general case, if S is not minimally generated it is because either γ X or γ Y are elements in the canonical bases of N r or N t , respectively. To avoid this situation, it is sufficient to take γ X and γ Y satisfying From the above result we obtain a characterization of the glued semigroups: S is a glued semigroup if and only if ker S has a basis as (3) satisfying the condition (6).
We compute their associated lattices This semigroup verifies that ker S is generated by the rows of the above matrix, and it is the gluing of the semigroups B 1 and B 2 . The ideal I S ⊂ C[x 1 , . . . , x 4 , y 1 , . . . , y 3 ] is generated 3 by then S is really a glued semigroup.
In this way, we provide a procedure that allows us the construction of (glued) semigroups that are complete intersections. Regarding the following Lemma, it is sufficient that the semigroups T 1 and T 2 are complete intersections in order to be S as well. Next we give an algorithm to generate many examples of complete intersection semigroups.
Lemma 12. T 1 and T 2 are two complete intersection semigroups if and only if S is complete intersection semigroup.

Generating affine glued semigroups
As one can check in Example 11, the semigroup S is not necessarily torsionfree. In general, a semigroup T is affine, i.e. it is torsion-free, if and only if the invariant factors 4 of the matrix whose rows are a basis of ker T are equal to one. We suppose that if the Smith Normal Form, D, of a matrix has some zero-columns, which are on the right side of D.
We use this fact to give conditions for S being torsion-free. Let P 1 , P 2 , Q 1 and Q 2 some matrices with determinant ±1 (i.e. unimodular matrices) such that D 1 = P 1 L 1 Q 1 and D 2 = P 2 L 2 Q 2 are the Smith Normal Form of L 1 and L 2 , respectively.
If T 1 and T 2 are two affine semigroups, the invariant factors of L 1 and L 2 are equal to 1. Then where γ X = γ X Q 1 and γ Y = −γ Y Q 2 . Let s 1 and s 2 be the numbers of zerocolumns of D 1 and D 2 (s 1 , s 2 > 0 because T 1 and T 2 are reduced).

Lemma 13. S is an affine semigroup if and only if
Proof. With the conditions imposed to T 1 , T 2 and (γ X , γ Y ), gcd {γ Xi } r i=r−s1 ∪ {γ Y i } t i=t−s2 = 1 is a necessary and sufficient condition for the invariant factors of A to be all equal to one.
In the following Corollary we give the explicit conditions that γ X and γ Y must fulfill to construct an affine semigroup. 1. T 1 and T 2 are two affine semigroups.
Proof. Trivial by construction, Corollary 10 and Lemma 13. Therefore, to construct an affine glued semigroup, it is enough to take two affine semigroups, and any solution, (γ X , γ Y ), of the equations of the above corollary.
Example 15. Let T 1 and T 2 be the semigroups of Example 11.