The lattice of ideals of a numerical semigroup and its frobenius restricted variety associated

149 (2024) MATHEMATICA BOHEMICA No. 3, 439–454 THE LATTICE OF IDEALS OF A NUMERICAL SEMIGROUP AND ITS FROBENIUS RESTRICTED VARIETY ASSOCIATED Maria Angeles Moreno-Frías, Puerto Real, José Carlos Rosales, Granada, Received March 8, 2023. Published online October 23, 2023. Communicated by Sándor Radeleczki Abstract. Let ∆ be a numerical semigroup. In this work we show that J(∆) = {I∪ {0}:Iis an ideal of ∆}is a distributive lattice, which in addition is a Frobenius restricted variety. We give an algorithm which allows us to compute the set Ja(∆) = {S∈ J(∆): max(∆\S) = a}for a given a∈∆.As a consequence, we obtain another algorithm that computes all the elements of J(∆) with a fixed genus. Keywords: numerical semigroup; ideal; Frobenius restricted variety; embedding dimension; Frobenius number; restricted Frobenius number; genus; multiplicity; Arf numerical semigroup; saturated semigroup MSC 2020: 20M14, 11Y16 1. Introduction Let Zbe the set of integer numbers and N={x∈Z:x⩾0}. A numerical semigroup is a subset Sof N, which is closed by the sum, 0∈Sand N\S= {x∈N:x /∈S}is finite. If Ais a nonempty subset of N, we denote by hAithe submonoid of (N,+) generated by A, that is, hAi={λ1a1+...+λnan:n∈N\ {0},{a1,...,an} ⊆ A and {λ1,...,λn} ⊆ N}.By Lemma 2.1 from [12], we know that hAiis a numerical semigroup if and only if gcd(A) = 1. The work was partially supported by Junta de Andalucía groups FQM-298 and FQM-343, Proyecto de Excelencia de la Junta de Andalucía ProyExcel 00868, Proyecto de investigación del Plan Propio–UCA 2022-2023 (PR2022-011) and Proyecto de investigación del Plan Propio–UCA 2022-2023 (PR2022-004). c The author(s) 2023. This is an open access article under the CC BY-NC-ND licence cbnd DOI: 10.21136/MB.2023.0038-23 439 If Sis a numerical semigroup and S=hAi, then we say that Ais a system of generators of S. Moreover, if S6=hBifor all B!A, then we say that Ais a minimal system of generators of S. In [12], Corollary 2.8, it is shown that every numerical semigroup has a unique minimal system of generators which, in addition, is finite. We denote by msg(S)the minimal system of generators of S. The cardinality of msg(S)is called the embedding dimension of Sand is denoted by ed(S). If Sis a numerical semigroup, then F(S) = max(Z\S),g(S) = ♯(N\S), where ♯A denotes the cardinality of set A, and m(S) = min(S\ {0}). They are three important invariants of Swhich we call Frobenius number,genus and multiplicity of S, respectively. Let ∆be a numerical semigroup. An ideal of ∆is a nonempty subset Iof ∆such that I+ ∆ = {a+b:a∈Iand b∈∆} ⊆ I. If Iis an ideal of ∆, then I∪ {0}is a numerical semigroup. This fact induces us to give the following definition. A numerical semigroup Sis an I(∆)-semigroup if S\{0}is an ideal of ∆. We denote J(∆) = {S:Sis an I(∆)-semigroup}.The main aim of this manuscript is to study the set J(∆). In Section 2, we recall some basic notions and results of the theory of ideals of numerical semigroups. In Section 3, we show that J(∆) is closed under union and intersection, and so it is a distributive lattice. Moreover, we show that if S∈ J (∆) and x= max(∆ \S),then S∪ {x} ∈ J (∆) and consequentely, we have that J(∆) is a Frobenius restricted variety. We say that an ideal Iof a numerical semigroup ∆is principal if there exists a∈∆such that I={a}+ ∆. A P(∆)-semigroup is a numerical semigroup with the form ({a}+∆)∪{0}and a∈∆.In Section 4, we illustrate that every I(∆)-semigroup can be expressed as a finite and irredundant union of P(∆)-semigroups. By Proposition 2.10 from [12], we know that if Sis a numerical semigroup, then ed(S) ⩽m(S).A MED-semigroup is a numerical semigroup Ssuch that ed(S) = m(S).This class of numerical semigroups has been widely studied, see for instance [3]. In Section 4, we show that if Sis an P(∆)-semigroup and S6= ∆, then Sis a MEDsemigroup. Inspired by [1], Lipman introduces and motivates in [6] the study of Arf rings. The characterization of these rings via their value semigroups yields the notion of Arf numerical semigroup. Every Arf numerical semigroup is a MED-semigroup. In Section 4, we show that ∆is an Arf numerical semigroup if and only if every P(∆)- semigroup is an Arf numerical semigroup. A particularly interesting type of numerical semigroups are called saturated numerical semigroups. The idea of saturation of singularities were introduced in three different ways by: Zariski in [13]–[15], Pham-Teissier in [9], and Campillo in [4]. As for the Arf property, saturated numerical semigroups come into scene after a char440 acterization of saturated rings in terms of their value semigroups (see [5], [8]). In Section 4, we show that a numerical semigroup ∆is satured if and only if there is at least one P(∆)-satured semigroup. Let ∆be a numerical semigroup. We say that an ideal is irreducible if it cannot be expressed as the intersection of two ideals properly containing it. If a∈∆, then we denote B(a) = {s∈∆: a−s∈∆}.As a consequence of Lemma 3.1 from [2] we have that Iis an irreducible ideal of ∆if and only if I= ∆ or I= ∆ \B(a)for some a∈∆.AD(∆)-semigroup is a numerical semigroup with the form (∆ \B(a)) ∪ {0} for some a∈∆.As a consequence of Theorem 3.3 from [2], we have that every I(∆)-semigroup can be expressed as a unique, finite and irredundant intersection of D(∆)-semigroups. If S(Tare numerical semigroups, then the Frobenius number of Srestricted to Tis FT(S) = max(T\S).If ∆is a numerical semigroup and a∈∆, then we put Ja(∆) = {S:Sis an I(∆)-semigroup and F∆(S) = a}.In Section 5, we order the elements of the set Ja(∆) in the form of a tree with root (∆ \B(a)) ∪ {0}.This fact allows us in Section 6 to give an algorithm which computes all the elements of the set Ja(∆).Finally and based on the previous algorithm, we show another one, of different nature and with complexity not comparable to the algorithm presented in [7], that allows us to compute the set J(∆, k) = {S:Sis an I(∆)-semigroup and g(∆) = g(∆) + k}for all k∈N. 2. Basic concepts and results Let ∆be a numerical semigroup. An ideal of ∆is a nonempty subset Iof ∆such that I+ ∆ ⊆I. The following result has an easy proof. Proposition 2.1. If Iand Jare ideals of a numerical semigroup ∆, then I∪J and I∩Jare also ideals of ∆. It is clear that if Iis an ideal of ∆, then ∆\Iis finite. Therefore, if I6= ∆, then there exists max(∆ \I). Proposition 2.2. Let ∆be a numerical semigroup, let Ibe an ideal of ∆such that I6= ∆ and x= max(∆ \I).Then I∪ {x}is an ideal of ∆. P r o o f. By maximility of x, we have {x}+∆ ⊆I∪{x}.Therefore, (I∪{x})+∆ ⊆ I∪ {x}. The following result is Proposition 1 from [7]. Proposition 2.3. If ∆is a numerical semigroup and Xis a nonempty subset of ∆, then X+ ∆ is an ideal of ∆.Moreover, every ideal of ∆has this form. 441 If ∆is a numerical semigroup, then we define over Zthe following order relation: a⩽∆bif and only if b−a∈∆.We say that a nonempty subset Xof ∆is a ∆- incomparable set if a−b /∈∆for all (a, b)∈X×Xsuch that a6=b. The following result is Theorem 5 from [7]. Theorem 2.4. Let ∆be a numerical semigroup. Then the set {X+ ∆: Xis a ∆-incomparable set} is the set formed by all the ideals of S. Moreover, if Xand Yare different ∆- incomparable sets, then X+ ∆ 6=Y+ ∆. If Iis an ideal of a numerical semigroup ∆and I=X+ ∆,then we say that X is an ideal system of generators of I. Moreover, if Xis a ∆-incomparable set, then we say that Xis the ideal minimal system of generators of I. By Theorem 2.4, we know that every ideal Iof ∆admits a unique ideal minimal system of generators. We denote this system by imsg∆(I). The following result is Proposition 2.6 from [7]. Proposition 2.5. Let ∆be a numerical semigroup and let Ibe an ideal of ∆. Then imsg∆(I) = Minimals⩽∆(I). The following result is Proposition 7 from [7]. Proposition 2.6. If ∆is a numerical semigroup and let Xbe a ∆-incomparable set, then Xis finite. As a consequence of Propositions 2.5 and 2.6, the cardinal of imsg∆(I)is an integer positive number. This number is called the ideal dimension of Iin ∆and it is denoted by dim∆(I). The following result is Proposition 8 from [7]. Proposition 2.7. If Iis an ideal of ∆,then: (1) I= ∆ if and only if 0∈I, (2) I∪ {0}is a numerical semigroup. The following result is Proposition 9 from [7]. Proposition 2.8. Let ∆be a numerical semigroup and let Ibe an ideal of ∆ such that I6= ∆. Then imsg∆(I) = Minimals⩽∆(msg(I ∪ {0})). As an immediate consequence of Proposition 2.8, we have the following result. Corollary 2.9. Let ∆be a numerical semigroup and let Ibe an ideal of ∆. Then dim∆(I)⩽ed(I ∪ {0}). 442 It is well known that if ∆is a numerical semigroup and x∈∆, then ∆\ {x}is a numerical semigroup if and only if x∈msg(∆). The following result is easy to prove. Proposition 2.10. If Iis an ideal of ∆and x∈I, then I\ {x}is an ideal of ∆ if and only if x∈imsg∆(I). 3. I(∆)-semigroups Let ∆be a numerical semigroup. By Proposition 2.7, we know that if Iis an ideal of ∆, then I∪ {0}is a numerical semigroup. An I(∆)-semigroup is a numerical semigroup Ssuch that S\ {0}is an ideal of ∆.We put J(∆) = {S:Sis an I(∆)-semigroup}. E x a m p l e 3.1. It is clear that Xis an N-incomparable set if and only if X={n} for every n∈N.Hence, by applying Theorem 2.4, J(N) = {{0, n, →}:n∈N}(the symbol →means that every integer greater than nbelongs to the set). The numerical semigroups with the form {0, n, →} are called ordinary numerical semigroups. So the concepts of I(N)-semigroup and ordinary numerical semigroup are equivalent. If Sand Tare numerical semigroups and S⊆T, the Frobenius number of S restricted to Tis FT(S) = max(T\S).By definition FT(T) = −1. By applying Propositions 2.1 and 2.2, we can easily deduce the following result. Theorem 3.2. Let ∆be a numerical semigroup. Then: (1) If {S, T } ⊆ J (∆),then {S∪T, S ∩T} ⊆ J (∆). (2) ∆is the maximum element (with respect to set inclusion) of J(∆). (3) If S∈ J (∆) and S6= ∆, then S∪ {F∆(S)} ∈ J (∆). Alattice is an algebraic structure (L, ∨,∧)consisting of a set Land two binary operations ∨and ∧over Lsatisfying the properties: commutative, associative, idempotent and absorption. If, in addition, it verifies the distributive property, then the lattice is called distributive. As an immediate consequence of Theorem 3.2, we have the following result. Corollary 3.3. If ∆is a numerical semigroup, then (J(∆),∪,∩)is a distributive lattice. AFrobenius restricted variety (see [10]) is a nonempty family Fof numerical semigroups verifying the following conditions: 443 (1) Fhas a maximum element (and we denote it ∆(F)). (2) If {S, T } ⊆ F,then S∩T∈ F. (3) If S∈ F and S6= ∆(F),then S∪ {F∆(F)(S)} ∈ F. As an immediate consequence of Theorem 3.2, we have the following result. Corollary 3.4. If ∆is a numerical semigroup, then J(∆) is a Frobenius restricted variety. 4. P(∆)-semigroups In the rest of this work ∆denotes a numerical semigroup. An ideal Iof ∆is principal if dim∆(I) = 1.So the set formed by all the principal ideals of ∆is {{a}+ ∆: a∈∆}. Proposition 4.1. If Iis an ideal of ∆, then the next conditions are equivalent: (1) Iis a principal ideal. (2) Icannot be expressed as the union of two ideals of ∆strictly contained in I. P r o o f. (1) ⇒(2): Let Jand Kbe ideals of ∆such that J⊆I, K ⊆I and I=J∪K. As Iis a principal ideal of ∆, then there exits a∈∆such that I={a}+ ∆.Then a∈I=J∪Kand hence a∈Jor a∈K. If a∈J, then I={a}+ ∆ ⊆J+ ∆ ⊆Jand so I=J. (2) ⇒(1): If Iis not a principal ideal of ∆, then dim∆(I) = n⩾2.Therefore, there exists {a1, a2,...,an}a∆-incomparable set such that {a1,...,an}+ ∆ = I. Let J={a1}+ ∆ and K={a2,...,an}+ ∆.Then Jand Kare ideals of ∆such that J⊆I, K ⊆Iand I=J∪K. Moreover, applying that {a1, a2,...,an}is a ∆-incomparable set, we deduce that J(Iand K(I.  AP(∆)-semigroup is a numerical semigroup with the shape ({a}+ ∆) ∪ {0}for some a∈∆.We put P(∆) = {S:Sis a P(∆)-semigroup}. Proposition 4.2. Let ∆be a numerical semigroup. (1) If {S1, S2,...,Sn} ⊆ P(∆),then S1∪S2∪...∪Sn∈ J (∆). (2) If S∈ J (∆) and dim∆(S\ {0}) = n, then there exists {S1, S2,...,Sn} ⊆ P(∆) such that S=S1∪S2∪. . . ∪Sn. P r o o f. (1) It is a consequence from Theorem 3.2. (2) If dim∆(S\{0}) = n, then there exists {x1, x2,...,xn} ⊆ ∆such that S\{0}= {x1,...,xn}+ ∆.For every i∈ {1,...,n}, let Si= ({xi}+ ∆) ∪ {0}.It is clear that Si∈ P(∆) for all i∈ {1,...,n}and S=S1∪...∪Sn. 444 We say that a union S i∈{1,...,n} Aiof the sets Aiis irredundant if for every j∈ {1,...,n},it is verified that S i∈{1,...,n} Ai6=S i∈{1,...,n}\{j} Ai.The following result has an easy proof. Proposition 4.3. Every I(∆)-semigroup can be expressed in a unique way as a finite and irredundant union of P(∆)-semigroups. The following result is deduced from [11], Proposition 2. Proposition 4.4. If Sis a P(∆)-semigroup and S6= ∆, then Sis a MEDsemigroup. The following result can be easily deduced from [11], Proposition 9. Proposition 4.5. If ∆6=N, a ∈∆\ {0}and S= ({a}+ ∆) ∪ {0},then F(S) = a+ F(∆),g(S) = a−1 + g(∆) and m(S) = a. As an immediate consequence of the previous proposition we have the following result. Corollary 4.6. If {S, T } ⊆ P(∆), then the following conditions are equivalent: (1) S=T, (2) m(S) = m(T), (3) F(S) = F(T), (4) g(S) = g(T). Note that as a consequence of Proposition 4.5 and Corollary 4.6, the number of elements of P(∆) with Frobenius number F, genus g, multiplicity m, respectively, is 1 or 0depending on whether there exists a∈∆such that F = F(∆)+a, g = g(∆)+a−1, m = a, respectively. A numerical semigroup Sis Arf if x+y−z∈Sfor every x, y, z ∈Ssuch that z⩽y⩽x. If Sis an Arf numerical semigroup, then by [12], Proposition 3.12, we can deduce that Sis a MED-semigroup. The following result follows from [11], Corollary 38. Proposition 4.7. ∆is an Arf numerical semigroup if and only if all the elements of the set P(∆) are Arf numerical semigroups. If A⊆Nand a∈A\{0}, then we denote dA(a) = gcd{x∈A:x⩽a}.A numerical semigroup is saturated if s+ dS(s)∈Sfor all s∈S\ {0}. By Lemma 3.31 from [12], we know that every saturated numerical semigroup is an Arf numerical semigroup. The following result is deduced from [11], Corollary 43. 445 Proposition 4.8. ∆is a saturated numerical semigroup if and only if P(∆)\{∆} contains at least a saturated numerical semigroup. 5. D(∆)-semigroups Let ∆be a numerical semigroup. An ideal is irreducible if it cannot be expressed as the intersection of two ideals properly containing it. If a∈∆, then we denote B(a) = {s∈∆: a−s∈∆}.The following result follows from [2], Lemma 3.1. Proposition 5.1. Iis an irreducible ideal of ∆if and only if I= ∆ \B(a)for some a∈∆or I= ∆. AD(∆)-semigroup is a numerical semigroup with the form (∆\B(a))∪{0}for some a∈∆.We put D(∆) = {S:Sis a D(∆)-semigroup}.We say that an intersection T i∈{1,...,n} Aiof the sets Aiis irredundant if T i∈{1,...,n} Ai6=T i∈{1,...,n}\{j} Aifor every j∈ {1,...,n}. The following result is deduced from [2], Theorem 3.3. Proposition 5.2. Every I(∆)-semigroup can be expressed as a unique finite and irredundant intersection of D(∆)-semigroups. If ∆is a numerical semigroup and a∈∆, then we put S(∆, a) = (∆ \B(a)) ∪ {0} ∈ D(∆). The following result has an easy proof. Proposition 5.3. If ∆is a numerical semigroup and a∈∆\ {0}, then F∆(S(∆, a)) = aand g(S(∆, a)) = g(∆) + #B(a)−1. R e m a r k 5.4. ⊲Observe that S(∆,0) = ∆ and so F∆(S(∆,0)) = F∆(∆) = −1.Therefore, {F∆(S(∆, a)): a∈∆}= (∆ \ {0})∪ {−1}. ⊲We propose the study of the set {#B(a): a∈∆}as an open problem. Theorem 5.5. Let Sbe a numerical semigroup. Then Sis a D(∆)-semigroup if and only if Sis a maximal element (with respect to set inclusion) of the set {T:is an I(∆)-semigroup and F∆(T) = F∆(S)}. P r o o f. Necessity. If Sis not maximal, then there exists an I(∆)-semigroup T such that S(Tand F∆(T) = F∆(S).By Theorem 3.2, we know that S∪ {F∆(S)} is an I(∆)-semigroup. Then S= (S∪ {F∆(S)})∩Tand so we have been able to write Sas an intersection of two I(∆)-semigroups properly containing S. Hence, S is not a D(∆)-semigroup. 446 Sufficiency. Let T= (∆\B(F∆(S)))∪{0}.It is clear that Tis an I(∆)-semigroup, S⊆Tand F∆(T) = F∆(S).By applying the maximility of S, we obtain that S=T. Therefore, Sis a D(∆)-semigroup.  As a consequence of Propositions 5.1 and 5.3, we deduce the following result. Proposition 5.6. If Sis an I(∆)-semigroup, then there exists a unique D(∆)- semigroup Tsuch that S⊆Tand F∆(T) = F∆(S).Moreover, T= S(∆,F∆(S)) if S6= ∆ and T= ∆ if S= ∆. If ∆is a numerical semigroup and a∈∆\ {0}, then we put Ja(∆) = {S: Sis an I(∆)-semigroup and F∆(S) = a}. Proposition 5.7. Let Sbe an I(∆)-semigroup such that S6= ∆ and F∆(S) = a. Then S= S(∆, a)if and only if {h∈∆\S:h /∈B(a)}=∅. P r o o f. If S= S(∆, a),then S= (∆ \B(a)) ∪ {0}and so {h∈∆\S: h /∈B(a)}=∅.Conversely, if {h∈∆\S:h /∈B(a)}=∅,then it is clear that S= S(∆, a). If S∈ Ja(∆) and S6=S(∆, a),then we put α(S) = max{h∈∆\S:h /∈B(a)}. By definition, α((S, a)) = 0. Proposition 5.8. If a∈∆and S∈ Ja(∆),then S∪ {α(S)} ∈ Ja(∆). P r o o f. By the maximality of α(S),we deduce that {α(S)}+ ∆ ⊆S∪ {α(S)}. From this result one easily deduces that S∪ {α(S)} ∈ Ja(∆). If S∈ Ja(∆),then the previous proposition can be used to define recursively the following sequence of elements of Ja(∆): ⊲ S0=S, ⊲ Sn+1 =Sn∪ {α(Sn)}for all n∈N. As a consequence of Propositions 5.7 and 5.8, we have the following result. Proposition 5.9. If a∈∆and S∈ Ja(∆),then there is p∈Nsuch that S=S0(S1(...(Sp=S(∆, a). Agraph Gis a pair (V, E)where Vis a nonempty set and E⊆ {(u, v)∈V×V: u6=v}. The elements of Vand Eare called vertices and edges, respectively. A path of length nconnecting the vertices xand yof graph Gis a sequence of different edges of the form (v0, v1),(v1, v2),...,(vn−1, vn)such that v0=xand vn=y. A graph is a tree if G= (V, E),where there exists a vertex r(known as the root of G) such that for any other vertex xof Gthere exists a unique path connecting x and r. If (u, v)is an edge of a tree, then we say that uis a child of v. 447 [15] O. Zariski: General theory of saturation and of saturated local rings III. Saturation in arbitrary dimension and, in particular, saturation of algebroid hypersurfaces. Am. J. Math. 97 (1975), 415–502. zbl MR doi Authors’ addresses:Maria Angeles Moreno-Frías (corresponding author), Dpto. de Matemáticas, Facultad de Ciencias, Universidad de Cádiz, E-11510, Puerto Real, Cádiz, Spain, e-mail: [email protected];José Carlos Rosales, Dpto. de Álgebra, Facultad de Ciencias, Universidad de Granada, E-18071, Granada, Spain, e-mail: jrosales@ ugr.es. 454