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J. I. García-García, M. A. Moreno-Frías, A. Vigneron-Tenorio, "Combinatorial Properties and Characterization of Glued Semigroups", Abstract and Applied Analysis, vol. 2014, Article ID 436417, 8 pages, 2014. https://doi.org/10.1155/2014/436417
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.
Let be a finitely generated commutative semigroup with zero element which is reduced (i.e., ) and cancellative (if and then ). Under these settings if is torsion-free, then it is isomorphic to a subsemigroup of which means it is an affine semigroup (see ). From now on assume that all the semigroups appearing in this work are finitely generated, commutative, reduced, and cancellative, but not necessarily torsion-free.
Let be a field and the polynomial ring in indeterminates. This polynomial ring is obviously an -graded ring (by assigning the -degree to the indeterminate , the -degree of is ). It is well known that the ideal generated by is an -homogeneous binomial ideal called semigroup ideal (see  for details). If is torsion-free, the ideal obtained defines a toric variety (see  and the references therein). By Nakayama’s lemma, all the minimal generating sets of have the same cardinality and the -degrees of its elements can be determinated.
The main goal of this work is to study the semigroups which result from the gluing of other two. This concept was introduced by Rosales in  and it is closely related to complete intersection ideals (see  and the references therein). A semigroup minimally generated by (with and ) is the gluing of and if there exists a set of generators of of the form , where , are generating sets of and , respectively, , and the supports of and verify and . Equivalently, is the gluing of and if . A semigroup is a glued semigroup when it is the gluing of other two.
As seen, glued semigroups can be determinated by the minimal generating sets of which can be studied by using combinatorial methods from certain simplicial complexes (see [6–8]). In this work the simplicial complexes used are defined as follows: for any , set and the simplicial complex with as the greatest common divisor of the monomials in .
Furthermore, some methods which require linear algebra and integer programming are given to obtain examples of glued semigroups.
The content of this work is organized as follows. Section 2 presents the tools to generalize to nontorsion-free semigroups a classical characterization of affine gluing semigroups (Proposition 2). In Section 3, the nonconnected simplicial complexes associated with glued semigroups are studied. By using the vertices of the connected components of these complexes we give a combinatorial characterization of glued semigroups as well as their glued degrees (Theorem 6). Besides, in Corollary 7 we deduce the conditions for the ideal of a glued semigroup to have a unique minimal system of generators. Finally, Section 4 is devoted to the construction of glued semigroups (Corollary 10) and affine glued semigroups (Section 4.1).
2. Preliminaries and Generalizations about Glued Semigroups
A binomial of is called indispensable if it is an element of all systems of generators of (up to a scalar multiple). This kind of binomials was introduced in  and they have an important role in Algebraic Statistics. In  the authors characterize indispensable binomials by using simplicial complexes . Note that if is generated by its indispensable binomials then it is minimally generated, up to scalar multiples, in an unique way.
With the above notation, the semigroup is associated with the lattice formed by the elements such that . Given a system of generators of , the lattice is generated by the elements with and also verifies that if and only if is reduced. If is a minimal generating set of , denote by the set of elements whose -degree is equal to and by the set of the -degrees of the elements of . When is minimally generated by elements, the semigroup is called a complete intersection semigroup.
Let be the number of connected components of . The cardinality of is equal to (see Remark 2.6 in  and Theorem 3 and Corollary 4 in ) and the complexes associated with the elements in are nonconnected.
Construction 1 (see [7, Proposition 1]). For each the set is obtained by taking binomials with monomials in different connected components of satisfying that two different binomials do not have their corresponding monomials in the same components and fulfilling that there is at least a monomial of every connected component of . This let us construct a minimal generating set of in a combinatorial way.
Let be minimally (we consider a minimal generator set of and in the other case is trivially the gluing of the semigroup generated by one of its nonminimal generators and the semigroup generated by the others) generated by with and . From now on, identify the sets and with the matrices and . Denote by and the polynomial rings and , respectively. A monomial is a pure monomial if it has indeterminates only in or only in ; otherwise it is a mixed monomial. If is the gluing of and , then the binomial is a glued binomial if is a generating set of and in this case the element is called a glued degree.
It is clear that if is a glued semigroup, the lattice has a basis of the form where the supports of the elements in are in , the supports of the elements in are in , by considering only the coordinates in or of , and . Moreover, since is reduced, one has that . Denote by the elements in and by the elements in .
The following proposition generalizes [4, Theorem 1.4] to nontorsion-free semigroups.
Proposition 2. The semigroup is the gluing of and if and only if there exists such that , where , , and are the associated commutative groups of , , and , respectively.
Proof. Assume that is the gluing of and . In this case, is generated by the set (4). Since , the element is equal to and . Let be in ; then there exists such that . Therefore because and so there exist , , satisfying
and . We conclude that with .
Conversely, suppose that there exists such that . We see that . Trivially, . Let be a binomial in . Its -degree is . Using , there exists such that and . We have the following cases.(i)If , (ii)If , Using that the binomial belongs to .(iii)The case is solved similarly.
We conclude that .
From the above proof it is deduced that given the partition of the system of generators of the glued degree is unique.
3. Glued Semigroups and Combinatorics
Glued semigroups by means of nonconnected simplicial complexes are characterized. For any , redefine from (2) as and consider the sets of vertices and the simplicial complexes where and as in Section 2. Trivially, the relations between , , and are
The following result shows an important property of the simplicial complexes associated with glued semigroups.
Lemma 3. Let be the gluing of and and . Then all the connected components of have at least a pure monomial. In addition, all mixed monomials of are in the same connected component.
Proof. Suppose that there exists , a connected component of only with mixed monomials. By Construction 1 in all generating sets of there is at least a binomial with a mixed monomial, but this does not occur in with as a glued binomial.
Since is a glued semigroup, has a system of generators as (4). Let be two monomials such that . In this case, and there exist satisfying (i)If , , and , then , , and .(ii)If , , and then .(iii)The case is solved likewise. In any case, and are in the same connected component of .
We now describe the simplicial complexes that correspond to the -degrees which are multiples of the glued degree.
Lemma 4. Let be the gluing of and , the glued degree, and . Then if and only if . Furthermore, the simplicial complex has at least one connected component with elements in and .
Proof. If there exist , then . Hence, .
Conversely, let with and and let be a glued binomial. It is easy to see that and thus and belong to .
Lemma 5. Let be the gluing of and and the glued degree. Then the elements of are pure monomials and .
Proof. The order defined by if is a partial order on .
Assume that there exists a mixed monomial . By Lemma 3, there exists a pure monomial in such that (the proof is analogous if we consider with ). Now take and . Both monomials are in , where is equal to minus the -degree of . By Lemma 4, if , then , but since this is not possible. So, if is a mixed monomial and , then . If there exists a pure monomial in connected to a mixed monomial in , we perform the same process obtaining with as a mixed monomial and . This process can be repeated if there existed a pure monomial and a mixed monomial in the same connected component. By degree reasons this cannot be performed indefinitely and an element verifying that is not connected having a connected component with only mixed monomials is found. This contradicts Lemma 3.
After examining the structure of the simplicial complexes associated with glued semigroups, we enunciate a combinatorial characterization by means of the nonconnected simplicial complexes .
Theorem 6. The semigroup is the gluing of and if and only if the following conditions are fulfilled.(1)For all , any connected component of has at least a pure monomial.(2)There exists a unique such that and the elements in are pure monomials.(3)For all with , .
Besides, the above is the glued degree.
Proof. If is the gluing of and , the result is obtained from Lemmas 3, 4, and 5.
Conversely, by hypotheses 1 and 3, given that , the set is constructed from and from as in Construction 1. Analogously, if , the set is obtained from the union of , and the binomial with and . Finally is a generating set of and is the gluing of and .
Corollary 7. Let be the gluing of and and a glued binomial with -degree . The ideal is minimally generated by its indispensable binomials if and only if the following conditions are fulfilled.(i)The ideals and are minimally generated by their indispensable binomials.(ii)The element is an indispensable binomial of .(iii)For all , the elements of are pure monomials.
Proof. Suppose that is generated by its indispensable binomials. By [10, Corollary 6], for all the simplicial complex has only two vertices. By Construction 1 and by Theorem 6 for all the simplicial is equal to or . In any case, is an indispensable binomial, and , are generated by their indispensable binomials.
Conversely, suppose that is not generated by its indispensable binomials. Then, there exists such that has more than two vertices in at least two different connected components. By hypothesis, there are not mixed monomials in and thus(i)if is equal to (or ), then (or ) is not generated by its indispensable binomials;(ii)otherwise, and by Lemma 4, with , therefore which contradicts the hypothesis.
We conclude that is generated by its indispensable binomials.
The following example taken from  illustrates the above results.
Example 8. Let be the semigroup generated by the set
In this case, is
Using the appropriated notation for the indeterminates in the polynomial ring (, , , and for the first four generators of and , , , for the others), the simplicial complexes associated with the elements in are those that appear in Figure 1. From Figure 1 and by using Theorem 6, the semigroup is the gluing of and and the glued degree is . From Corollary 7, the ideal is not generated by its indispensable binomials ( has only four indispensable binomials).
4. Generating Glued Semigroups
In this section, an algorithm to obtain examples of glued semigroups is given. Consider and as two minimal generator sets of the semigroups and and let be a basis of with . Assume that and are nontrivial proper ideals of their corresponding polynomial rings. Consider and be two nonzero elements in and , respectively, (note that and because these semigroups are reduced) and the integer matrix Let be a semigroup such that is the lattice generated by the rows of matrix . This semigroup can be computed by using the Smith Normal Form (see [1, Chapter 3]). Denote by , two sets of cardinality and , respectively, satisfying and is generated by the rows of .
The following proposition shows that the semigroup satisfies one of the necessary conditions to be a glued semigroup.
Proposition 9. The semigroup verifies with .
Because may not be a minimal generating set, this condition does not assure that is a glued semigroup. For instance, taking the numerical semigroups , , and , the matrix obtained from formula (17) is and is not a minimal generating set. The following result solves this issue.
Corollary 10. The semigroup is a glued semigroup if
Proof. Suppose that the set of generators of is nonminimal and thus one of its elements is a natural combination of the others. Assume that this element is the first of and then there exist such that . By Proposition 9, there exists satisfying . Since , and thus
with the following cases.(i)If , then is not minimally generated which it is not possible by hypothesis.(ii)If , then , but this is not possible because is a reduced semigroup.(iii)If , then and
If for some , then is not a reduced semigroup. This implies that for all .
We have just proved that . In the general case, if is not minimally generated it is because either or are elements in the canonical bases of or , respectively. To avoid this situation, it is sufficient to take and satisfying and .
Example 11. Let and be two reduced affine semigroups. We compute their associated lattices If we take and , the matrix is and the semigroup is generated by The semigroup is the gluing of the semigroups and and is generated by the rows of the above matrix. The ideal is generated (see  to compute when has torsion) by then is really a glued semigroup.
4.1. Generating Affine Glued Semigroups
From Example 11 it be can deduced that the semigroup is not necessarily torsion-free. In general, a semigroup is affine (or equivalently it is torsion-free) if and only if the invariant factors (the invariant factors of a matrix are the diagonal elements of its Smith Normal Form (see [13, Chapter 2] and [1, Chapter 2])) of the matrix whose rows are a basis of are equal to one. Assume that zero-columns of the Smith Normal Form of a matrix are located on its right side. We now show conditions for being torsion-free.
Take and as the matrices whose rows form a basis of and , respectively, and let , , , and be some matrices with determinant (i.e., unimodular matrices) such that and are the Smith Normal Form of and , respectively. If and are two affine semigroups, the invariant factors of and are equal to 1. Then where and . Let and be the numbers of zero-columns of and ( because and are reduced, see [1, Theorem 3.14]).
Lemma 12. The semigroup is an affine semigroup if and only if
Proof. With the conditions fulfilled by , , and , the necessary and sufficient condition for the invariant factors of to be all equal to one is .
The following corollary gives the explicit conditions that and must satisfy to construct an affine semigroup.
Corollary 13. The semigroup is an affine glued semigroup if and only if(1) and are two affine semigroups;(2);(3);(4)there exist such that
Therefore, to obtain an affine glued semigroup it is enough to take two affine semigroups and any solution of the equations of the above corollary.
Example 14. Let and be the semigroups of Example 11. We compute two elements and in order to obtain an affine semigroup. First of all, we perform a decomposition of the matrix as (26) by computing the integer Smith Normal Form of and :
Second, by Corollary 13, we must find a solution to the system: with . Such solution is computed (in less than a second) using FindInstance of Wolfram Mathematica (see ):
We now take and , and construct the matrix We have the affine semigroup which is minimally generated by satisfying that is generated by the rows of and it is the result of gluing the semigroups and . The ideal is generated by therefore, is a glued semigroup.
All glued semigroups have been computed by using our program Ecuaciones which is available in  (this program requires Wolfram Mathematica 7 or above to run).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
J. I. García-García was partially supported by MTM2010-15595 and Junta de Andalucía group FQM-366. M. A. Moreno-Frías was partially supported by MTM2008-06201-C02-02 and Junta de Andalucía group FQM-298. A. Vigneron-Tenorio was partially supported by Grant MTM2007-64704 (with the help of FEDER Program), MTM2012-36917-C03-01, and Junta de Andalucía group FQM-366.
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