On the ideal of some sumset semigroups
aa r X i v : . [ m a t h . N T ] F e b On the ideal of some sumset semigroups
J. I. Garc´ıa-Garc´ıa ∗ D. Mar´ın-Arag´on † A. Vigneron-Tenorio ‡ Abstract
A sumset semigroup is a non-cancellative commutative monoid ob-tained by sumset of finite non-negative integer sets. In this work, analgorithm for computing the ideal associated to some sumset semi-groups is provided. This approach links computational commutativealgebra with additive number theory.
Key words: h -fold sumset, non-cancellative semigroup, power monoid, semi-group ideal, sumset.2020 Mathematics Subject Classification:
Introduction
Additive number theory is the subfield of number theory concerning thestudy of subsets of integers and their behaviour under addition. More ab-stractly, the field of additive number theory includes the study of abeliangroups and commutative semigroups with an operation of addition. Twoprincipal objects of study are the sumset of two subsets A and B of ele-ments form an abelian group G , A + B = { a + b | a ∈ A, b ∈ B } , and todetermine the structure and properties of the h -fold sumset hA when the set A is known. In an inverse problem, we start with the sumset hA and try todeduce information about the underlying set A . A more modern referencefor inverse problems can be found in [9, Chapter 5]. There is a beautiful andstraightforward solution of the direct problem of describing the structure of ∗ Departamento de Matem´aticas/INDESS (Instituto Universitario para el DesarrolloSocial Sostenible), Universidad de C´adiz, E-11510 Puerto Real (C´adiz, Spain). E-mail:[email protected]. † Departamento de Matem´aticas, Universidad de C´adiz, E-11510 Puerto Real (C´adiz,Spain). E-mail: [email protected]. ‡ Departamento de Matem´aticas/INDESS (Instituto Universitario para el DesarrolloSocial Sostenible), Universidad de C´adiz, E-11406 Jerez de la Frontera (C´adiz, Spain).E-mail: [email protected]. h -fold sumset hA for any finite set A of integers and for all sufficientlylarge h (see [6, Theorem 1.1]). This result has implications on the study ofWeierstrass semigroups such as is shown in [2].Here, we consider the commutative semigroup whose elements are the fi-nite subsets of N , denoted by (FS( N ) , +) with + the operation defined as be-fore. This semigroup is the power monoid of N (see [3], [4] and the referencestherein). A sumset semigroup is a semigroup generated by a finite numberof elements of FS( N ). It is well-known that finitely generated semigroupsare finitely presented (see [5]). That is, there exists p ∈ N and a congruence σ in N p × N p such that the semigroup S is isomorphic to N p /σ . Equiva-lently, there exists a binomial ideal I S in the polynomial ring k [ x , . . . , x p ]such that S is isomorphic to the set of monomials in k [ x , . . . , x p ] /I S withthe product operation. The presentation of S (or a system of generators ofthe semigroup ideal I S ) provides us with a way to obtain the expressions ofan element of the semigroup in terms of its generators. This can be donewith Gr¨obner bases and related techniques. For instance, we can check ifthe h -fold of any A ∈ S can be expressed as sumset of others elements.By using these techniques, we can build a bridge between computationalcommutative algebra and additive number theory.In this work, we show some properties of the semigroup FS( N ) and givethe ideal of some types of sumset semigroups. The work is organised asfollows. In Section 1, we make available some definitions and results onGr¨obner basis. Section 2 is devoted to present the semigroup FS( N ), its op-erations and properties. In Section 3 and 4, by using algebraic commutativealgebra tools, the ideal of some families of sumset semigroups is studied.This allows us to introduce Algorithm 1 and to provide some examples. For a field k , and a set of indeterminates { x , . . . , x t } , the polynomial ring k [ x , . . . , x t ] (also denoted by k [ X ]) is the set of polynomials in { x , . . . , x t } with coefficients in k ; that is, the set { P mi =1 a i x α · · · x α t t | m ∈ N , a i ∈ k , α = ( α , . . . , α t ) ∈ N t } . We denote by X α the monomial x α · · · x α t t .In this work, some results use Gr¨obner basis theory and the EliminationTheorem. The readers can find the necessary results about them in [1, § § ≺ on k [ X ] is a multiplicative total order onthe set of monomials if for each two monomials X α , X β such that X α ≺ X β ,then X α X γ ≺ X β X γ for every monomial X γ .For a fixed monomial order ≺ on k [ X ], In ≺ ( I ) denotes the set of leading2erms of non-zero elements of I , and h In ≺ ( I ) i the monomial ideal generatedby In ≺ ( I ). A subset G of I is a Gr¨obner basis of I if h In ≺ ( I ) i = h{ In ≺ ( g ) | g ∈ G }i , where In ≺ ( g ) is the leading term of g . An algorithm to computea Gr¨obner basis for I is showed in [1, Chapter 2, § f and g , their S -polinomial is defined by S ( f, g ) = aX α In ≺ ( f ) f − aX α In ≺ ( g ) g , where aX α is least common multiple of In ≺ ( f ) and In ≺ ( g ).Besides, for k non-zero polynomials f , . . . , f k , we say that S ( f i , f j ) = P kl =1 g l f l has an lcm representation if the least common multiple of themonomial leaders of f i and f j is bigger than In ≺ ( g l f l ) (respect ≺ ) whenever g l f l = 0. Thus, we obtain another equivalent definition of Gr¨obner basis. Theorem 1. [1, Chapter 2, §
9, Theorem 6] A basis { f , . . . f k } of an ideal I is a Gr¨obner basis if and only if for every i = j , the S -polynomial S ( f i , f j ) has an lcm representation. Above Theorem allows us to prove the next result. We use this lemmain the following sections.
Lemma 2.
Let I ⊂ k [ X, Y ] and J ⊂ k [ Z, X, Y ] be two binomial ideals with J generated by G = { z − M , . . . , z t − M t } , where each M i is a monomialin k [ X, Y ] . Fix ≺ a monomial order on k [ Z, X, Y ] such that x j , y k ≺ z i , forevery x j , y k and z i . Then, the union of G and a Gr¨obner basis of I respect ≺ is a Gr¨obner basis of I + J respect ≺ . Moreover, if a binomial L − T belongs to ( I + J ) ∩ k [ X, Y ] , then L − T ∈ I .Proof. Note that G is a Gr¨obner basis of J respect ≺ , and consider G ′ = { g , . . . , g h } ⊂ k [ X, Y ] a Gr¨obner basis of I . Thus, S ( f, f ′ ) and S ( g, g ′ ) havean lcm representation for every f, f ′ ∈ G and g, g ′ ∈ G ′ . Let L − T be abinomial in G ′ , z i − M i ∈ G , and assume L ≻ T . Hence, S ( z i − M i , L − T ) = L ( z i − M i ) − z i ( L − T ) = T ( z i − M i ) − M i ( L − T ), that is to say, S ( z i − M i , L − T )has an lcm representation. Therefore, G ∪ G ′ is a Gr¨obner basis of I + J .Consider any L − T ∈ I + J with L − T ∈ k [ X, Y ]. By [1, Chapter 2, §
3, Theorem 3] and [1, Chapter 2, §
6, Corollary 2], we have that L − T = P hi =1 f i g i , with In ≺ ( L − T ) (cid:23) In ≺ ( f i g i ), for i = 1 , . . . , h . Hence, every f i belongs to k [ X, Y ].A method for computing the ideal I ∩ k [ x l +1 , . . . , x t ] (for t > l ≥
1) iscalled
The Elimination Theorem . Theorem 3. [1, Chapter 3] Let I ⊂ k [ x , . . . , x t ] be an ideal and let G bea Gr¨obner basis of I with respect to lex order where x > x > · · · > x t .Then, for every ≤ l ≤ t , the set G ∩ k [ x l +1 , . . . , x t ] is a generating set ofthe ideal I ∩ k [ x l +1 , . . . , x t ] . Furthermore, G ∩ k [ x l +1 , . . . , x t ] is a Gr¨obnerbasis of I ∩ k [ x l +1 , . . . , x t ] . A = { a , . . . , a t } ⊂ S such that S = h A i := { α a + · · · + α t a t | α , . . . , α t ∈ N } ( αa denotes P αi =1 a ). For a field k , S has associated the binomial ideal in k [ x , . . . , x t ], I S = Dn X α − X β | t X i =1 α i a i = t X i =1 β i a i oE . This ideal is usually called the semigroup ideal of S , and it has an importantrole in studying some properties of the semigroup. Note that I S codifies therelationships between the elements of S .When the semigroup S is a subset of N such that N \ S is finite, S iscalled a numerical semigroup, and it is finitely generated. In [7], the authorsintroduce some algorithms for computing the ideals of numerical semigroups. Let FS( N ) be the set whose elements are the finite nonempty subsets of N . Recall that on FS( N ), the binary operation + is defined as A + B = { a + b | a ∈ A, b ∈ B } for all A, B ∈ FS( N ). The pair (FS( N ) , +) isa commutative monoid with identity element equal to { } . Every finitelygenerated submonoid of (FS( N ) , +) is called a sumset semigroup. If A ∈ FS( N ) and α ∈ N , denote by α ⊗ A to the sumset P αi =1 A .The monoid FS( N ) satisfies the following interesting properties: • Since { , } + { , , } = { , , } + { , , } , it is non-cancellative. • It has no units, that is, there are no elements
A, B ∈ FS( N ) such that A + B = { } . • Since 2 ⊗ { , , , } = 2 ⊗ { , , , , } , it is not torsion free.The operation ⊗ has good properties showed in the following lemma. Lemma 4.
Let A , B be in FS( N ) and α, β ∈ N then:1. α ⊗ ( β ⊗ A ) = ( αβ ) ⊗ A .2. α ⊗ ( A + B ) = α ⊗ A + α ⊗ B . Note that for the sumset semigroup generated by { A , . . . , A t } , its asso-ciated ideal is I S = Dn X α − X β | t X i =1 α i ⊗ A i = t X i =1 β i ⊗ A i oE ⊂ k [ x , . . . , x t ] . A = { a < · · · < a n } ∈ FS( N ), then A = { a } + { , a − a , . . . , a n − a } (denote { , a − a , . . . , a n − a } by ˜ A ). The semigroup S = h A , . . . , A t i , with A i = { a i < · · · < a it i } , is a submonoid of h{ a } , . . . , { a t } , ˜ A , . . . , ˜ A t i .Trivially, the sumset semigroup h{ a } , . . . , { a t }i is isomorphic to the semi-group h a , . . . , a t i , thus I h a ,...,a s i = I h{ a } ,..., { a t }i . Proposition 5.
For every ˜ S = (cid:8) { a } , . . . , { a s } , ˜ A , . . . , ˜ A t (cid:9) , with a i = 0 and min ˜ A i = 0 , we have I ˜ S = I h a ,...,a s i + I h ˜ A ,..., ˜ A t i , where I h a ,...,a s i ⊂ k [ x , . . . , x s ] and I h ˜ A ,..., ˜ A t i ⊂ k [ y , . . . , y t ] .Proof. Trivially, I h a ,...,a s i , I h ˜ A ,..., ˜ A t i ⊂ I ˜ S .Let X α Y β − X γ Y δ ∈ I ˜ S , then P si =1 α i ⊗ { a i } + P ti =1 β i ⊗ ˜ A i = P si =1 γ i ⊗{ a i } + P ti =1 δ i ⊗ ˜ A i . Since min ˜ A i = 0, we then have P si =1 α i ⊗ { a i } = P si =1 γ i ⊗ { a i } , and P ti =1 β i ⊗ ˜ A i = P ti =1 δ i ⊗ ˜ A i . That is to say, X α − X γ ∈ I h a ,...,a s i , and Y β − Y δ ∈ I h ˜ A ,..., ˜ A t i . Note that X α Y β − X γ Y δ = Y β ( X α − X γ ) + X γ ( Y β − Y δ ) ∈ I h a ,...,a s i + I h ˜ A ,..., ˜ A t i ⊂ k [ x , . . . x s , y . . . , y t ].There exist algorithms for computing I h a ,...,a s i , since the semigroup h a , . . . , a s i is isomorphic to a numerical semigroup. Thus, to compute apresentation of ˜ S we need an algorithm to calculate I h ˜ A ,..., ˜ A t i . Next sec-tions provide some algorithms for computing the ideals of some families ofsumset semigroups. In this section we give explicitly the ideals associated to the sumset semi-groups generated by the elements { , k a } and { , k b } , where a < b aretwo positive co-prime integers, and k ∈ N \ { } . These semigroups are keyto provide an algorithm to compute the semigroup ideals of more types ofsumset semigroups.Fixed a < b two positive co-prime integers, k ∈ N \ { } , consider thesemigroup S = h k a , k b i , and the sumset semigroup e S minimally generatedby A = { , k a } and B = { , k b } . We prove that I e S ⊂ k [ x, y ] is a principalideal providing its generator. Note that I e S ⊂ I S = h x b − y a i . Lemma 6.
Set x α y β − x γ y δ ∈ I e S \{ } . Then, α = γ , β = δ , and α · β · γ · δ ≥ .Proof. Note that, since f = x α y β − x γ y δ ∈ I e S , α ⊗ { , k a } + β ⊗ { , k b } = γ ⊗ { , k a } + δ ⊗ { , k b } .Suppose α = γ , we then have αk a + βk b = max (cid:8) α ⊗ { , k a } + β ⊗{ , k b } (cid:9) = max (cid:8) α ⊗ { , k a } + δ ⊗ { , k b } (cid:9) = αk a + δk b , and β = δ . Since5 = 0, this is not possible and therefore α = γ . Analogously, it can beproved that β = δ .Assumed α = 0. Then, k b = min (cid:8) β ⊗ { , k b } \ { } (cid:9) = min (cid:8) ( γ ⊗{ , k a } + δ ⊗ { , k b } ) \ { } (cid:9) . If γ is non-zero, then k b = min (cid:8) ( γ ⊗ { , k a } + δ ⊗ { , k b } ) \ { } (cid:9) = k a . Therefore, the integers γ , β , and δ are zero and f = 0. In a similar way, α, β, γ, δ ≥ α > γ ≥
1, and x α y β − x γ y δ ∈ I e S \ { } . Since I e S ⊂ h x b − y a i , α ≥ b and δ ≥ a . Lemma 7. If α > γ , then δ > β . Besides, there exists a positive integer n such that α = n b + γ and δ = n a + β .Proof. Since x α y β − x γ y δ ∈ I e S , α ⊗ { , k a } + β ⊗ { , k b } = γ ⊗ { , k a } + δ ⊗ { , k b } , and αk a + βk b = max (cid:8) α ⊗ { , k a } + β ⊗ { , k b } (cid:9) = max (cid:8) γ ⊗{ , k a } + δ ⊗ { , k b } (cid:9) = γk a + δk b . So, ( δ − β ) k b = ( α − γ ) k a > α − γ ) / ( δ − β ) = b / a . Since gcd( a , b ) = 1, there exist twopositive integers n and m , such that α = n b + γ and δ = m a + β . From theequality αk a + βk b = γk a + δk b , we deduce that n = m . Lemma 8.
Let x α y β − x γ y δ ∈ I e S \ { } . Then, γ ≥ b − , β ≥ a − , andthere is a positive integer n ∈ N such that x α y β − x γ y δ = x γ y β ( x n b − y n a ) .Proof. Assume γ < b −
1. Take ( γ + 1) k a + βk b in α ⊗ { , k a } + β ⊗ { , k b } (recall α > γ ). For that element, there exist two integers i ∈ [0 , γ ], and j ∈ [0 , δ ] such that ( γ + 1) k a + βk b = ( γ − i ) k a + ( δ − j ) k b ∈ γ { , k a } + δ { , k b } ,hence (1 + i ) k a = ( δ − β − j ) k b . Since gcd( a , b ) = 1, i + 1 ≥ b , and i ≥ b − > γ , but i ≤ γ which is a contradiction. Analogously, the fact β ≥ a − n ∈ N \ { } such that x α y β − x γ y δ = x n b + γ y β − x γ y n a + β = x γ y β ( x n b − y n a ) . Theorem 9.
Let ≤ a < b be two co-prime integers, k ∈ N \ { } , and e S bethe sumset semigroup h{ , k a } , { , k b }i . The ideal I e S ⊂ k [ x, y ] is principaland it is generated by x b − y a − ( x b − y a ) .Proof. Observe that x b − y a − ( x b − y a ) = x b − y a − − x b − y a − . To provethis theorem, we describe explicitly the sets A = (2 b − ⊗ { , k a } + ( a − ⊗ { , k b } and B = ( b − ⊗ { , k a } + (2 a − ⊗ { , k b } associated to themonomials x b − y a − and x b − y a − (respectively), to achieve that A = B .6ote that the first set A = (2 b − ⊗ { , k a } + ( a − ⊗ { , k b } is equalto { , k a , . . . , (2 b − k a } + { , k b , . . . , ( a − k b } == k (cid:16) { , a , . . . , (2 b − a } ∪ { , b , . . . , ( a − b }∪{ a + b , . . . , a + ( a − b } ∪ { a + b , . . . , a + ( a − b } ∪ · · ·∪ { (2 b − a + b , . . . , (2 b − a + ( a − b } (cid:17) = k (cid:16) { , a , . . . , ( b − a , ba , ( b + 1) a , . . . , (2 b − a }∪{ , b , . . . , ( a − b } ∪ { a + b , . . . , a + ( a − b } ∪ · · ·∪ { ( b − a + b , . . . , ( b − a + ( a − b }∪{ ba + b , . . . , ba + ( a − b }∪{ ( b + 1) a + b , . . . , ( b + 1) a + ( a − b } ∪ · · ·∪ { (2 b − a + b , . . . , (2 b − a + ( a − b } (cid:17) . We denote C = { , a , . . . , ( b − a } , C = { , b , . . . , ( a − b } , C = ∪ b − i =1 { i a + b , . . . , i a + ( a − b } , C = { ab } ∪ { ba + b , . . . , ba + ( a − b } , C = { ( b + 1) a , . . . , (2 b − a } , and C = ∪ b − i = b +1 { i a + b , . . . , i a + ( a − b } . The set A is the union ∪ i =1 kC i .The set B = ( b − ⊗ { , k a } + (2 a − ⊗ { , k b } is { , k a , . . . , ( b − k a } + { , k b , . . . , (2 a − k b } == k (cid:16) { , a , . . . , ( b − a } ∪ { , b , . . . , ( a − b , ab , . . . , (2 a − b }∪{ a + b , . . . , a + ( a − b , a + ab , . . . , a + (2 a − b } ∪ · · ·∪{ ( b − a + b , . . . , ( b − a +( a − b , ( b − a + ab , . . . , ( b − a +(2 a − b } (cid:17) = ∪ i =1 kC i . Thus, A = B , and x b − y a − ( x b − y a ) ∈ I S .To finish the proof, we use Lemma 8. If n = 1, then x α y β − x γ y δ = x γ − b +1 y β − a +1 x b − y a − ( x b − y a ). In case n >
1, by factorizing the binomial x n b − y n a , we obtain x α y β − x γ y δ = x γ − b +1 y β − a +1 ( x ( n − b + x ( n − b y + · · · + xy ( n − a + y ( n − a ) x b − y a − ( x b − y a ) . In any case, I e S = (cid:10) x b − y a − ( x b − y a ) (cid:11) .7 orollary 10. Let a , . . . , a s be positive integers, and S be the sumset semi-group generated by (cid:8) { a } , . . . , { a s } , { , k a } , { , k b } (cid:9) . Then, I S = I h a ,...,a s i + (cid:10) x b − y a − ( x b − y a ) (cid:11) ⊂ k [ x , . . . , x s , x, y ] . Proof.
From Proposition 5 and Theorem 9, we obtain the result.
The aim of this section is to determinate an algorithm for computing theideal associated to some families of sumset semigroups. As in previous sec-tion, we consider a < b two positive co-prime integers, k ∈ N \ { } , thesemigroup S = h k a , k b i , and the sumset semigroup e S = h{ , k a } , { , k b }i .For any two non-negative integers n and m , A nm denotes (cid:8) αk a + βk b | α ∈ { , . . . , n } , β ∈ { , . . . , m } (cid:9) = n ⊗ k { , a } + m ⊗ k { , b } . Theorem 11.
Let { ( n i , m i ) | n i , m i ∈ N , n i + m i > , i = 1 , . . . , t } bea nonempty subset of N , b , . . . , b p , a , . . . , a s ∈ N \ { } with s ≤ t , andconsider S the sumset semigroup generated by (cid:8) { b } , . . . , { b p } , { a } + A n m , . . . , { a s } + A n s m s , A n s +1 m s +1 , . . . , A n t m t (cid:9) , and the sumset semigroup S ′ = (cid:10) { b } , . . . , { b p } , { a } , . . . , { a s } , { , k a } , { , k b } (cid:11) .Then, I S ∈ k [ x , . . . , x p , z , . . . , z t ] is (cid:16) I S ′ + (cid:10) z − w x n y m , . . . , z s − w s x n s y m s , z s +1 − x n s +1 y m s +1 , . . . , z t − x n t y m t (cid:11)(cid:17)\ k [ x , . . . , x p , z , . . . , z t ] , with I S ′ ⊂ k [ x , . . . , x p , w , . . . , w s , x, y ] .Proof. Denote J to the ideal (cid:10) z − w x n y m , . . . , z s − w s x n s y m s , z s +1 − x n s +1 y m s +1 , . . . , z t − x n t y m t (cid:11) . Any monomials Z β and Z δ can be rewritten as follow. Denote by f i = Q tk = i z β j i , ˆ f i = Q tk = i z δ j i , g ji = Q jk = i ( w k x n k y m k ) β k , ˆ g ji = Q jk = i ( w k x n k y m k ) δ k , h ji = Q jk = i ( x n k y m k ) β k , ˆ h ji = Q jk = i ( x n k y m k ) δ k , and suppose β = δ = 0 wehave that Z β = z β · · · z β t t = s − X i =0 f i +2 g i (cid:0) z β i +1 i +1 − g i +1 i +1 (cid:1) + t X i = s +1 f i +1 g s h i − s +1 (cid:0) z β i i − h ii (cid:1) + g s h ts +1 , and Z δ = z δ · · · z δ t t = s − X i =0 ˆ f i +2 ˆ g i (cid:0) z δ i +1 i +1 − ˆ g i +1 i +1 (cid:1) + t X i = s +1 ˆ f i +1 ˆ g s ˆ h i − s +1 (cid:0) z δ i i − ˆ h ii (cid:1) +ˆ g s ˆ h ts +1 . F = X α g s h ts +1 − X γ ˆ g s ˆ h ts +1 .Since any binomial u n − v n is equal to ( u − v )( u n − + u n − v + · · · + uv n − + v n − ), the binomials z ni − ( w i x n i y m i ) n and z nj − ( x n j y m j ) n belong to J for every non-negative integer n , and all i = 1 , . . . , s , and j = s + 1 , . . . , t .Let X α Z β − X γ Z δ be a binomial belonging to I S ′ + J , so F = X α Z β − X γ Z δ + q ∈ I S ′ + J, with q ∈ J . By Lemma 2, we have F ∈ I S ′ , thence p X i =1 α i ⊗{ b i } + s X i =1 β i ⊗{ a i } + t X i =1 (cid:16) β i ⊗ (cid:0) n i ⊗{ , k a } (cid:1) + β i ⊗ (cid:0) m i ⊗{ , k b } (cid:1)(cid:17) = p X i =1 γ i ⊗ { b i } + s X i =1 δ i ⊗ { a i } + t X i =1 (cid:16) δ i ⊗ (cid:0) n i ⊗ { , k a } (cid:1) + δ i ⊗ (cid:0) m i ⊗ { , k b } (cid:1)(cid:17) . Therefore, p X i =1 α i ⊗ { b i } + s X i =1 β i ⊗ (cid:0) { a i } + A n i m i (cid:1) + t X i = s +1 β i A n i m i = p X i =1 γ i ⊗ { b i } + s X i =1 δ i ⊗ (cid:0) { a i } + A n i m i (cid:1) + t X i =1 δ i A n i m i , and X α Z β − X γ Z δ ∈ I S .Analogously, if X α Z β − X γ Z δ ∈ I S , then F ∈ I S ′ , and X α Z β − X γ Z δ ∈ I S ′ + J . This completes the proof.The above proof can also be done by using [8, Proposition 4]. In ourproof, we employ the language of polynomials, ideals, and Gr¨obner basis,avoiding congruences.From Theorem 11, we obtain an algorithm (Algorithm 1) for comput-ing the ideal of the sumset semigroup generated by (cid:8) { b } , . . . , { b p } , { a } +9 n m , . . . , { a s } + A n s m s , A n s +1 m s +1 , . . . , A n t m t (cid:9) . Algorithm 1:
Computation of I S . Data: (cid:8) { b } , . . . , { b p } , { a } + A n m , . . . , { a s } + A n s m s , A n s +1 m s +1 , . . . , A n t m t (cid:9) , the generating set of S . Result: G , a generating set of the semigroup ideal of S . begin S ′ ← (cid:8) { b } , . . . , { b p } , { a } , . . . , { a s } , { , k a } , { , k b } (cid:9) ; S ← (cid:8) { b } , . . . , { b p } , { a } , . . . , { a s } (cid:9) ; G ← a generating set of I h S i ⊂ k [ X, W ]; G ′ ← G ⊔ { x b − y a − ( x b − y a ) } ⊂ k [ X, W, x, y ], it is agenerating set of I h S ′ i (Corollary 10); G ← G ′ ⊔ { z − w x n y m , . . . , z s − w s x n s y m s , z s +1 − x n s +1 y m s +1 , . . . , z t − x n t y m t } ⊂ k [ X, W, x, y, Z ] ; G ← a Gr¨obner basis of G respect to a monomial order with x, y, w q > x i and x, y, w q > z j for every i = 1 , . . . , p , j = 1 , . . . , t , and q = 1 , . . . , s ; G ← { X α Z β − X γ Z δ | X α Z β − X γ Z δ ∈ G } , it is a generatingset (Gr¨obner basis) of I S ; return G ;We show how this algorithm works with an example. Example . Let S be the sumset semigroup generated by (cid:8) { } , { } , { , } , { , , } , { , , , } (cid:9) . Then, from the first steps of Algorithm 1, • S ′ = (cid:8) { } , { } , { } , { } , { , } , { , } (cid:9) , • S = (cid:8) { } , { } , { } , { } (cid:9) .If we compute a generating set of the ideal of S , then we get the followingone, G = (cid:8) w − w , w x − w , w x − w , w x − w , x − w , w x − w x , w x − w x , x x − w , x − w (cid:9) . Therefore, G ′ = G ∪ (cid:8) x ( x − y ) (cid:9) and G = G ′ ∪ (cid:8) z − w y, z − w x , z − x (cid:9) . Now, we compute a Gr¨obner basis of G respect to the lex order where10 > y > w i > x j > z k for all i, j, and k , and we obtain G = (cid:8) z z − z z , z z − z z , x z z − z z , x z z − z z ,x z z − z z , x z z − z , x z z − z z , x z z − z z ,x z z − z z , x z z − z z , x z z − z z , x z z − z z ,x z − z z , x z z − x z z , x z − x z z , x z z − z z ,x z z − z , x x z − z z , x z z − x z z , x z z − x z ,x x z − z z , x z z − x z z , x z z − x z , x x z − z ,x − x , w − x x , w − x , yz − x x z , yz z − x x z ,yz z − x z , yx z z − x z z , yx z z − z z , yx − x z ,yx x z − z z , yx x z − z , yx − z , y z − x x z , y z z − x z ,y z − z , xz − y z , xz − x x z , xz − x z , xx z − x z z ,xx x z − yz , xy − z , x z − yz , x x − x z , x x x − z , x − z (cid:9) . Finally, the output of the algorithm is (cid:8) z z − z z , z z − z z , x z z − z z , x z z − z z ,x z z − z z , x z z − z , x z z − z z , x z z − z z ,x z z − z z , x z z − z z , x z z − z z , x z z − z z ,x z − z z , x z z − x z z , x z − x z z , x z z − z z , x z z − z ,x x z − z z , x z z − x z z , x z z − x z , x x z − z z ,x z z − x z z , x z z − x z , x x z − z , x − x (cid:9) . Since the binomial x z z − x z z ∈ I S , { } + { , , } + { , , , } = { } + { , } + { , , , } , but { } + { , , } 6 = { } + { , } , the semigroup S is not cancellative.The last example introduces an algorithm to obtain an expression of aninteger set in function of others, if possible. In particular, the i -fold sumsetof a set is studied. Example . We now use the above presentation of S to check whether theelement i ⊗ { , , } can be expressed in terms of the others generators ofthe semigroup S . We compute the Gr¨obner basis with respect to the ordergiven by the matrix A = , G A = (cid:8) x − x , x x z − z z , x z − x z z , z z − x x z ,x z z − x z z , x z z − x z z , z z − x z , x z − x z z ,x z z − x z z , x z − x z z , z − x x z (cid:9) . In Table 1, we show some elements z i that reduced with respect the basis G A are expressed only by using the variables x , x , z and z , and the expressionof i ⊗{ , , } in terms of the elements of the set (cid:8) { } , { } , { , } , { , , , } (cid:9) . i Reduction of z i i ⊗ { , , } =3 x x z ⊗ { } + 3 ⊗ { } + 2 ⊗ { , , , } x x z z ⊗ { } + 4 ⊗ { } + 1 ⊗ { , , } + 2 ⊗ { , , , } x x z z ⊗ { } + 5 ⊗ { } + 2 ⊗ { , , } + 2 ⊗ { , , , } x z z ⊗ { } + 3 ⊗ { , } + 2 ⊗ { , , , } x x z z ⊗ { } + 4 ⊗ { } + 4 ⊗ { , , } + 2 ⊗ { , , , } x x z z ⊗ { } + 5 ⊗ { } + 5 ⊗ { , , } + 2 ⊗ { , , , } Table 1: Expressions of some elements i ⊗ { , , } in terms of the subsetof generators (cid:8) { } , { } , { , } , { , , , } (cid:9) .In general, the reduction of z i with respect to G A is x (2 − i ) mod 41 x ⌊ i +14 ⌋ +2+ (cid:0) ( i −
3) mod 4 (cid:1) z i − z . Therefore, for every i ≥ i ⊗ { , , } = (cid:0) (2 − i ) mod 4 (cid:1) ⊗ { } + (cid:16) ⌊ i + 14 ⌋ + 2 + (cid:0) ( i −
3) mod 4 (cid:1)(cid:17) ⊗ { } + ( i − ⊗ { , } + 2 ⊗ { , , , } . Acknowledgement . The authors were partially supported by Junta deAndaluc´ıa research group FQM-366 and by the project MTM2017-84890-P(MINECO/FEDER, UE).
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