AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 436417 10.1155/2014/436417 436417 Research Article Combinatorial Properties and Characterization of Glued Semigroups García-García J. I. 1 Moreno-Frías M. A. 1 Vigneron-Tenorio A. 2 Gandarias Maria 1 Departamento de Matemáticas Universidad de Cádiz Puerto Real 11510 Cádiz Spain uca.es 2 Departamento de Matemáticas Universidad de Cádiz Jerez de la Frontera 11405 Cádiz Spain uca.es 2014 1742014 2014 26 09 2013 12 10 2013 17 4 2014 2014 Copyright © 2014 J. I. García-García et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

Let S = n 1 , , n l be a finitely generated commutative semigroup with zero element which is reduced (i.e., S ( - S ) = { 0 } ) and cancellative (if m , n , n S and m + n = m + n then n = n ). Under these settings if S is torsion-free, then it is isomorphic to a subsemigroup of p 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 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 1 α 1 X l α l is i = 1 l α i n i S ). It is well known that the ideal I S generated by (1) { X α - X β i = 1 l α i n i = i = 1 l β i n i } 𝕂 [ X 1 , , X l ] is an S -homogeneous binomial ideal called semigroup ideal (see  for details). If S 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 I S have the same cardinality and the S -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 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 ρ = ρ 1 ρ 2 { X γ - X γ } , where ρ 1 , ρ 2 are generating sets of I S 1 and I S 2 , respectively, X γ - X γ I S , and the supports of γ and γ verify supp ( γ ) { 1 , , r } and supp ( γ ) { r + 1 , , l } . Equivalently, S is the gluing of S 1 and S 2 if I S = I S 1 + I S 2 + X γ - X γ . 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 I S which can be studied by using combinatorial methods from certain simplicial complexes (see ). In this work the simplicial complexes used are defined as follows: for any m S , set (2) C m = { X α = X 1 α 1 X l α l i = 1 l α i n i = m } , and the simplicial complex (3) m = { F C m gcd ( F ) 1 } , with gcd ( F ) as the greatest common divisor of the monomials in F .

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 m 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 I S is called indispensable if it is an element of all systems of generators of I S (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 m . Note that if I S 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 S is associated with the lattice ker S formed by the elements α = ( α 1 , , α l ) l such that i = 1 l α i n i = 0 . Given G a system of generators of I S , the lattice ker S is generated by the elements α - β with X α - X β G and ker S also verifies that ker S l = { 0 } if and only if S is reduced. If ( I S ) is a minimal generating set of I S , denote by ( I S ) m ( I S ) the set of elements whose S -degree is equal to m S and by Betti ( S ) the set of the S -degrees of the elements of ( I S ) . When I S is minimally generated by rank ( ker S ) elements, the semigroup S is called a complete intersection semigroup.

Let 𝒞 ( m ) be the number of connected components of m . The cardinality of ( I S ) m is equal to 𝒞 ( m ) - 1 (see Remark  2.6 in  and Theorem 3 and Corollary  4 in ) and the complexes associated with the elements in Betti ( S ) are nonconnected.

Construction 1 (see [<xref ref-type="bibr" rid="B4">7</xref>, Proposition 1]).

For each m Betti ( S ) the set ( I S ) m is obtained by taking 𝒞 ( m ) - 1 binomials with monomials in different connected components of m 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 m . This let us construct a minimal generating set of I S in a combinatorial way.

Let S be minimally (we consider a minimal generator set of S and in the other case S is trivially the gluing of the semigroup generated by one of its nonminimal generators and the semigroup generated by the others) generated by A 1 A 2 with A 1 = { a 1 , , a r } and A 2 = { b 1 , , b t } . From now on, identify the sets A 1 and A 2 with the matrices ( a 1 | | a r ) and ( b 1 | | b t ) . Denote by 𝕂 [ A 1 ] and 𝕂 [ A 2 ] the polynomial rings 𝕂 [ X 1 , , X r ] and 𝕂 [ Y 1 , , Y t ] , respectively. A monomial is a pure monomial if it has indeterminates only in X 1 , , X r or only in Y 1 , , Y t ; otherwise it is a mixed monomial. If S is the gluing of S 1 = A 1 and S 2 = A 2 , then the binomial X γ X - Y γ Y I S is a glued binomial if ( I S 1 ) ( I S 2 ) { X γ X - Y γ Y } is a generating set of I S and in this case the element d = S -degree ( X γ X ) S is called a glued degree.

It is clear that if S is a glued semigroup, the lattice ker S has a basis of the form (4) { L 1 , L 2 , ( γ X , - γ Y ) } r + t , 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 } , ker S i = L i ( i = 1,2 ) by considering only the coordinates in { 1 , , r } or { r + 1 , , r + t } of L i , and ( γ X , γ Y ) r + t . Moreover, since S is reduced, one has that L 1 r + t = L 2 r + t = { 0 } . Denote by { ρ 1 i } i the elements in L 1 and by { ρ 2 i } i the elements in L 2 .

The following proposition generalizes [4, Theorem 1.4] to nontorsion-free semigroups.

Proposition 2.

The semigroup S is the gluing of S 1 and S 2 if and only if there exists d ( S 1 S 2 ) { 0 } such that G ( S 1 ) G ( S 2 ) = d , where G ( S 1 ) , G ( S 2 ) , and d are the associated commutative groups of S 1 , S 2 , and { d } , respectively.

Proof.

Assume that S is the gluing of S 1 and S 2 . In this case, ker S is generated by the set (4). Since ( γ X , - γ Y ) ker S , the element d is equal to A 1 γ X = A 2 γ Y S and d S 1 S 2 G ( S 1 ) G ( S 2 ) . Let d be in G ( S 1 ) G ( S 2 ) ; then there exists ( δ 1 , δ 2 ) r × t such that d = A 1 δ 1 = A 2 δ 2 . Therefore ( δ 1 , - δ 2 ) ker S because ( A 1 A 2 ) ( δ 1 , - δ 2 ) = 0 and so there exist λ , λ i ρ 1 , λ i ρ 2 satisfying (5) ( δ 1 , 0 ) = i λ i ρ 1 ρ 1 i + λ ( γ X , 0 ) , ( 0 , δ 2 ) = - i λ i ρ 2 ρ 2 i + λ ( 0 , γ Y ) , and d = A 1 δ 1 = i λ i ρ 1 ( A 1 0 ) ρ 1 i + λ A 1 γ X = λ d . We conclude that G ( S 1 ) G ( S 2 ) = d with d S 1 S 2 .

Conversely, suppose that there exists d ( S 1 S 2 ) { 0 } such that G ( S 1 ) G ( S 2 ) = d . We see that I S = I S 1 + I S 2 + X γ X - Y γ Y . Trivially, I S 1 + I S 2 + X γ X - Y γ Y I S . Let X α Y β - X γ Y δ be a binomial in I S . Its S -degree is A 1 α + A 2 β = A 1 γ + A 2 δ . Using A 1 ( α - γ ) = A 2 ( β - δ ) G ( S 1 ) G ( S 2 ) = d , there exists λ such that A 1 α = A 1 γ + λ d and A 2 δ = A 2 β + λ d . We have the following cases.

If λ = 0 , (6) X α Y β - X γ Y δ = X α Y β - X γ Y β + X γ Y β - X γ Y δ = Y β ( X α - X γ ) + X γ ( Y β - Y δ ) I S 1 + I S 2 .

If λ > 0 , (7) X α Y β - X γ Y δ = X α Y β - X γ X λ γ X Y β + X γ X λ γ X Y β - X γ X λ γ Y Y β + X γ X λ γ Y Y β - X γ Y δ = Y β ( X α - X γ X λ γ X ) + X γ Y β ( X λ γ X - Y λ γ Y ) + X γ ( Y λ γ Y Y β - Y δ ) .

Using that (8) X λ γ X - Y λ γ Y = ( X γ X - Y γ Y ) ( i = 0 λ - 1 X ( λ - 1 - i ) γ X Y i γ Y ) ,

the binomial X α Y β - X γ Y δ belongs to I S 1 + I S 2 + X γ X - Y γ Y .

The case λ < 0 is solved similarly.

We conclude that I S = I S 1 + I S 2 + X γ X - Y γ Y .

From the above proof it is deduced that given the partition of the system of generators of S the glued degree is unique.

3. Glued Semigroups and Combinatorics

Glued semigroups by means of nonconnected simplicial complexes are characterized. For any m S , redefine C m from (2) as (9) C m = { X α Y β = X 1 α 1 X r α r Y 1 β 1 Y t β t i = 1 r α i a i + i = 1 t β i b i = m } , and consider the sets of vertices and the simplicial complexes (10) C m A 1 = { X 1 α 1 X r α r i = 1 r α i a i = m } , m A 1 = { F C m A 1 gcd ( F ) 1 } , C m A 2 = { Y 1 β 1 Y t β t i = 1 t β i b i = m } , m A 2 = { F C m A 2 gcd ( F ) 1 } , where A 1 = { a 1 , , a r } and A 2 = { b 1 , , b t } as in Section 2. Trivially, the relations between m A 1 , m A 2 , and m are (11) m A 1 = { F m F C m A 1 } , m A 2 = { F m F C m A 2 } .

The following result shows an important property of the simplicial complexes associated with 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.

Suppose that there exists C , a connected component of m only with mixed monomials. By Construction 1 in all generating sets of I S there is at least a binomial with a mixed monomial, but this does not occur in ( I S 1 ) ( I S 2 ) { X γ X - Y γ Y } with X γ X - Y γ Y as a glued binomial.

Since S is a glued semigroup, ker S has a system of generators as (4). Let X α Y β , X γ Y δ C m be two monomials such that gcd ( X α Y β , X γ Y δ ) = 1 . In this case, ( α , β ) - ( γ , δ ) ker S and there exist λ , λ i ρ 1 , λ i ρ 2 satisfying (12) ( α - γ , 0 ) = i λ i ρ 1 ρ 1 i + λ ( γ X , 0 ) , ( 0 , β - δ ) = i λ i ρ 2 ρ 2 i - λ ( 0 , γ Y ) .

If λ = 0 , α - γ ker S 1 , and β - δ ker S 2 , then A 1 α = A 1 γ , A 2 β = A 2 δ , and X α Y δ C m .

If λ > 0 , ( α , 0 ) = i λ i ρ 1 ρ 1 i + λ ( γ X , 0 ) + ( γ , 0 ) , and (13) A 1 α = i λ i ρ 1 ( A 1 0 ) ρ 1 i + λ A 1 γ X + A 1 γ = λ d + A 1 γ ,

then X λ γ X X γ Y β C m .

The case λ < 0 is solved likewise.

In any case, X α Y β and X γ Y δ are in the same connected component of m .

We now describe the simplicial complexes that correspond to the S -degrees which are multiples of the glued degree.

Lemma 4.

Let S be the gluing of S 1 and S 2 , d S the glued degree, and d S { d } . Then C d A 1 C d A 2 if and only if d ( d ) { 0 } . Furthermore, the simplicial complex d has at least one connected component with elements in C d A 1 and C d A 2 .

Proof.

If there exist X α , Y β C d , then d = i = 1 r α i a i = i = 1 t β i b i S 1 S 2 G ( S 1 ) G ( S 2 ) = d . Hence, d d .

Conversely, let d = j d with j and j > 1 and let X γ X - Y γ Y I S be a glued binomial. It is easy to see that X j γ X , Y j γ Y C d and thus { X j γ X , X ( j - 1 ) γ X Y γ Y } and { X ( j - 1 ) γ X Y γ Y , Y j γ Y } belong to d .

The following lemma is a combinatorial version of [11, Lemma 9] and it is a necessary condition of Theorem 6.

Lemma 5.

Let S be the gluing of S 1 and S 2 and d S the glued degree. Then the elements of C d are pure monomials and d Betti ( S ) .

Proof.

The order S defined by m S m if m - m S is a partial order on S .

Assume that there exists a mixed monomial T C d . By Lemma 3, there exists a pure monomial Y b in C d such that { T , Y b } d (the proof is analogous if we consider X a with { T , X a } d ). Now take T 1 = gcd ( T , Y b ) - 1 T and Y b 1 = gcd ( T , Y b ) - 1 Y b . Both monomials are in C d , where d is equal to d minus the S -degree of gcd ( T , Y b ) . By Lemma 4, if C d A 1 , then d d , but since d S d this is not possible. So, if T 1 is a mixed monomial and C d A 1 = , then C d A 2 . If there exists a pure monomial in C d A 2 connected to a mixed monomial in C d , we perform the same process obtaining T 2 , Y b 2 C d ′′ with T 2 as a mixed monomial and d ′′ S d . 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 d ( i ) Betti ( S ) verifying that d ( i ) 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 m .

Theorem 6.

The semigroup S is the gluing of S 1 and S 2 if and only if the following conditions are fulfilled.

For all d Betti ( S ) , any connected component of d has at least a pure monomial.

There exists a unique d Betti ( S ) such that C d A 1 C d A 2 and the elements in C d are pure monomials.

For all d Betti ( S ) { d } with C d A 1 C d A 2 , d d .

Besides, the above d Betti ( S ) is the glued degree.

Proof.

If S is the gluing of S 1 and S 2 , the result is obtained from Lemmas 3, 4, and 5.

Conversely, by hypotheses 1 and 3, given that d Betti ( S ) { d } , the set ( I S 1 ) d is constructed from C d A 1 and ( I S 2 ) d from C d A 2 as in Construction 1. Analogously, if d Betti ( S ) , the set ( I S ) d is obtained from the union of ( I S 1 ) d , ( I S 2 ) d and the binomial X γ X - Y γ Y with X γ X C d A 1 and Y γ Y C d A 2 . Finally (14) m Betti ( S ) ( ( I S 1 ) m ( I S 2 ) m ) { X γ X - Y γ Y } is a generating set of I S and S is the gluing of S 1 and S 2 .

From Theorem 6 we obtain an equivalent property to Theorem  12 in  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 . The ideal I S is minimally generated by its indispensable binomials if and only if the following conditions are fulfilled.

The ideals I S 1 and I S 2 are minimally generated by their indispensable binomials.

The element X γ X - Y γ Y is an indispensable binomial of I S .

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 [10, Corollary 6], for all m Betti ( S ) the simplicial complex m has only two vertices. By Construction 1 d = { { X γ X } , { Y γ Y } } and by Theorem 6 for all d Betti ( S ) { d } the simplicial d is equal to d A 1 or d A 2 . In any case, X γ X - Y γ Y I S is an indispensable binomial, and I S 1 , I S 2 are generated by their indispensable binomials.

Conversely, suppose that I S is not generated by its indispensable binomials. Then, there exists d Betti ( S ) { d } such that d has more than two vertices in at least two different connected components. By hypothesis, there are not mixed monomials in d and thus

if d is equal to d A 1 (or d A 2 ), then I S 1 (or I S 2 ) is not generated by its indispensable binomials;

otherwise, C d A 1 C d A 2 and by Lemma 4, d = j d with j , therefore X ( j - 1 ) γ X Y γ Y C d which contradicts the hypothesis.

We conclude that I S is generated by its indispensable binomials.

The following example taken from  illustrates the above results.

Example 8.

Let S 2 be the semigroup generated by the set (15) { ( 13,0 ) , ( 5,8 ) , ( 2,11 ) , ( 0,13 ) , ( 4,4 ) , ( 6,6 ) , ( 7,7 ) , ( 9,9 ) } . In this case, Betti ( S ) is (16) { ( 15,15 ) , ( 14,14 ) , ( 12,12 ) , ( 18,18 ) , ( 10,55 ) , ( 15,24 ) , ( 13,52 ) , ( 13,13 ) } .

Using the appropriated notation for the indeterminates in the polynomial ring 𝕂 [ x 1 , , x 4 , y 1 , , y 4 ] ( x 1 , x 2 , x 3 , and x 4 for the first four generators of S and y 1 , y 2 , y 3 , y 4 for the others), the simplicial complexes associated with the elements in Betti ( S ) are those that appear in Figure 1. From Figure 1 and by using Theorem 6, the semigroup S is the gluing of ( 13,0 ) , ( 5,8 ) , ( 2,11 ) , ( 0,13 ) and ( 4,4 ) , ( 6,6 ) , ( 7,7 ) , ( 9,9 ) and the glued degree is ( 13,13 ) . From Corollary 7, the ideal I S is not generated by its indispensable binomials ( I S has only four indispensable binomials).

Nonconnected simplicial complexes associated with Betti ( S ) .

4. Generating Glued Semigroups

In this section, an algorithm to obtain examples of glued semigroups is given. Consider A 1 = { a 1 , , a r } and A 2 = { b 1 , , b t } as two minimal generator sets of the semigroups T 1 and T 2 and let L j = { ρ j i } i be a basis of ker T j with j = 1,2 . Assume that I T 1 and I T 2 are nontrivial proper ideals of their corresponding polynomial rings. Consider γ X and γ Y be two nonzero elements in r and t , respectively, (note that γ X ker T 1 and γ Y ker T 2 because these semigroups are reduced) and the integer matrix (17) A = ( L 1 0 0 L 2 γ X - γ Y ) . Let S be a semigroup such that ker S is the lattice generated by the rows of matrix A . This semigroup can be computed by using the Smith Normal Form (see [1, Chapter 3]). Denote by B 1 , B 2 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 necessary conditions to be a glued semigroup.

Proposition 9.

The semigroup S verifies G ( B 1 ) G ( B 2 ) = ( B 1 γ X ) = ( B 2 γ Y ) with d = B 1 γ X B 1 B 2 .

Proof.

Use that ker S has a basis as (4) and proceed as in the proof of the necessary condition of Proposition 2.

Because B 1 B 2 may not be a minimal generating set, this condition does not assure that S is a glued semigroup. For instance, taking the numerical semigroups T 1 = 3,5 , T 2 = 2,7 , and ( γ X , γ Y ) = ( 1,0 , 2,0 ) , the matrix obtained from formula (17) is (18) ( 5 - 3 0 0 0 0 7 - 2 1 0 - 2 0 ) , and B 1 B 2 = { 12,20,6 , 21 } is not a minimal generating set. The following result solves this issue.

Corollary 10.

The semigroup S is a glued semigroup if (19) i = 1 r γ X i > 1 , i = 1 t γ Y i > 1 .

Proof.

Suppose that the set of generators B 1 B 2 of S is nonminimal and thus one of its elements is a natural combination of the others. Assume that this element is the first of B 1 B 2 and then there exist λ 2 , , λ r + t such that B 1 ( 1 , - λ 2 , , - λ r ) = B 2 ( λ r + 1 , , λ r + t ) G ( B 1 ) G ( B 2 ) . By Proposition 9, there exists λ satisfying B 1 ( 1 , - λ 2 , , - λ r ) = B 2 ( λ r + 1 , , λ r + t ) = B 1 ( λ γ X ) . Since B 2 ( λ r + 1 , , λ r + t ) S , λ 0 and thus (20) ν = ( 1 - λ γ X 1 , - λ 2 - λ γ X 2 , , - λ r - λ γ X r 0 ) ker ( B 1 ) = ker T 1 , with the following cases.

If λ γ X 1 = 0 , then T 1 is not minimally generated which it is not possible by hypothesis.

If λ γ X 1 > 1 , then 0 > ν ker T 1 , but this is not possible because T 1 is a reduced semigroup.

If λ γ X 1 = 1 , then λ = γ X 1 = 1 and (21) ν = ( 0 , - λ 2 - γ X 2 , , - λ r - γ X r 0 ) ker T 1 .

If λ i + γ X i 0 for some i = 2 , , r , then T 1 is not a reduced semigroup. This implies that λ i = γ X i = 0 for all i = 2 , , r .

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 r or t , respectively. To avoid this situation, it is sufficient to take γ X and γ Y satisfying i = 1 r γ X i > 1 and i = 1 t γ Y i > 1 .

From the above result we obtain a characterization of glued semigroups: S is a glued semigroup if and only if ker S has a basis as (4) satisfies Condition (19).

Example 11.

Let T 1 = ( - 7,2 ) , ( 11,1 ) , ( 5,0 ) , ( 0,1 ) 2 and T 2 = 3,5 , 7 be two reduced affine semigroups. We compute their associated lattices (22) ker T 1 = ( 1,2 , - 3 , - 4 ) , ( 2 , - 1,5 , - 3 ) , ker T 2 = ( - 4,1 , 1 ) , ( - 7,0 , 3 ) . If we take γ X = ( 2,0 , 2,0 ) and γ Y = ( 1,2 , 1 ) , the matrix A is (23) ( 1 2 - 3 - 4 0 0 0 2 - 1 5 - 3 0 0 0 0 0 0 0 - 4 1 1 0 0 0 0 - 7 0 3 2 0 2 0 - 1 - 2 - 1 ) and the semigroup S 4 × 2 is generated by (24) { ( 1 , - 5,35 ) , ( 3,12 , - 55 ) , ( 1,5 , - 25 ) , ( 0,1 , 0 ) B 1 , ( 2,0 , 3 ) , ( 2,0 , 5 ) , ( 2,0 , 7 ) B 2 } . The semigroup S is the gluing of the semigroups B 1 and B 2 and ker S is generated by the rows of the above matrix. The ideal I S [ x 1 , , x 4 , y 1 , , y 3 ] is generated (see  to compute I S when S has torsion) by (25) { x 1 2 x 3 2 - y 1 5 y 2    g l u e d    b i n o m i a l    x 1 x 3 8 x 4 - x 2 3 , x 1 x 2 2 - x 3 3 x 4 4 , x 1 2 x 3 5 - x 2 x 4 3 , x 1 3 x 2 x 3 2 - x 7 7 , y 1 y 3 - y 2 2 , y 1 3 y 2 - y 3 2 , y 1 4 - y 2 y 3 , x 1 2 x 3 2 - y 1 5 y 2 glued    binomial } ; then S is really a glued semigroup.

4.1. Generating Affine Glued Semigroups

From Example 11 it be can deduced that the semigroup S is not necessarily torsion-free. In general, a semigroup T 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 ker T 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 S being torsion-free.

Take L 1 and L 2 as the matrices whose rows form a basis of ker T 1 and ker T 2 , respectively, and let P 1 , P 2 , Q 1 , and Q 2 be 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 (26) ( D 1 0 0 D 2 γ X γ Y ) = ( P 1 0 0 0 P 2 0 0 0 1 ) ( L 1 0 0 L 2 γ X - γ Y ) = : A ( Q 1 0 0 Q 2 ) , where γ X = γ X Q 1 and γ Y = - γ Y Q 2 . Let s 1 and s 2 be the numbers of zero-columns of D 1 and D 2 ( s 1 , s 2 > 0 because T 1 and T 2 are reduced, see [1, Theorem 3.14]).

Lemma 12.

The semigroup S is an affine semigroup if and only if (27) gcd ( { γ X i } i = r - s 1 r { γ Y i } i = t - s 2 t ) = 1 .

Proof.

With the conditions fulfilled by T 1 , T 2 , and ( γ X , γ Y ) , the necessary and sufficient condition for the invariant factors of A to be all equal to one is gcd ( { γ X i } i = r - s 1 r { γ Y i } i = t - s 2 t ) = 1 .

The following corollary gives the explicit conditions that γ X and γ Y must satisfy to construct an affine semigroup.

Corollary 13.

The semigroup S is an affine glued semigroup if and only if

T 1 and T 2 are two affine semigroups;

( γ X , γ Y ) r + t ;

i = 1 r γ X i , i = 1 t γ Y i > 1 ;

there exist f r - s 1 , , f r , g t - s 2 , , g t such that (28) ( f r - s 1 , , f r ) · ( γ X ( r - s 1 ) , , γ X r ) + ( g t - s 2 , , g t ) · ( γ Y ( t - s 2 ) , , γ Y t ) = 1 .

Proof.

It is trivial by the given construction, Corollary 10 and Lemma 12.

Therefore, to obtain 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 14.

Let T 1 and T 2 be the semigroups of Example 11. We compute two elements γ X = ( a 1 , a 2 , a 3 , a 4 ) and γ Y = ( b 1 , b 2 , b 3 ) 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 L 1 and L 2 : (29) ( 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 a 1 a 1 - 2 a 2 - a 3 - 7 a 1 + 11 a 2 + 5 a 3 2 a 1 + a 2 + a 4 - b 1 b 1 + 2 b 2 + 3 b 3 - 3 b 1 - 5 b 2 - 7 b 3 ) = ( 1 0 0 0 0 2 - 1 0 0 0 0 0 - 2 1 0 0 0 7 - 4 0 0 0 0 0 1 ) ( 1 2 - 3 - 4 0 0 0 2 - 1 5 - 3 0 0 0 0 0 0 0 - 4 1 1 0 0 0 0 - 7 0 3 a 1 a 2 a 3 a 4 - b 1 - b 2 - b 3 ) ( 1 1 - 7 2 0 0 0 0 - 2 11 1 0 0 0 0 - 1 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 - 1 3 0 0 0 0 0 - 2 5 0 0 0 0 0 - 3 7 ) .

Second, by Corollary 13, we must find a solution to the system: (30) a 1 + a 2 + a 3 + a 4 > 1 , b 1 + b 2 + b 3 > 1 , f 1 , f 2 , g 1 , f 1 ( - 7 a 1 + 11 a 2 + 5 a 3 ) + f 2 ( 2 a 1 + a 2 + a 4 ) + g 1 ( - 3 b 1 - 5 b 2 - 7 b 3 ) = 1 , with a 1 , a 2 , a 3 , a 4 , b 1 , b 2 , b 3 . Such solution is computed (in less than a second) using FindInstance of Wolfram Mathematica (see ): (31) F i n d I n s t a n c e [ ( - 7 a 1 + 11 a 2 + 5 a 3 ) * f 1 + ( 2 a 1 + a 2 + a 4 ) * f 2 + ( - 3 b 1 - 5 b 2 - 7 b 3 ) * g 1 = = 1 && a 1 + a 2 + a 3 + a 4 > 1 && b 1 + b 2 + b 3 > 1 && a 1 0 && a 2 0 && a 3 0 && a 4 0 && b 1 0 && b 2 0 && b 3 0 , { a 1 , a 2 , a 3 , a 4 , b 1 , b 2 , b 3 , f 1 , f 2 , g 1 } , ( - 7 a 1 + 11 a 2 + 5 a 3 ) * f 1 Integers ] { { a 1 0 , a 2 0 , a 3 3 , a 4 0 , b 1 1 , b 2 1 , b 3 0 , f 1 1 , f 2 0 , g 1 0 } } .

We now take γ X = ( 0,0 , 3,0 ) and γ Y = ( 1,1 , 0 ) , and construct the matrix (32) A = ( 1 2 - 3 - 4 0 0 0 2 - 1 5 - 3 0 0 0 0 0 0 0 - 4 1 1 0 0 0 0 - 7 0 3 0 0 3 0 - 1 - 1 0 ) . We have the affine semigroup S 2 which is minimally generated by (33) { ( 2 , - 56 ) , ( 1,88 ) , ( 0,40 ) , ( 1,0 ) B 1 , ( 0,45 ) , ( 0,75 ) , ( 0,105 ) B 2 } satisfying that ker S is generated by the rows of A and it is the result of gluing the semigroups B 1 and B 2 . The ideal I S is generated by (34) { x 3 3 - y 1 y 2    g l u e d    b i n o m i a l    x 3 3 - y 1 y 2    glued    binomial    x 1 x 3 8 x 4 - x 2 3 , x 1 x 2 2 - x 3 3 x 4 4 , x 1 2 x 3 5 - x 2 x 4 3 , x 1 3 x 2 x 3 2 - x 4 7 , y 1 y 3 - y 2 2 , y 1 3 y 2 - y 3 2 , y 1 4 - y 2 y 3 , x 3 3 - y 1 y 2 glued    binomial } ; therefore, S 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.

Acknowledgments

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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