Cyclotomic exponent sequences of numerical semigroups
Alexandru Ciolan, Pedro A. García-Sánchez, Andrés Herrera-Poyatos, Pieter Moree
aa r X i v : . [ m a t h . A C ] J a n CYCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS
ALEXANDRU CIOLAN, PEDRO A. GARC´IA-S ´ANCHEZ, ANDR´ES HERRERA-POYATOS,AND PIETER MOREE
Abstract.
We study the cyclotomic exponent sequence of a numerical semigroup S, and we com-pute its values at the gaps of S, the elements of S with unique representations in terms of minimalgenerators, and the Betti elements b ∈ S for which the set { a ∈ Betti( S ) : a ≤ S b } is totally orderedwith respect to ≤ S (we write a ≤ S b whenever a − b ∈ S, with a, b ∈ S ). This allows us to char-acterize certain semigroup families, such as Betti-sorted or Betti-divisible numerical semigroups, aswell as numerical semigroups with a unique Betti element, in terms of their cyclotomic exponentsequences. Our results also apply to cyclotomic numerical semigroups , which are numerical semi-groups with a finitely supported cyclotomic exponent sequence. We show that cyclotomic numericalsemigroups with certain cyclotomic exponent sequences are complete intersections, thereby makingprogress towards proving the conjecture of Ciolan, Garc´ıa-S´anchez and Moree (2016) stating that S is cyclotomic if and only if it is a complete intersection. Introduction A numerical semigroup S is a submonoid of N (the set of non-negative integers) under addition,with finite complement in N . The non-negative integers that are not in S are its gaps , and the setof gaps is denoted by G( S ). The largest gap is the Frobenius number of S , denoted by F( S ). Thenumber of gaps of S , also known as the genus of S , is denoted by g(S). A numerical semigroupadmits a unique minimal generating system; its elements are called minimal generators , and itscardinality the embedding dimension , denoted by e( S ). The smallest positive integer in S is the multiplicity of S and is denoted by m( S ). For an introduction to the theory of numerical semigroupsthe reader is referred, e.g., to [17].To a numerical semigroup S we can associate its Hilbert series , defined as the formal power seriesH S ( x ) = P s ∈ S x s ∈ Z [[ x ]] , and its semigroup polynomial , given by P S ( x ) = (1 − x ) P s ∈ S x s . (Indeed,since all elements larger than F( S ) are in S and F( S ) is not, P S ( x ) is a monic polynomial of degreeF( S ) + 1 . ) In the sequel we say that a formal identity of the form A ( x ) = B ( x ) is true if it holdsin Z [[ x ]]. For notational convenience we will often denote the infinite sum 1 + x d + x d + · · · by(1 − x d ) − , where d ∈ N . It is not difficult to conclude that the coefficients of P S are in {− , , } and that consecutive non-zero coefficients alternate in sign. On noting the formal identity H S ( x ) = (1 − x ) − − P s ∈ G( S ) x s ,we have(1) P S ( x ) = 1 + ( x − X s ∈ G( S ) x s , Mathematics Subject Classification.
Key words and phrases.
Numerical semigroups, cyclotomic polynomials, Betti elements, complete intersections.The second author is supported by the project MTM2017–84890–P, which is funded by Ministerio de Econom´ıa yCompetitividad and Fondo Europeo de Desarrollo Regional FEDER, and by the Junta de Andaluc´ıa Grant NumberFQM–343.The third author was supported by an Initiation to Research Fellowship from the University of Granada in theacademic year 2017-2018. He is currently supported by an Oxford-DeepMind Graduate Scholarship and an EPSRCDoctoral Training Partnership. and so P S (1) = 1. In addition, P S (0) = 1 and, by [7, Lemma 11] (see also Lemma 3.1), there exist unique integers e j such that the formal identity(2) P S ( x ) = ∞ Y j =1 (1 − x j ) e j holds. We call the sequence e = { e j } j ≥ the cyclotomic exponent sequence of S and we will use thisnotation throughout the paper.Cyclotomic exponent sequences were introduced in [7] as a tool for studying cyclotomic numericalsemigroups; we will come back to this later in this section. The purpose of this paper is to initiate thestudy of any numerical semigroup by means of its cyclotomic exponent sequence. Our ultimate goalis to characterize special families of numerical semigroups in terms of properties of their cyclotomicexponent sequences.Our first main result determines the exponent sequence of S at gaps and minimal generators. Theorem 1.1. If S = N is a numerical semigroup and e is its cyclotomic exponent sequence, then a) e = 1;b) e j = 0 for every j ≥ not in S ;c) e j = − for every minimal generator j of S ;d) e j = 0 for every j ∈ S that has only one factorization and is not a minimal generator. Our second main result determines e at certain Betti elements (see Section 2.3 for a definitionof the latter and the related notion of R -classes). In order to introduce our findings, we need thefollowing definitions.Let ( X, ≤ ) be a partially ordered set. We define the set U( X ) asU( X ) = { x ∈ X : ↓ x is totally ordered } , where ↓ x = { y ∈ X : y ≤ x } . We note that(3) Minimals ≤ X = Minimals ≤ U( X ) . We write a ≤ S b if b − a ∈ S . Since S is a cancellative monoid free of units, the relation ≤ S defines an order relation on Z . Moreover, for any s ∈ S , the set ↓ s (considered in ( S, ≤ S )) is finite.If S is a numerical semigroup, we define the set E ( S ) = { d ∈ N : d ≥ , e d = 0 , d is not a minimal generator } , notation which we will use throughout.Our next result relates the partially ordered sets (Betti( S ) , ≤ S ) and ( E ( S ) , ≤ S ). Theorem 1.2.
Let S be a numerical semigroup with cyclotomic exponent sequence e . Then U(Betti( S )) = U( E ( S )) . Moreover, for every b ∈ U(Betti( S )) , the exponent e b is equal to thenumber of R -classes of b minus . A direct consequence of (3) and Theorem 1.2 is that Minimals ≤ S Betti( S ) = Minimals ≤ S E ( S ).In order to prove Theorem 1.2 we need to understand the graph of factorizations ∇ b of the elements b in U(Betti( S )) (see Section 2.3 for a definition of ∇ b ). Our main technical result on this matteris Theorem 5.9, which shows that when b ∈ U(Betti( S )) \ Minimals ≤ S Betti( S ), the graph ∇ b hasexactly one connected component that is not a singleton.As a consequence of Theorem 1.2 we are able to characterize some families of numerical semigroupssolely in terms of their cyclotomic exponent sequences. Before stating our next result, let us definethese families. In what follows, S is a numerical semigroup. We say that S is Betti-sorted if Betti( S )is totally ordered with respect to ≤ S , and that S is Betti-divisible if Betti( S ) is totally ordered withrespect to the divisibility order in N . These two families of numerical semigroups were introduced in[10], where the authors showed that they are complete intersections (see Section 2.4 for a definition). YCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS 3
The third family we consider is that of numerical semigroups with a unique Betti element, which isobviously a subset of each of the two previous families. This family was studied in [10] and [11].
Theorem 1.3.
For a numerical semgiroup S the following assertions hold: a) The semigroup S is Betti-sorted if and only if E ( S ) is totally ordered by ≤ S . b) The semigroup S is Betti-divisible if and only if E ( S ) is totally ordered by the divisibility order. c) The semigroup S has a unique Betti element if and only if E ( S ) is a singleton. Our work also has some consequences for cyclotomic numerical semigroups.
Definition 1.1.
A cyclotomic numerical semigroup is a numerical semigroup whose cyclotomicexponent sequence e has finite support; that is, there exists N ∈ N such that e j = 0 for every j ≥ N . These semigroups were introduced and studied in [7] using a different, but equivalent, definition(see Section 2.1). It turns out that every complete intersection numerical semigroup is cyclotomic,the former being a numerical semigroup such that the cardinality of its minimal presentation equalsits embedding dimension minus one (see Section 2.3 for a brief recap on the concept of mini-mal presentations). In [7] the authors made the following conjecture, which they checked to betrue for numerical semigroups with Frobenius number not exceeding 70 , using the GAP package numericalsgps [8, 9].
Conjecture 1.4 ([7, Conjecture 1]) . A numerical semigroup is a complete intersection if and onlyif it is cyclotomic.
A version of this conjecture has been established for certain graded algebras, see [6], but thenumerical semigroup version remains open. Conjecture 1.4 is equivalent with saying that a numericalsemigroup S is a complete intersection if and only if its cyclotomic exponent sequence e has finitesupport . This establishes an equivalence between an algebraic property of a numerical semigroupand one that only involves its cyclotomic exponent sequence. Note that e has finite support if andonly if E ( S ) is finite. Recall that Theorem 1.3 deals with the case where E ( S ) is a singleton.As a consequence of our results, we make further progress towards proving Conjecture 1.4 byshowing that all members of a certain family of cyclotomic numerical semigroups are completeintersections. More precisely, if the Hasse diagrams of Betti( S ) and E ( S ) with respect to ≤ S areforests, that is, U(Betti( S )) = Betti( S ) and U( E ( S )) = E ( S ), then we are able to deduce that S is acomplete intersection (Corollary 7.6). Computations suggest that such forests arise very frequently;for instance, there are 197 complete intersection numerical semigroups with Frobenius number 101(equivalently, with genus equal to 52), and for 170 of them the Hasse diagram of their set of Bettielements with respect to ≤ S is a forest. Here we should mention that, for any complete intersectionnumerical semigroup S , we have Betti( S ) = E ( S ), as explained in Section 2.4.The paper is organized as follows. In Section 2 we gather some preliminary material used inthe rest of the paper. In Section 3 we introduce cyclotomic exponent sequences and establish someelementary properties. In Section 4 we prove Theorem 1.1. In Section 5 we give the proof ofTheorem 1.2, which comes in two parts, and we discuss a few tools needed for this purpose, such asminimal Betti elements and restricted factorizations (as this section is the longest, we kindly ask inadvance for the reader’s patience). In Section 6 we give the proof of Theorem 1.3, while Section 7is dedicated to applications to cyclotomic numerical semigroups, open questions, and concludingremarks. 2. Preliminaries
Here we recall a few properties and notions that are needed throughout the paper. References inthe subsection headers give suggestions for further reading.Section 2.1 is exceptional in that it is not needed for the rest of the paper. Its purpose is to showthat the original definition of a cyclotomic numerical semigroup S , given in [7] through saying that CIOLAN, GARC´IA-S ´ANCHEZ, HERRERA-POYATOS, AND MOREE P S admits a factorization into cyclotomic polynomials as in (5), is equivalent with the definitionused here.2.1. Cyclotomic numerical semigroups and cyclotomic polynomials [7, 20] . The semigrouppolynomial and the Frobenius number of a numerical semigroup of embedding dimension two canbe easily determined (see, for instance, [15]).
Lemma 2.1 ([15, Theorem 1]) . If ≤ a < b are coprime integers, then P h a,b i ( x ) = (1 − x )(1 − x ab )(1 − x a )(1 − x b ) . Corollary 2.2 (Sylvester, 1884) . If ≤ a < b are coprime integers, then F( h a, b i ) = ab − a − b . Lemma 2.1 shows that S = h a, b i is a cyclotomic numerical semigroup, since its exponent sequencehas finite support. The factorization of P h a,b i into irreducibles is easily found by using the well-knownfactorization(4) x n − Y d | n Φ d ( x )of x n − Q ). In the special case where a and b are prime numbers, we find that P h a,b i ( x ) = Φ ab ( x ), which then gives a very natural proof ofthe classical fact that the coefficients of Φ ab ( x ) are all in {− , , } and that consecutive non-zerocoefficients alternate in sign.The following two results describe some basic properties of the cyclotomic exponent sequenceattached to a cyclotomic numerical semigroup. Proposition 2.3.
Let S be a numerical semigroup and let e be its cyclotomic exponent sequence.If S is cyclotomic, then P j ≥ e j = 0 .Proof. Let N be the largest index j such that e j = 0 . Then we haveP S ( x ) = (1 − x ) P j ≤ N e j G S ( x ) , for some rational function G S ( x ) satisfying G S (1)
6∈ { , ∞} (in fact G S (1) = Q j ≤ N j e j ). SinceP S (1) = 1, it follows that P j ≥ e j = 0. (cid:3) Proposition 2.4.
Let S be a numerical semigroup. Then S is cyclotomic if and only if P S ( x ) factorizes in the form (5) P S ( x ) = Y d ∈D Φ h d d , where D is a finite set and h d are positive integers.Proof. Let e be the exponent sequence of S and let N be the largest index j such that e j = 0 . ByProposition 2.3 we have P ≤ j ≤ N e j = 0 and soP S ( x ) = N Y j =1 (1 − x j ) e j = N Y j =1 ( x j − e j . By (4) it then follows that P S ( x ) can be written as in (5), where a priori some of the integers h d may be negative. As the complex zeros of the cyclotomic polynomials are all different, this wouldlead to P S having a pole, contradicting the fact that P S is a polynomial. YCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS 5
For the other direction, we use the M¨obius function µ ( n ), which is equal to zero for non-square freeintegers n and to ( − r otherwise, where r is the number of prime factors in the prime decompositionof n . By applying M¨obius inversion to (4) one obtainsΦ n ( x ) = Y d | n ( x d − µ ( n/d ) . Using the fact that P d | n µ ( d ) = 0 for n ≥
2, this can be rewritten for n ≥ n ( x ) = Y d | n (1 − x d ) µ ( n/d ) . Since P S (1) = 0, Φ (1) = 0, and Φ d (1) = 0 for d ≥
2, we have 1
6∈ D , and so, using the latteridentity, it follows that there are integers e , e , . . . such thatP S ( x ) = ∞ Y d =1 (1 − x d ) e d , where e d = 0 for d > max D , which means that e has finite support. (cid:3) Recall that a polynomial f ( x ) of degree d is self-reciprocal if f ( x ) = x d f (1 /x ). The cyclotomicpolynomial Φ n is self-reciprocal for n ≥
2. As a consequence of this fact and Proposition 2.4,it follows that if S is cyclotomic, then P S ( x ) is self-reciprocal. It is not difficult to show that anumerical semigroup is symmetric (that is, for every n ∈ Z , either n or F( S ) − n is in S ) if and onlyif P S is self-reciprocal [15]. Therefore, every cyclotomic numerical semigroup is symmetric. Theconverse is generally not true; for instance, it can be shown that for every positive integer e ≥ e that is symmetric but not cyclotomic[12, 18].Proposition 2.4 raises the question whether one can classify cyclotomic numerical semigroups forwhich P S decomposes into a small number of irreducible factors. This and similar questions areaddressed in [5], where the authors show, for example, that P S = Φ n if and only if n = pq and S = h p, q i for distinct prime numbers p and q .2.2. Ap´ery sets [17] . Let S be a numerical semigroup and m ∈ Z . The setAp( S ; m ) = { s ∈ S : s − m S } is called the Ap´ery set of m in S . Given any arithmetic progression modulo m , the numbers in itthat are large enough will be in S , whereas the numbers that are small enough will not be in S. Therefore, among them we will find at least one element from Ap( S ; m ), and so | Ap( S ; m ) | ≥ m .In the remainder of this subsection we assume that m ∈ S , in which case S = Ap( S ; m ) + m N and | Ap( S ; m ) | = m . It then follows that every integer z can be uniquely written as z = km + w with k ∈ Z and w ∈ Ap( S ; m ), and that z ∈ S if and only if k ≥
0. We will use this fact severaltimes.From S = Ap( S ; m ) + m N we infer that H S ( x ) = P w ∈ Ap( S ; m ) x w P ∞ k =0 x km , hence(6) (1 − x m ) H S ( x ) = X w ∈ Ap( S ; m ) x w , with the right-hand side being the Ap´ery polynomial of m in S , see [16]. CIOLAN, GARC´IA-S ´ANCHEZ, HERRERA-POYATOS, AND MOREE
Minimal presentations and Betti elements [1, 17] . Let S be a numerical semigroupminimally generated by { n , . . . , n e } . There is a natural epimorphism ϕ : N e → S , defined as ϕ ( a , . . . , a e ) = P ei =1 a i n i . The set ker ϕ = { ( a, b ) ∈ N e × N e : ϕ ( a ) = ϕ ( b ) } is a congruence, that is,an equivalence relation compatible with addition; hence, S is isomorphic, as a monoid, to N e / ker ϕ .A presentation for S is a system of generators of ker ϕ as a congruence. A presentation is minimal if none of its proper subsets generates ker ϕ . It can be shown that all minimal presentations of anumerical semigroup have the same (finite) cardinality (see, for instance, [17, Chapter 7]).Given ρ ⊆ N e × N e , denote by cong( ρ ) the congruence generated by ρ , that is, the intersection ofall congruences containing ρ . Define ρ = ρ ∪ { ( y, x ) : ( x, y ) ∈ ρ } and ρ = { ( x + u, y + u ) : ( x, y ) ∈ ρ , u ∈ N e } . It turns out that cong( ρ ) is the transitive closure of ρ .A minimal presentation of S can be constructed as follows. For s ∈ S , let Z( s ) be the set offactorizations of s in S , that is, the fiber ϕ − ( s ) (we use Z( S ) to denote the set of all factorizationsof elements in S , which equals N e( S ) ). Define ∇ s to be the graph with vertices Z( s ) and with edges xy so that x · y = 0 (dot product; that is, edges join factorizations having minimal generators incommon). The connected components of ∇ s are called the R-classes of s . The element s ∈ S is a Betti element if ∇ s is not connected. We denote by Betti( S ) the set of Betti elements of S , and bync( ∇ s ) the number of connected components of ∇ s .Assume that s ∈ Betti( S ) and let C , . . . , C r be the connected components of ∇ s (thus r =nc( ∇ s )). Pick x i ∈ C i for all i ∈ { , . . . , r } , and set ρ ( s ) = { ( x , x ) , ( x , x ) , . . . , ( x r − , x r ) } . Then ρ = S s ∈ Betti( S ) ρ ( s ) is a minimal presentation of S . All minimal presentations can be constructedby using the following idea. Think of ( x i , x j ) as a link connecting C i and C j . Then you need allconnected components to be connected with these links. The minimal possible choice is to havea spanning tree connecting them all, once the x i have been chosen. Different choices of x i in C i and different spanning trees will yield different minimal presentations, but they all have the samecardinality (see, for instance, [1, Chapter 4]). As a consequence, all minimal presentations havecardinality equal to P s ∈ Betti( S ) (nc( ∇ s ) − Complete intersection numerical semigroups [17] . Let S be a numerical semigroup withembedding dimension e . It can be shown that the cardinality of any minimal presentation of S has e − complete intersections .Let S and S be two numerical semigroups, and a , a be two coprime integers such that a ∈ S , a ∈ S and neither a , nor a is a minimal generator. The set a S + a S is a numerical semigroupknown as the gluing of S and S . We will write S = a S + a a a S .A complete intersection numerical semigroup S is either N or a gluing a S + a S with both S and S complete intersection numerical semigroups (see [17, Chapter 8]). It turns out thatBetti( S ) = { a a } ∪ { a b : b ∈ Betti( S ) } ∪ { a b : b ∈ Betti( S ) } , see [2].It is well-known (see [2]) that(7) H a S + a a a S ( x ) = (1 − x a a ) H S ( x a ) H S ( x a ) , which, in terms of semigroup polynomials, can be written as(8) P a S + a a a S ( x ) = (1 − x )(1 − x a a )(1 − x a )(1 − x a ) P S ( x a ) P S ( x a ) . Consequently, a formula for P S in terms of the minimal generators and Betti elements of S can begiven. If S = n N + b n N + · · · + b e − n e N (with { n , . . . , n e } the minimal generating system of S and with b i not necessarily distinct integers), then [2, Theorem 4.8] states that(9) H S ( x ) = Q e − i =1 (1 − x b i ) Q ei =1 (1 − x n i ) . YCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS 7 If S = a S + a a a S is a gluing of S and S , then every minimal presentation of S comes fromthe union of a minimal presentation of S , a minimal presentation of S and a pair of factorizationsof a a , one in a S and the other in a S ; see, for instance, [17, Chapter 8]. Thus, by (7) and thefact that every minimal presentation of S has cardinality P b ∈ Betti( S ) (nc( ∇ b ) − b i in the numerator of (9) is precisely nc( ∇ b i ) − S ( x ) = Q b ∈ Betti( S ) (1 − x b ) nc( ∇ b ) − Q ei =1 (1 − x n i ) . Indeed, this identity characterizes complete intersection numerical semigroups.
Proposition 2.5.
Let S be a numerical semigroup. Then S is a complete intersection numericalsemigroup if and only if H S satisfies (10) .Proof. We prove that if S verifies (10), then S is a complete intersection numerical semigroup;the other implication also holds, as we have just seen. Recall that P S ( x ) = (1 − x ) H S ( x ) is apolynomial and that by (1) we have P S (1) = 1. Thus the factors 1 − x of the numerator anddenominator of P S ( x ) must cancel each other out and we find P b ∈ Betti( S ) (nc( ∇ b ) −
1) = e( S ) − S has cardinality e( S ) −
1, which means that S is acomplete intersection. (cid:3) One of our aims is to prove that the Hilbert series of a cyclotomic numerical semigroup alwayssatisfies (10). In this paper we do so for some particular classes of cyclotomic numerical semigroups.2.5.
Other series and polynomials associated to numerical semigroups [19] . This subsec-tion is dedicated to introducing a few other objects that arise naturally in connection to numericalsemigroups. However, the only reults needed in the sequel are the upcoming definitions and the ac-companying identity (11). The reader may therefore choose to omit the discussion on the polynomial K S , which we make here for sake of completeness, and directly skip to Section 2.6.Let S be a numerical semigroup minimally generated by a set A . The denumerant of s ∈ S ,denoted by d ( s ), is the cardinality of Z( s ), the set of factorizations of s in S . We can consider the denumerant series P s ∈ S d ( s ) x s , which verifies the equality(11) X s ∈ S d ( s ) x s = Y n ∈ A ∞ X j =0 x jn = Y n ∈ A − x n . This equality is widely used in our work and its proof is straightforward. We note that every Bettielement has denumerant exceeding one.Let A = { n < · · · < n e } be the minimal system of generators of S . Sz´ekely and Wormald [19]were the first to study the function(12) K S ( x ) = (1 − x n ) · · · (1 − x n e ) H S ( x ) , which, on writing K S ( x ) = (1 + x + · · · + x n − )(1 − x n ) · · · (1 − x n e ) P S ( x ) , turns out to be a polynomial of degree F( S ) + P ej =1 n j .Let S be a complete intersection numerical semigroup. From (10) we derive(13) K S ( x ) = Y b ∈ Betti( S ) (1 − x b ) nc( ∇ b ) − . Corollary 2.6.
Let S be a complete intersection numerical semigroup minimally generated by { n , . . . , n e } . Then F( S ) + e X j =1 n j = X b ∈ Betti( S ) b (nc( ∇ b ) − . CIOLAN, GARC´IA-S ´ANCHEZ, HERRERA-POYATOS, AND MOREE
Proof.
The result follows from taking degrees in (13). (cid:3)
The polynomial K S has been explicitly computed for several families of numerical semigroups.For instance, an expression is given in [4] for numerical semigroups of embedding dimension three,and for those of embedding dimension four that are symmetric or pseudo-symmetric. In that paper, K S is related to the Betti numbers of the semigroup ring associated to S (see [3] for a differentapproach).2.6. Isolated factorizations [10] . Let S be a numerical semigroup and let s ∈ S . We say that afactorization z of s is isolated if z · x = 0 for every factorization x of s different from z . Thus, z is an isolated factorization if and only if { z } is an R -class of ∇ s . This means either that s has aunique factorization, or that s is a Betti element with one of its R -classes being a singleton. Wedenote by I( s ) the set of isolated factorizations of s , and by I(Λ) the set of isolated factorizationsof the elements of Λ ⊆ S . Thus I(Λ) = I s (Λ) ∪ I b (Λ) , where I s (Λ) is the set of isolated factorizations coming from elements with a unique factorization,and I b (Λ) = I(Λ) ∩ Z(Betti( S )) that of the isolated factorizations of the Betti elements in Λ. Wealso denote the cardinality of I( s ) by i( s ) and we define IBetti( S ) as the set of Betti elements withan isolated factorization. Isolated factorizations can be characterized as in Lemma 2.7. First, weneed some notation. Let x, y ∈ N e . We say that x ≤ y if x j ≤ y j for every j . This gives an orderrelation on N e , known as the cartesian product order . Recall that x < y when x ≤ y and x = y . Lemma 2.7.
Let S be a numerical semigroup and s ∈ S . A factorization z ∈ Z( s ) is not isolatedif and only if there exists x ∈ I b ( S ) such that x < z . In particular, I b ( S ) = Minimals ≤ Z( { s ∈ S : d ( s ) ≥ } ) . As a consequence, any factorization z ∈ N e(S) can be written as z = w + x + · · · + x l with w ∈ I s ( S ) and x , . . . , x l ∈ I b ( S ) .Proof. The first assertion is merely a rephrasing of [10, Lemma 3.1].Assume that z ∈ N e( S ) = Z( S ). If ϕ ( z ) has a unique factorization, then z = w ∈ I s ( S ). Otherwise,there exists x ∈ I b ( S ) such that x < z . We consider now z − x and start anew. This processmust end either with a 0 or with a factorization that is the unique factorization of an element inthe semigroup. (cid:3) We say that an element s ∈ S is Betti-minimal if s ∈ Minimals ≤ S Betti( S ). As a consequence ofLemma 2.7, one can characterize Betti-minimal elements as in Proposition 2.8. Proposition 2.8 ([10, Proposition 3.6]) . Let S be a numerical semigroup and s ∈ S . The followingstatements are equivalent: a) s is Betti-minimal; b) s is a minimal element of IBetti( S ) with respect to ≤ S ; c) s has at least two factorizations and all of them are isolated, that is, nc( ∇ s ) = i( s ) ≥ . The following result is a particular case of [10, Corollary 3.8] and characterizes the elementshaving a unique factorization in terms of Ap´ery sets.
Corollary 2.9 ([10, Corollary 3.8]) . Let S be a numerical semigroup. Then { m ∈ S : d ( m ) = 1 } = \ b ∈ Betti( S ) Ap( S ; b ) = \ b ∈ Minimals ≤ S Betti( S ) Ap( S ; b ) . The next lemma allows us to deal with sequences of Betti elements of the form b ≤ S · · · ≤ S b t ,and will be useful for the study of U(Betti( S )). Lemma 2.10 ([10, Lemma 3.12]) . Let S be a numerical semigroup. If b and b are two Bettielements of S such that b < S b , then x · y = 0 for every x ∈ Z( b ) and y ∈ I( b ) . YCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS 9 Cyclotomic exponent sequences
In this section we show how to compute, both theoretically and practically, the cyclotomic ex-ponent sequence of a numerical semigroup, and we give some examples. Practically, they can becomputed with the function
CyclotomicExponentSequence or, alternatively, with the function
WittCoefficients of the
GAP [9] package numericalsgps [8], which implements the method givenin the upcoming Lemma 3.5.Let a ( x ) , b ( x ) ∈ Z [[ x ]] and p ( x ) ∈ Z [ x ]. We use the notation a ( x ) ≡ b ( x ) (mod p ( x )) to indicatethat a ( x ) − b ( x ) ∈ p ( x ) Z [ x ]. Note that ≡ is an equivalence relation.The next lemma, together with the fact that P S ( x ) ≡ x ), shows that for any numericalsemigroup there is an expansion of the form (2), where the exponents e j are uniquely determinedintegers. Lemma 3.1.
Let f ( x ) ∈ Z [[ x ]] and suppose that f ( x ) ≡ x ) . Then there exist unique integers e , e , . . . such that, in Z [[ x ]] , (14) f ( x ) = ∞ Y k =1 (1 − x k ) e k . Proof.
We show how to successively determine the integers e , e , . . . , e m such that (14) holds modulo x m +1 . By assumption, f ( x ) = 1 − e x (mod x ) for some e ∈ Z . This then gives f ( x )(1 − x ) − e ≡ x ). Let m ≥
2. Suppose that we have found integers e , . . . , e m − such that f ( x ) m − Y k =1 (1 − x k ) − e k ≡ x m ) . As the right-hand side is of the form 1 − e m x m (mod x m +1 ) for some integer e m , we infer that f ( x ) m Y k =1 (1 − x k ) − e k ≡ x m +1 ) . We now turn our attention to the uniqueness claim. For the sake of contradiction, suppose thereexists a different sequence of integers f n such that f ( x ) = Q ∞ k =1 (1 − x k ) f k . Let m be the smallestinteger such that f m = e m . Put h ( x ) = Q m − k =1 (1 − x k ) e k . We then have f ( x ) ≡ h ( x ) (1 − x m ) e m (mod x m +1 )on the one hand, and f ( x ) ≡ h ( x ) (1 − x m ) f m (mod x m +1 )on the other. As the two expressions have different coefficients in front of x m , we have reached acontradiction, concluding the proof. (cid:3) Example . Let α be an integer. We have 1 − αx = Q ∞ k =1 (1 − x k ) M ( α,k ) , with M ( α, k ) = k P j | k µ ( k/j ) α j . This is the so-called cyclotomic identity , see, e.g., [13]. In case p is a primenumber, the fact that M ( α, p ) must be an integer implies Fermat’s Little Theorem stating that α p ≡ α (mod p ). Remark . Expansions of the form (14) arise in quite different areas such as automata, group,graph and Lie algebra theory; see [14] for some references.
Remark . Let f ( x ) ∈ Z [ x ] and suppose that f ( x ) ≡ x ). By (14) we have f ( x ) ≡ Q nk =1 (1 − x k ) e k (mod x n +1 ). This identity allows one to determine the first n coefficients of f . Ontaking n = deg( f ), we can even reconstruct f completely. Although the proof of Lemma 3.1 is constructive, it is computationally slow. If instead of a formalseries we are given a polynomial, then the following “polynomial version” of Lemma 3.1 provides afaster way to calculate the exponents in (14). We can thus use it in the particular case f = P S inorder to obtain the exponent sequence of a numerical semigroup S . Lemma 3.5 ([14, Lemma 1]) . Let f ( x ) = 1 + a x + · · · + a d x d ∈ Z [ x ] be a polynomial with a d = 0 ,and let α , . . . , α d be its roots. Then the numbers s f ( k ) = α − k + · · · + α − kd are integers satisfyingthe recursion (15) s f ( k ) + a s f ( k −
1) + · · · + a k − s f (1) + ka k = 0 , with a m = 0 for m > d . In particular, for k > d , the integer s f ( k ) is given by the linear recurrence s f ( k ) = − a s f ( k − − · · · − a d s f ( k − d ) . Over Z [[ x ]] one has f ( x ) = ∞ Y k =1 (1 − x k ) e f ( k ) , with (16) e f ( k ) = 1 k X j | k s f ( j ) µ (cid:18) kj (cid:19) ∈ Z . Proof.
This follows from Lemma 1 of [14] on taking ˆ F ( x ) = f ( x ) and noting that the reciprocal F of ˆ F has α − , . . . , α − d as roots. (Note that 0 does not occur as root of f .) (cid:3) Combining Lemmas 3.1 and 3.5 with Remark 3.4 gives rise to the following.
Proposition 3.6.
A numerical semigroup S has a unique cyclotomic exponent sequence e = { e j } j ≥ with e j = e P S ( j ) for every j ≥ . Conversely, given a cyclotomic exponent sequence coming from anumerical semigroup, there is a unique numerical semigroup corresponding to it. Let S be a complete intersection numerical semigroup. The cyclotomic exponent sequence of S is provided by (10). We havea) e = 1;b) e j = − j is a minimal generator of S ;c) e j = nc( ∇ j ) − j is a Betti element of S ;d) e j = 0 otherwise. Example . As illustrated below, cyclotomic exponent sequences of cyclotomic and non-cyclotomicnumerical semigroups can differ greatly in their behavior, although they may share some commonproperties, as observed in Theorems 1.1 and 1.2.a) The semigroup S = h , , i is a complete intersection, hence cyclotomic. Indeed, we haveP S ( x ) = x − x + x − x + x − x + x − x + 1= (1 − x )(1 − x ) − (1 − x ) − (1 − x ) − (1 − x )(1 − x ) , and the cyclotomic exponent sequence of S is given by1 , , , − , , − , , , − , , , , , , , , , , , . . . . b) Let S = h , , i , with semigroup polynomial P S ( x ) = x − x + x − x + 1 . The first 100 entriesof its cyclotomic exponent sequence are
YCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS 11
1, 0, -1, 0, -1, 0, -1, 0, 0, 1, 0, 1, 0, 1, 0, 0, -1, 0, -1, 0, 0, 1, 0,1, 0, 1, -1, 0, -2, 0, -2, 1, -1, 3, 0, 3, -1, 3, -3, 1, -5, 1, -5, 3, -3,7, -2, 8, -4, 7, -9, 4, -14, 6, -14, 12, -10, 22, -9, 25, -16, 23, -30, 17,-42, 23, -43, 41, -36, 66, -37, 76, -60, 73, -100, 66, -133, 91, -139, 148,-129, 219, -146, 252, -222, 252, -340, 255, -438, 346, -469, 524, -473,731, -564, 846, -820, 887, -1183, 973, -1488, 1309, -1635, 1889, -1756,2530, -2157, 2947, -3026, 3214, -4181, 3701, -5187, 4922, -5839, 6834,-6563, 8905, -8200, 10467, -11195, 11807, -14992, 14052, -18463, 18510,-21237, 24982, -24675, 31960, -31101, 37904, -41573, 43905, -54450, 53343,-66840, 69606, -78312, 91968, -93176, 116272, -117909, 139142, -155059,164573, -199918, 202659, -245305, 262345, which suggests that S is not cyclotomic, and this is indeed the case (for otherwise, if S werecyclotomic, the roots of P S would be of absolute value 1; since deg(P S ) = 5, we would have atleast one real root, which can only be ±
1, a contradiction).One can rapidly check with the help of a computer that the roots of P S ( x ) in Example 3.7b satisfythe hypothesis of Corollary 3.8, hence explaining why the cyclotomic exponents grow exponentially. Corollary 3.8.
Suppose that the roots α i of f are ordered in such a way that their absolute valueis non-decreasing and, in addition, | α | < | α | . Then we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e f ( k ) − α − k k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ d ( k ) k ( | α | − k/ + deg( f ) | α | − k ) , with d ( k ) the number of divisors of k . In case f is linear, the term involving α can be omitted.Proof. From (16) and s f ( j ) = α − j + · · · + α − j deg( f ) , we infer that e f ( k ) = k α − k + E ( k ) with k | E ( k ) | ≤ X j | k, j In this section we prove Theorem 1.1 by combining the upcoming Lemmas 4.1, 4.2, 4.3 and 4.5.We start with the trivial observation that S = N if and only if P S ( x ) = 1. That is, N is the onlynumerical semigroup whose cyclotomic exponent sequence is constantly zero. We deal with the case S = N for the remainder of this section.Throughout this section we let A be the minimal generating system of S , and e = { e j } j ≥ itscyclotomic exponent sequence. Lemma 4.1. For every j ∈ { , . . . , m( S ) − } , we have e j = 0 . Moreover, e = 1 and e m( S ) = − . Proof. Let m = m( S ). From the congruence H S ( x ) ≡ x m (mod x m +1 ) we see thatP S ( x ) ≡ − x + x m ≡ (1 − x )(1 + x m ) ≡ (1 − x )(1 − x m ) − (mod x m +1 ) . The latter congruence determines e , . . . , e m uniquely, see the proof of Lemma 3.1. (cid:3) It follows that we have the formal identity(17) H S ( x ) = ∞ Y j =m( S ) (1 − x j ) e j . Lemma 4.2. For every gap g of S with g > , we have e g = 0 .Proof. We proceed by contradiction. Let g > S having a non-zero exponent.Considering the series H S ( x ) modulo x g +1 , we obtain(18) H S ( x ) ≡ (1 − x g ) e g g − Y j =m( S ) (1 − x j ) e j ≡ − e g x g + g − Y j =m( S ) (1 − x j ) e j (mod x g +1 ) . Since, by assumption, we have e j = 0 for every gap j of S with 2 ≤ j < g , the series expansion of Q g − j =m( S ) (1 − x j ) e j is of the form P s ∈ S a s x s , and so in particular the coefficient of x g is zero. Oncomparing the coefficients of x g in both sides of (18), we obtain 0 = − e g , a contradiction. (cid:3) Lemma 4.3. For every n ∈ A we have e n = − . Proof. Let n be a minimal generator of S . By considering H S ( x ) modulo x n +1 , we have(19) H S ( x ) ≡ (1 − x n ) e n n − Y j =m( S ) (1 − x j ) e j ≡ − e n x n + n − Y j =m( S ) (1 − x j ) e j (mod x n +1 ) . By Lemma 4.2, the series expansion of Q n − j =m( S ) (1 − x j ) e j is of the form P s ∈ S a s x s . Since n cannotbe written as a sum of two or more non-zero elements of S , we have a n = 0. On comparing thecoefficients of x n in both sides of (18), we obtain 1 = − e n . (cid:3) Remark . Let S be a numerical semigroup minimally generated by A , and let e be its cyclotomicexponent sequence. By Lemmas 4.1, 4.2 and 4.3, we have { } ∪ A ⊆ { j ∈ N : e j = 0 } and { j ∈ N : e j = 0 , j ≥ } ⊆ S . Hence, E ( S ) = { j ∈ N : e j = 0 , j ≥ } \ A ⊆ S. In light of (11), we obtain(20) H S ( x ) = X s ∈ S d ( s ) x s Y d ∈E ( S ) (1 − x d ) e d . Lemma 4.5. For every d ∈ E ( S ) , we have d ( d ) ≥ . Moreover, if α ∈ Minimals ≤ S E ( S ) , then e α = d ( α ) − .Proof. Let d ∈ E ( S ) and α ∈ Minimals ≤ S E ( S ) with α ≤ S d . By comparing the coefficients of x α inboth sides of (20) we find that 1 = d ( α ) − e α . Since 1 ≤ d ( α ) and e α = 0, we obtain e α > 0. Inparticular, we find that d ( α ) = 1 + e α ≥ d ( d ) ≥ (cid:3) YCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS 13 Proof of Theorem 1.1. The first statement in Lemma 4.5 is equivalent to the fact that e j = 0 forevery j ∈ S \ A with d ( j ) = 1. Theorem 1.1 now follows by combining Lemmas 4.1–4.3 and 4.5. (cid:3) We illustrate this theorem with an example. Example . Let S = h , , i be the numerical semigroup considered in Example 3.7b. We haveG( S ) = { , , } and Betti( S ) = { , , } . Moreover, the elements of S that are not minimalgenerators, but yet have only one factorization, are 6 , , , 11. Thus we have e j = 0 for j ∈{ , , , , , } , which is consistent with the entries of the exponent sequence given in Example 3.7b.5. Cyclotomic exponent sequences and Betti elements The goal of this section is to prove Theorem 1.2. This result relates the partially ordered sets(Betti( S ) , ≤ S ) and ( E ( S ) , ≤ S ). Let ( X, ≤ ) be a partially ordered set. In the introduction we definedthe set U( X ) as U( X ) = { x ∈ X : ↓ x is totally ordered } , where ↓ x = { y ∈ X : y ≤ x } . Note that Minimals ≤ X is a subset of U( X ). Recall that Theorem 1.2states that U(Betti( S )) = U( E ( S )), and that it also determines the cyclotomic exponents of theseBetti elements.The proof of Theorem 1.2 is carried out by induction and is rather technical and elaborate. Forthe benefit of the reader, it is divided into several parts. We start by showing, in Section 5.1,that Minimals ≤ s Betti( S ) = Minimals ≤ S E ( S ). The proof of this result is simple and elegant, andserves as motivation and warm-up for the work carried out in this section. Moreover, the fact thatMinimals ≤ s Betti( S ) = Minimals ≤ S E ( S ) is used in the base case of the induction of the proof ofTheorem 1.2. In Section 5.2 we explain the main idea behind this induction. In Section 5.3 we studythe graph ∇ b of factorizations of the elements b ∈ U(Betti( S )) and develop the technical resultsthat we need to tackle Theorem 1.2. Finally, in Section 5.4 we use our insights on ∇ b to completethe proof of Theorem 1.2.5.1. Minimal Betti elements. In this section we relate the Betti-minimal elements of a numericalsemigroup to its cyclotomic exponent sequence. The following result was established in [7]. Lemma 5.1 ([7, Lemma 15]) . If S is a cyclotomic numerical semigroup such that { d : e d < } isits minimal system of generators and max { d : e d < } < min E ( S ) , then min E ( S ) = min Betti( S ) . In order to prove [7, Lemma 15] the authors show that min E ( S ) is the smallest element of S having at least two representations in terms of the minimal generators of S . Here we reach a muchmore general conclusion without any assumptions on S .Let m and b be two positive integers. The Taylor series centered at 0 of the complex function(1 − z b ) − m is given by(21) (1 − z b ) − m = ∞ X j =0 (cid:18) m + j − j (cid:19) z jb = (cid:18) ∞ X j =0 z jb (cid:19) m and its radius of convergence is 1. Therefore, the expression (20) for H S ( x ) can be rewritten as(22) H S ( x ) = X s ∈ S d ( s ) x s Y d ∈E ( S ) e d < ∞ X j =0 (cid:18) − e d + j − j (cid:19) x jd Y d ∈E ( S ) e d > e d X j =1 (cid:18) e d j (cid:19) ( − j x jd . Proposition 5.2. Let S be a numerical semigroup. For every s ∈ S with d ( s ) ≥ , there exists d ∈ E ( S ) such that d ≤ S s .Proof. From (22) we infer that if there is no d ∈ E ( S ) with d ≤ S s for some s ∈ S , then d ( s ) = 1. (cid:3) Theorem 5.3. Let S be a numerical semigroup. Then Minimals ≤ S Betti( S ) = Minimals ≤ S E ( S ) .Moreover, we have e α = d ( α ) − α ) − for every α ∈ Minimals ≤ S E ( S ) .Proof. Let β be a Betti element minimal with respect to ≤ S . Then d ( β ) ≥ α ∈ Minimals ≤ S E ( S ) such that α ≤ S β . Since d ( α ) ≥ β ′ such that β ′ ≤ S α . The minimality of β forces β ′ = α = β .Hence, we have β ∈ Minimals ≤ S E ( S ). The other inclusion is proved similarly. The value e α is foundby invoking Lemma 4.5 and Proposition 2.8. (cid:3) Proof idea of Theorem 1.2. In this section we describe our approach to proving Theorem 1.2.We focus on the inclusion U( E ( S )) ⊆ U(Betti( S )).Let η ∈ U( E ( S )) and Λ = ↓ η . We want to show that η ∈ U(Betti( S )). Since η ∈ U( E ( S )), we canwrite Λ = { b ≤ S . . . ≤ S b l } with b l = η . Note that b ∈ Minimals ≤ S E ( S ), so b is Betti-minimalby Theorem 5.3. Let i be an integer with 2 ≤ i ≤ l . We define(23) X s ∈ S r i − ( s ) x s = H S ( x ) i − Y j =1 (1 − x b j ) − e bj . In light of (20), we have X s ∈ S r i − ( s ) x s = X s ∈ S d ( s ) x s Y d ∈E ( S ) \ Λ i − (1 − x d ) e d . We can rewrite this expression as(24) X s ∈ S r i − ( s ) x s = X s ∈ S d ( s ) x s X k ∈hE ( S ) \ Λ i − i c k x k , for certain coefficients c k with c = 1 and c d = − e d for any d ∈ Minimals ≤ S ( E ( S ) \ Λ i − ). Therefore,we have(25) r i − ( s ) = d ( s ) when { d ∈ E ( S ) : d ≤ S s } ⊆ Λ i − ,r i − ( s ) = d ( s ) − e s when s ∈ Minimals ≤ S ( E ( S ) \ Λ i − ) . Most of our work is devoted to finding an algebraic interpretation of the coefficients r i − ( s ).Assuming that b , . . . , b i − ∈ U(Betti( S )), we want to show by induction on i that, when s ∈ Minimals ≤ S ( E ( S ) \ Λ i − ), we have 0 < r i − ( s ) and r i − ( s ) corresponds to the number of factoriza-tions of s that are not isolated. Combining this assertion with (25) for s = b i , we will conclude thatin this situation e b i is the number of isolated factorizations of b i , so b i is a Betti element (recall herethat d ( b i ) ≥ X s ∈ S r i ( s ) x s = (1 − x b i ) − e bi X s ∈ S r i − ( s ) x s can be rewritten as X s ∈ S r i ( s ) x s = ∞ X j =0 (cid:18) i( b i ) + j − j (cid:19) X s ∈ S r i − ( s ) x s , where we use e b i = i( b i ) and (21). Hence, for any s ∈ S , we obtain(26) r i ( s ) = q s X j =0 r i − ( s − jb i ) (cid:18) i( b i ) + j − j (cid:19) , where q s is the largest integer such that s − q s b i ∈ S .In order to study the connection between the coefficient r i ( s ) and the factorizations of s , weintroduce Betti restricted factorizations in Section 5.3, and we show that the number of Bettirestricted factorizations of certain elements satisfies a similar recursion to (26). In order to be able YCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS 15 to fill in the details of this proof, we will also need to understand the graph of factorizations ∇ b ofany b ∈ U(Betti( S )).5.3. Betti restricted factorizations and U(Betti( S )) . The main goal of this subsection is toprove Theorem 5.9, which describes the graph of factorizations ∇ b of any b ∈ U(Betti( S )), and todevelop the machinery that we need to complete the proof presented in Section 5.2. To this end,we introduce the concept of Betti restricted factorizations and we study their properties.Let S be a numerical semigroup and let Λ ⊆ IBetti( S ). We define the set of Betti restrictedfactorizations of an element s ∈ S with respect to Λ byB( s ; Λ) = { w + x + · · · + x l ∈ Z( s ) : w ∈ I s ( S ) , l ≥ x , . . . , x l ∈ I(Λ) } . If Λ = { b } , then we use the notation B( s ; b ) = B( s ; Λ). Note that B( s ; ∅ ) = ∅ when s has at leasttwo factorizations. Furthermore, if s only has one factorization, then B( s ; Λ) = Z( s ). Anothertrivial observation is that if Λ ⊆ Λ , then B( s ; Λ ) ⊆ B( s ; Λ ).Betti restricted factorizations allow us to obtain some information about the number of isolatedfactorizations of certain Betti elements, and play an essential role in the proof of Theorem 1.2. Firstwe need the following result, which shows that B( s ; Λ) and I b ( S ) are closely related. Lemma 5.4. Let S be a numerical semigroup. Let Λ ⊆ IBetti( S ) and s ∈ S \ Λ . Then B( s ; Λ) ⊆ Z( s ) \ I b ( S ) . Moreover, if { b ∈ IBetti( S ) : b < S s } ⊆ Λ , then B( s ; Λ) = Z( s ) \ I b ( S ) .Proof. If d ( s ) = 1, then Z( s ) ∩ I b ( S ) = ∅ and the result follows, thus we may assume that d ( s ) ≥ z ∈ B( s ; Λ). There is x ∈ I(Λ) such that x < z . By Lemma 2.7, z is not isolated, that is, z ∈ Z( s ) \ I b ( S ). Finally, if { b ∈ IBetti( S ) : b < S s } ⊆ Λ, then Lemma 2.7 asserts that every z ∈ Z( s ) \ I b ( S ) can be expressed as an element of B( s ; Λ). (cid:3) If s has at least two factorizations and B( s ; Λ) = Z( s ) \ I b ( S ), then we find that the number ofconnected components of ∇ s equals i( s ) plus the number of R -classes of B( s ; Λ), where i( s ) is thenumber of isolated factorizations of s . Hence, in this context, B( s ; Λ) is connected in ∇ s if and onlyif i( s ) = nc( ∇ s ) − 1. This observation plays an important role in the proof of Theorem 5.9. First,we prove that B( s ; b ) is always connected. Lemma 5.5. Let S be a numerical semigroup and let b ∈ IBetti( S ) . For every s ∈ S write s = ω s + q s b , with ω s ∈ Ap( S ; b ) and q s ∈ N . a) If Z( ω s ) = { w } , then B( s ; b ) = { w + x + · · · + x q s : x , . . . , x q s ∈ I( b ) } . b) If d ( ω s ) ≥ , then B( s ; b ) = ∅ .Proof. Let us assume that B( s ; b ) = ∅ . Let w + x + · · · + x l ∈ B( s ; b ) with w ∈ I s ( S ) and x , . . . , x l ∈ I( b ). By Corollary 2.9, we have ϕ ( w ) ∈ Ap( S ; b ), and thus s = ϕ ( w ) + lb . Hence, weobtain ϕ ( w ) = ω s and q s = l . Since w ∈ I s ( S ), we infer that Z( ω s ) = { w } andB( s ; b ) = { w + x + · · · + x q s : x , . . . , x q s ∈ I( b ) } . Consequently, if d ( ω s ) ≥ 2, then B( s ; b ) = ∅ . Finally, if Z( ω s ) = { w } , then we have w + q s x ∈ B( s ; b )for any x ∈ I( b ), which implies that B( s ; b ) is not empty, and the result follows. (cid:3) The base case of the induction presented in Section 5.2 requires us to prove that r ( s ) = | B( s, b ) | for some elements s . This identity follows from the following corollary. Corollary 5.6. Let S be a numerical semigroup and let b ∈ IBetti( S ) . For every s ∈ S we have | B( s ; b ) | = ((cid:0) i( b )+ q s − q s (cid:1) if d ( ω s ) = 1;0 otherwise , where q s and ω s are as in Lemma 5.5. Proof. The result is a consequence of Lemma 5.5. Let s ∈ S . If d ( ω s ) ≥ 2, then | B( s ; b ) | = 0.Otherwise, since the isolated factorizations of b are disjoint, we find that every element of B( s ; b )is uniquely determined by q s isolated factorizations of b . The proof is completed by counting thenumber of combinations with repetitions of size q s from a set with i( b ) elements. (cid:3) We will use the following observation several times in the proof of Lemma 5.8, which showsconnectivity of B( s ; Λ) in ∇ s under some hypotheses. Lemma 5.7. Let S be a numerical semigroup and let b be a Betti-minimal element of S . Let Λ ⊆ IBetti( S ) with b ∈ Λ . For each s ∈ S such that b is the only Betti-minimal element b of S with b ≤ S s , and for each x ∈ I(Λ) such that ϕ ( x ) ≤ S s , there exists z ∈ B( s ; Λ) with x < z .Proof. Let s ∈ S be such that b is the only Betti-minimal element of S below s with respect to ≤ S ,and let x ∈ I(Λ) such that ϕ ( x ) ≤ S s . Set b = ϕ ( x ). Write s − b = ω + qb , where ω ∈ Ap( S ; b )and q ∈ N . By Corollary 2.9, ω has only one factorization. We can choose z = w + x + qy ∈ B( s ; Λ),where Z( ω ) = { w } and y ∈ I( b ). Here we have used that Betti-minimal elements have isolatedfactorizations by Proposition 2.8. (cid:3) Recall that U(Betti( S )) is the set of b ∈ Betti( S ) such that ↓ b = { b ′ ∈ Betti( S ) : b ′ ≤ S b } istotally ordered. In Lemma 5.8 we establish connectivity of B( s ; Λ) under some assumptions. Thisis the main ingredient of the proof of Theorem 5.9. Lemma 5.8. Let S be a numerical semigroup. Let u ∈ U(Betti( S )) and let Λ = ↓ u . If s ∈ S \ Λ issuch that u ≤ S b for all b ∈ Betti( S ) \ Λ with b ≤ S s , then B( s ; Λ) is connected in ∇ s .Proof. Assume that Λ = { b < S · · · < S b l } , and so u = b l . Write Λ i = ↓ b i = { b < S · · · < S b i } , for i ∈ { , . . . , l } . We proceed by induction on l , the size of Λ. Note that, by definition of ↓ u , b isBetti-minimal and the only Betti-minimal element with b ≤ S u . Let s ∈ S \ Λ be such that u ≤ S b for every b ∈ Betti( S ) \ Λ with b ≤ S s . If b ∈ Betti( S ) with b ≤ S s , then either b ∈ Λ and b ≤ S b ,or b Λ and b ≤ S u ≤ S b . In any case, we have shown that b ≤ S b for every b ∈ Betti( S ) with b ≤ S s . Hence, either d ( s ) = 1 or b is the only Betti-minimal element with b ≤ S s . We will usethis fact in our induction.First, we study the case l = 1. Note that either d ( s ) = 1 or b is the only minimal element of S with b ≤ S s . In the first case, we have B( s ; b ) = Z( s ). In the second case, Lemma 5.7, withΛ = { b } , yields that B( s ; b ) is non-empty. Its connectivity follows from Lemma 5.5.If l ≥ 2, let us assume that the result holds for l − 1. If B( s ; Λ) = B( s ; Λ l − ), then we are doneby the induction hypothesis. Let us consider the case B( s ; Λ) = B( s ; Λ l − ). In this case we have b l ≤ S s , so b is the only minimal Betti element of S with b ≤ S s . There are two cases dependingon the number of factorizations of s − b l . Case 1: d ( s − b l ) ≥ 2. Notice that, under this assumption, b is the only minimal Betti elementof S with b ≤ S s − b l . Let z ∈ B( s ; Λ) \ B( s ; Λ l − ). There is y ∈ I( b l ) such that y < z . Let x ∈ I( b ). Since b ≤ S s − b l , Lemma 5.7 provides us with z ∈ B( s − b l ; Λ) such that x < z . Thus,we have z = z + y ∈ B( s ; Λ l ) and z · z = 0. Moreover, there is z ∈ B( s ; b ) ⊆ B( s ; Λ l − ) with x < z (Lemma 5.5) and, in particular, z · z = 0. From the arbitrary choice of z and the factthat B( s ; Λ l − ) is connected, it follows that B( s ; Λ) is also connected. Case 2: d ( s − b l ) = 1. Set ω = s − b l , and let w be the unique factorization of ω . If z ∈ B( s ; Λ) \ B( s ; Λ l − ), then there is y ∈ I( b l ) with y < z . Note that z − y is a factorization of ω ,whence z = w + y . That is, we have shown thatB( s ; Λ) \ B( s ; Λ l − ) ⊆ { w + y : y ∈ I( b l ) } . Let us suppose that B( s ; Λ) has at least two R -classes in order to obtain a contradiction. SinceB( s ; Λ l − ) and B( s ; Λ) \ B( s ; Λ l − ) are connected, the only option is z · y = 0 for every z ∈ B( s ; Λ l − )and y ∈ B( s ; Λ) \ B( s ; Λ l − ). Write b l = ω + qb with ω ∈ Ap( S ; b ) and q ∈ N (this implies YCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS 17 Z( ω ) = { w } by Corollary 2.9). We have s = ω + ω + qb . There are two possible subcases, eachof which yields a contradiction. Subcase 2.1: d ( ω + ω ) = 1. The factorization z = w + w + qx is in B( s ; Λ) for any x ∈ I( b ),but it is not disjoint with any element of B( s ; Λ) \ B( s ; Λ l − ), a contradiction. Subcase 2.2: d ( ω + ω ) ≥ 2. In light of Lemma 2.7, there exist b ∈ Betti( S ) and y ∈ I( b )such that y ≤ w + w . For all x ∈ I(Λ l − ) we have ϕ ( x ) ≤ S b l ≤ S s and there is z ∈ B( s ; Λ l − )with x < z (Lemma 5.7). Under the standing assumption, z · w = 0. In particular, we have w · x = 0. It follows that the factorization w is disjoint with any isolated factorization of theBetti elements b , . . . , b l − . Since y ≤ w + w , y ∈ I( b ) and w , w ∈ I s ( S ), we have w · y = 0 and w · y = 0. Hence, it follows that y is not an isolated factorization of any of the elements b , . . . , b l − ,so b Λ l − . Since b ≤ S ω + ω ≤ S s , either b ∈ Λ or b l ≤ S b by hypothesis. We conclude that b l ≤ S b ≤ S ω + ω . Write ω + ω = ω + pb + b l with ω ∈ Ap( S ; b ) and p ≥ 0. Since s = ω + b l and s = ω + ω + qb = ω + ( p + q ) b + b l , we obtain ω = s − b l = ω + ( p + q ) b . Recall that d ( ω ) = 1. This forces p = 0 = q , a contradiction because b l = ω + qb and d ( b l ) ≥ (cid:3) We now have the ingredients necessary to prove Theorem 5.9, which determines the number ofisolated factorizations of any b ∈ U(Betti( S )). Theorem 5.9. Let S be a numerical semigroup and let b ∈ U(Betti( S )) . Then either b is minimaland all its factorizations are isolated, or the number of isolated factorizations of b equals its numberof R -classes minus .Proof. Let ↓ b = { b < S · · · < S b l } ⊆ Betti( S ). If l = 1, then b is Betti-minimal and its factorizationsare isolated (see Proposition 2.8). Otherwise, we apply Lemma 5.8 to b , . . . , b l , and conclude thatB( b ; Λ) is connected for Λ = { b , . . . , b l − } . By Lemma 5.4, we find that Z( b ) = B( b ; Λ) ∪ I( b ) , thusthe number of isolated factorizations of b is one less than the number of R -classes. (cid:3) Corollary 5.10. Let S a numerical semigroup. Write Betti( S ) = { b < b < · · · < b k } . Let usassume that k ≥ . a) If b − b S , then b is Betti-minimal; that is, all its factorizations are isolated. b) If b ≤ S b , then b has nc( ∇ b ) − isolated factorizations.Proof. This is a direct consequence of Theorem 5.9. (cid:3) Theorem 5.9 gives us some information about the smallest Betti elements. Let b = min Betti( S ).It is clear that b is Betti-minimal. Let us assume that there exists b = min(Betti( S ) \ { b } ). Theneither b is Betti-minimal ( b − b S ), or b ≤ S b . In the latter case we can apply Theorem 5.9to conclude that i( b ) = nc( ∇ b ) − 1. Therefore, b always has isolated factorizations. Example . Let S = h , , , , i . Then the Hasse diagram of (Betti( S ) , ≤ S ) looks asfollows: 48325730 363534Hence the minimal elements of Betti( S ) are 30 , , , , 36. The factorizations of these elementsare all isolated. Theorem 5.9 allows us to conclude that 48 has isolated factorizations and thatnc( ∇ ) = i(48) + 1. Note that this result does not provide information about the factorizations of57 ( ↓ 57 = { , , } ). In fact, one can check that 57 has an isolated factorization with the helpof the GAP package numericalsgps . Next, we want to give an expression for B( s ; Λ ∪ { b } ) in terms of B( s − jb ; Λ) , for suitable j, whichis meant to be of the same shape as that given in recursion (26). Lemma 5.12. Let S be a numerical semigroup. Let Λ ⊆ IBetti( S ) with Λ = ∅ and b ∈ IBetti( S ) \ Λ .Then for every s ∈ S we have B( s ; Λ ∪ { b } ) = q s [ j =0 B( s − jb ; Λ) + j X i =1 I( b ) ! , where q s is the largest integer such that q s b ≤ S s . In particular, | B( s ; Λ ∪ { b } ) | ≤ q s X j =0 | B( s − jb ; Λ) | (cid:18) i( b ) + j − j (cid:19) . Proof. It is clear that S q s j =0 (cid:16) B( s − jb ; Λ) + P ji =1 I( b ) (cid:17) ⊆ B( s ; Λ ∪ { b } ). Let z ∈ B( s ; Λ ∪ { b } ). If z B( s ; Λ), then there is y ∈ I( b ) such that y ≤ z and z − y ∈ B( s − b ; Λ ∪ { b } ). We can repeatthis argument a finite number of times until we find y , . . . , y j ∈ I( b ) and x ∈ B( s − jb ; Λ) such that z = x + y + · · · + y j . Hence, we obtain z ∈ B( s − jb ; Λ) + P ji =1 I( b ). (cid:3) Example . Let us consider the numerical semigroup S = h , , i . We have Betti( S ) = { , } ,Z(10) = { (1 , , , (0 , , } and Z(12) = { (3 , , , (0 , , } . Since 72 = 6 · 12, by Lemmas 5.4 and5.12 we find that Z(72) = B(72; { , } ) = [ j =0 B((6 − j )12; 10) + j X i =1 I(12) ! . Note that (6 , , ∈ Z(72) and(6 , , 8) = 6(1 , , 1) + (0 , , 2) = 2(3 , , 0) + 4(0 , , . This union is therefore not disjoint and the inequality given in Lemma 5.12 can be strict.The following lemma shows that, under the hypotheses of Theorem 5.9, the upper bound givenin Lemma 5.12 can be attained. Note that this recurrent expression has already arisen in (26). Lemma 5.14. Let S be a numerical semigroup. Let u ∈ U(Betti( S )) and let Λ = ↓ u . Then,for every s ∈ S and z ∈ B( s ; Λ) , there are unique w ∈ I s ( S ) and x , . . . , x t ∈ I(Λ) such that z = w + x + · · · + x t . Moreover, we have | B( s ; Λ) | = q s X j =0 | B( s − ju ; Λ \ { u } ) | (cid:18) i( u ) + j − j (cid:19) , where q s is the largest integer such that q s u ≤ S s .Proof. By Theorem 5.9, we have Λ ⊆ IBetti( S ). Let s ∈ S . If B( s ; Λ) = ∅ , then we are done. Letus assume that B( s ; Λ) = ∅ and let z ∈ B( s ; Λ). The definition of B( s ; Λ) ensures the existence of w ∈ I s ( S ) and x , . . . , x t ∈ I(Λ) such that z = w + x + · · · + x t . We show that this expression isunique. Let w , w ∈ I s ( S ) and, for each x ∈ I(Λ), let p x and q x be non-negative integers such that z = w + X x ∈ I(Λ) p x x = w + X x ∈ I(Λ) q x x. In light of Lemma 2.10, the supports of the elements of I(Λ) are disjoint. If there is x ∈ I(Λ)such that p x = q x , then either x < w , or x < w , contradicting the fact that w , w ∈ I s ( S ), seeLemma 2.7. Therefore we have p x = q x for every x ∈ Λ and w = w . YCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS 19 As a consequence, the union given in Lemma 5.12 is disjoint. Moreover, we have (cid:12)(cid:12)(cid:12) B( s − ju ; Λ \ { u } ) + X ji =1 I( u ) (cid:12)(cid:12)(cid:12) = | B( s − ju ; Λ \ { u } ) | (cid:18) i( u ) + j − j (cid:19) and the result follows. (cid:3) Completing the proof of Theorem 1.2. We now have all the ingredients necessary tocomplete the proof of Theorem 1.2. Proof of Theorem 1.2. First, we prove that if η ∈ U( E ( S )), then η ∈ U(Betti( S )) and e η is as inthe statement. The tools developed in this part of the proof will be helpful when proving the otherinclusion. As we follow the proof idea explained in Section 5.2, we recommend the reader to havea look at Section 5.2 before reading this proof. Let Λ = { d ∈ E ( S ) : d ≤ S η } . Since η ∈ U( E ( S )),we can write Λ = { b < S · · · < S b l } and b l = η . For each i ∈ { , . . . , l } , define Λ i = { b , . . . , b i } and r i − ( s ) as in (23), identity which we recall below for the convenience of the reader: X s ∈ S r i − ( s ) x s = H S ( x ) i − Y j =1 (1 − x b j ) − e bj . Note that from our hypothesis it follows that b is minimal in E ( S ). By Theorem 5.3, b is Betti-minimal. We prove by induction on i ∈ { , . . . , l } the following assertions:(a) if b ≤ S b i for some b ∈ Betti( S ), then b ∈ Λ i ;(b) b i ∈ U(Betti( S )) and e b i > r i ( s ) = | B( s ; Λ i ) | for every s ∈ S such that b is the only Betti-minimal element with b ≤ S s. First, we study the case i = 1. As a consequence of Theorem 5.3, b Betti-minimal and i( b ) = d ( b ) = e b + 1 ≥ 2. In particular, (a) and (b) hold for i = 1. Note that, by (6), we haveH S ( x ) = P ω ∈ Ap( S ; b ) x ω P ∞ j =0 x jb . In conjunction with (21) and i( b ) = e b + 1, this yields(27) ∞ X s =0 r ( s ) x s = X ω ∈ Ap( S ; b ) x ω (cid:18) ∞ X j =0 x jb (cid:19) i( b ) = X ω ∈ Ap( S ; b ) x ω ∞ X j =0 (cid:18) i( b ) + j − j (cid:19) x jb = X s ∈ S (cid:18) i( b ) + q s − q s (cid:19) x s , where q s is the unique non-negative integer such that s − q s b ∈ Ap( S ; b ). Let s ∈ S be suchthat b is the only Betti-minimal element with b ≤ S s . Then from Corollary 2.9 it follows that ω s = s − q s b has only one factorization. By combining Corollary 5.6 and (27), we conclude that r ( s ) = | B( s ; Λ ) | , as desired.Now assume that (a), (b) and (c) hold for i − ∈ { , . . . , l − } and let us prove that they alsohold for i . We prove each of the induction hypotheses for i separately. In doing so, we will use theidentities (25) several times, which, for the sake of readability, we also recall here: r i − ( s ) = d ( s ) when { d ∈ E ( S ) : d ≤ S s } ⊆ Λ i − ,r i − ( s ) = d ( s ) − e s when s ∈ Minimals ≤ S ( E ( S ) \ Λ i − ) . (a) We proceed by deriving a contradiction. Let us assume that there is b ∈ Betti( S ) \ Λ i − suchthat b < S b i . Then there exists an element β ∈ Minimals ≤ S { b ∈ Betti( S ) \ Λ i − : b < S b i } . Let D = { d ∈ Betti( S ) : d < S β } . Since β < S b i , from minimality in the choice of β it follows that D ⊆ Λ i − . If D = ∅ , then β is Betti-minimal and, by Theorem 5.3, β ∈ E ( S ). We thus obtain β ∈ { d ∈ E ( S ) : d < S b i } = Λ i − , but β Λ i − by definition, a contradiction. We conclude that ∅ 6 = D ⊆ Λ i − . Hence, we have b ∈ d , so b ≤ S β < S b i and b is the only Betti-minimal elementthat satisfies b ≤ S β . From our induction hypothesis, we obtain | B( β ; Λ i − ) | = r i − ( β ). Note that { d ∈ E ( S ) : d ≤ S β } ⊆ { d ∈ E ( S ) : d < S b i } = Λ i − . Hence, by (25), we find that r i − ( β ) = d ( β ),so | B( β ; Λ i − ) | = d ( β ). We can apply Lemma 5.8 with u = b i − and s = β , finding that B( β ; Λ i − )is connected in ∇ β . But we have shown that | B( b ; Λ i − ) | = d ( b ) or, equivalently, Z( β ) = B( β ; Λ i − ).This contradicts the fact that β ∈ Betti( S ).(b) From Lemma 5.4 and (a) it follows that Z( b i ) \ I b ( b i ), so | B( b i ; Λ i − ) | = d ( b i ) − i( b i ). Moreover, byour hypothesis we have | B( b i ; Λ i − ) | = r i − ( b i ). Note that b i ∈ Minimals ≤ S ( E ( S ) \ Λ i − ) by definitionof Λ i − . Hence, by (25) we obtain r i − ( b i ) = d ( b i ) − e b i . We conclude that d ( b i ) − i( b i ) = d ( b i ) − e b i ,that is, 0 ≤ i( b i ) = e b i . Since e b i = 0, we have i( b i ) ≥ b i ∈ Betti( S ) because b i has at leasttwo factorizations (Corollary 2.9). In view of (a), we have { b ∈ Betti( S ) : b ≤ S b i } = Λ i , which byhypothesis is totally ordered, so b ∈ U(Betti( S )).(c) Let s ∈ S such that b is the only Betti-minimal element with b ≤ S s . Note that thanks to (b)we can apply Lemma 5.14 with u = b i . Recall that in (26) we showed that r i ( s ) = q s X j =0 r i − ( s − jb i ) (cid:18) i( b i ) + j − j (cid:19) , where q s is the largest integer such that s − q s b i ∈ S . This equation in combination with Lemma 5.14yields r i ( s ) = q s X j =0 | B( s − jb i ; Λ i − ) | (cid:18) i( b i ) + j − j (cid:19) = | B( s ; Λ i ) | , which finishes the proof by induction.In the induction we have also shown that e b i = i( b i ), see the proof of our hypothesis (b). Since b i ∈ U(Betti( S )), by Theorem 5.9 we have nc( ∇ b i ) = i( b i ) + 1 = e b i + 1. The fact that e b =nc( ∇ b ) − S )) ⊆ U( E ( S )). Let u ∈ U(Betti( S )). Let us write ↓ u = { b ∈ Betti( S ) : b ≤ S u } = { b < S · · · < S b l = u } . We prove by induction on i that b i ∈ U( E ( S )).Note that b is Betti-minimal and, thus, b ∈ U( E ( S )) by Theorem 5.3. Let us assume that b , . . . , b i − ∈ U( E ( S )) and let us prove that b i ∈ U( E ( S )). Let Λ i − = { b , . . . , b i − } . We consider r i − ( s ) as in (23). In light of the induction hypothesis (c), we have r i − ( s ) = | B( s, Λ i − ) | for every s ∈ S such that b is the only Betti-minimal element with b ≤ S s . In particular, we have r i − ( b i ) = | B( b i , Λ i − ) | and, thus, r i − ( b i ) = d ( b i ) − i( b i ), where we used Lemma 5.4. From Theorem 5.9, wefind that i( b i ) > 0. Therefore, r i − ( b i ) < d ( b i ) and, by (24), there exists d ∈ E ( S ) \ Λ i − with d ≤ S b i .Hence, there is α ∈ Minimals ≤ S { d ∈ E ( S ) \ Λ i − } with α ≤ S b i . By (25) we have r i − ( α ) = d ( α ) − e α .From the induction hypothesis (c), we obtain r i − ( α ) = | B( α, Λ i − ) | . Since 0 < r ( s ) ≤ r i − ( s )by (27) and (26), we have B( α, Λ i − ) = ∅ . In view of Lemma 5.4, B( α ; Λ i − ) = Z( α ) \ I b ( α ), so1 ≤ r i − ( α ) = d ( α ) − i( α ). We find that i( α ) = e α = 0. We have 0 = e α = i( α ). We conclude that α is a Betti element with α ≤ S b i . Since α Λ i − by definition, we must have b i = α ∈ E ( S ) and { d ∈ E ( S ) : d ≤ S b i } = { b , . . . , b i } . We obtain b i ∈ U( E ( S )) , as wanted. (cid:3) Example . Here we can see Theorem 1.2 in action for a couple of numerical semigroups.a) We consider again the semigroup from Example 5.11. Let S = h , , , , i . Recall thatthe Hasse diagram of (Betti( S ) , ≤ S ) looks as follows: YCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS 21 S ) are 30 , , , 35 and 36. The set U(Betti( S )) consists ofthese Betti-minimal elements and the Betti element 48. Therefore, by Theorem 1.2, we concludethat U( E ( S )) = { , , , , , } and we can determine the exponents e s of these elementsfrom their number of isolated factorizations.b) An interesting application is finding Betti elements from E ( S ). Let us consider the semigroup S = h , , i of Example 3.7b. We gave the first entries of the cyclotomic exponent sequenceof S . The smallest elements of E ( S ) are 10 , , , , , . . . . Note that U( E ( S )) = { , , } since any other element in E ( S ) can be written as α + 3 j for some α ∈ { , , } and j ≥ S )) = { , , } = Minimals ≤ S Betti( S ), and each one of theseelements has only two factorizations (their cyclotomic exponents are 1).6. Betti-sorted and Betti-divisible numerical semigroups In this section we prove Theorem 1.3, which characterizes Betti-sorted and Betti-divisible nu-merical semigroups in terms of their cyclotomic exponent sequences. Recall that S is Betti-sortedif Betti( S ) is totally ordered by ≤ S , and that S is Betti-divisible if Betti( S ) is totally ordered bythe divisibility order in N . Our characterizations are consequences of Theorem 1.2. We will use thefollowing result on ordered sets. Lemma 6.1. Let ( X, ≤ ) be an ordered set. Then X is totally ordered if and only if U( X ) is totallyordered.Proof. First, if X is totally ordered, then any subset of X , and in particular U( X ), is totallyordered. Now let us assume that U( X ) is totally ordered. Suppose U( X ) = X in order to obtaina contradiction. Then we can choose α ∈ Minimals ≤ ( X \ U( X )). We have { a ∈ X : a < α } ⊆ U( X ) by minimality of α . Thus ↓ α is of the form { a < · · · < a k < α } for some k ≥ a , . . . , a k ∈ U( X ). We conclude that α ∈ U( X ), a contradiction. Therefore, X = U( X ) and X istotally ordered. (cid:3) Lemma 6.2. Let S be a numerical semigroup. Then S is Betti-sorted if and only if E ( S ) is totallyordered by ≤ S . Moreover, if this is the case, then Betti( S ) = E ( S ) .Proof. In view of Theorem 1.2 and Lemma 6.1, S is Betti-sorted if and only if U(Betti( S )) = U( E ( S ))is totally ordered by ≤ S or, equivalently, E ( S ) is totally ordered by ≤ S . (cid:3) This gives the following alternative proof of the fact that Betti-sorted numerical semigroups arecomplete intersections. For the original proof we refer to [10], where in fact the authors show thestronger result that Betti-sorted numerical semigroups are free. Corollary 6.3. If S is a Betti-sorted numerical semigroup, then S is a complete intersection.Proof. In view of Lemma 6.2, we have E ( S ) = U( E ( S )). By applying Theorem 1.2 we find that e b = nc( ∇ b ) − b ∈ Betti( S ) = E ( S ). Let A be the minimal system of generators of S .With the help of Theorem 1.1, we conclude thatH S ( x ) = Q b ∈ Betti( S ) (1 − x b ) nc( ∇ b ) − Q n ∈ A (1 − x n ) . Therefore, S is a complete intersection by Proposition 2.5. (cid:3) Lemma 6.4. Let S be a numerical semigroup. Then S is Betti-divisible if and only if E ( S ) is totallyordered by the divisibility order.Proof. Let S be a numerical semigroup such that either Betti( S ) or E ( S ) is totally ordered by thedivisibility order. Then, by Lemma 6.2, S is Betti-sorted and Betti( S ) = E ( S ). It follows thatBetti( S ) and E ( S ) are totally ordered by the divisibility order. (cid:3) Betti-divisible numerical semigroups are rare, but they have a very rich structure. In fact, itcan be shown that these are the numerical semigroups that are free for any arrangement of theirminimal generators, see [10, Theorem 7.10]. Lemma 6.5. Let S be a numerical semigroup minimally generated by A . Then S has a uniqueBetti element if and only if E ( S ) is a singleton.Proof. Let S be a numerical semigroup such that Betti( S ) or E ( S ) is a singleton. Then S is Betti-sorted by Lemma 6.2 and Betti( S ) = E ( S ), so both Betti( S ) and E ( S ) are singletons. (cid:3) Theorem 1.3 now follows by combining Lemmas 6.2, 6.4 and 6.5.7. Applications to cyclotomic numerical semigroups and open questions We can now use our freshly enriched insight on the connections between cyclotomic exponentsequences and Betti elements to prove that certain cyclotomic numerical semigroups are completeintersections. We do so by showing that these numerical semigroups satisfy the hypotheses ofProposition 2.5 and are, as such, complete intersections. This approach has already been carried outin Corollary 6.3, where we showed that Betti-sorted numerical semigroups are complete intersections.In fact, here we extend Corollary 6.3 to a larger family of numerical semigroups. First, let us considerthe following conjectures. Conjecture 7.1. Let S be a cyclotomic numerical semigroup and let e be its cyclotomic exponentsequence. Then n ∈ N is a minimal generator of S if and only if e n < . Conjecture 7.2. Let S be a cyclotomic numerical semigroup and let e be its cyclotomic exponentsequence. Then e b = nc( ∇ b ) − for all b ∈ Betti( S ) . In particular, we have Betti( S ) ⊆ E ( S ) . These conjectures are motivated by the following result. Proposition 7.3. Conjecture 1.4 holds if and only if Conjectures 7.1 and 7.2 hold.Proof. First, we note that Conjectures 7.1 and 7.2 are directly implied by Conjecture 1.4 and Propo-sition 2.5. Now let us assume that S is a cyclotomic numerical semigroup such that Conjectures 7.1and 7.2 hold for S and let us prove that Conjecture 1.4 holds for S or, equivalently, that S is acomplete intersection. Since Conjecture 7.1 holds for S , from Proposition 2.3 and Theorem 1.1 weobtain 0 = X d ≥ e d = − e( S ) + X d ≥ e d > e d . From these equalities, Conjecture 7.2 and the fact that e = 1 (by Theorem 1.1), we conclude thate( S ) = X d ≥ e d > e d ≥ X b ∈ Betti( S ) (nc( ∇ b ) − , which shows that the cardinality of any minimal presentation of S is bounded by e( S ) − , andtherefore that S is a complete intersection (see Section 2.4). (cid:3) Theorem 1.1 shows one direction of Conjecture 7.1. Here we show that this conjecture holds fora large set of cyclotomic numerical semigroups. YCLOTOMIC EXPONENT SEQUENCES OF NUMERICAL SEMIGROUPS 23 Corollary 7.4. Let S be a numerical semigroup minimally generated by A . If U( E ( S )) = E ( S ) ,then E ( S ) ⊆ Betti( S ) , S is cyclotomic and Conjecture 7.1 holds for S .Proof. From Theorem 1.2, we find that E ( S ) = U(Betti( S )) and that e b = nc( ∇ b ) − > b ∈ E ( S ). In particular, E ( S ) is finite and, thus, S is cyclotomic (Definition 1.1). Let n ∈ N with e n < 0. We have n 6∈ E ( S ) because e b > b ∈ E ( S ). Moreover, recall that e = 0 and e = 1 by Theorem 1.1, so n ≥ 2. Since E ( S ) is the set of positive integers j such that j ≥ e j = 0and j is not a minimal generator, we conclude that n is a minimal generator of S . (cid:3) As already mentioned in the introduction, computations suggest that these numerical semigroupsarise very frequently. Corollary 7.5. Let S be a cyclotomic numerical semigroup. If Betti( S ) = U(Betti( S )) and Con-jecture 7.1 holds for S , then S is a complete intersection.Proof. From Theorem 1.2 and Betti( S ) = U(Betti( S )), we obtain Betti( S ) ⊆ E ( S ). The result nowfollows from Proposition 7.3. (cid:3) Corollary 7.6. Let S be a numerical semigroup. If Betti( S ) = U(Betti( S )) and E ( S ) = U( E ( S )) ,then S is a complete intersection.Proof. This follows by combining Corollaries 7.4 and 7.5. (cid:3) Example . Let S = h , , , i . ThenP S ( x ) = (1 − x )(1 − x )(1 − x )(1 − x )(1 − x )(1 − x )(1 − x )(1 − x ) . Then E ( S ) = Betti( S ) = { , , } . The graph (Betti( S ) , ≤ S ) is depicted below.3650 24From this graph it follows that Betti( S ) = U(Betti( S )) and, thus, S is a complete intersection.Finally, let us make a few comments on Conjecture 7.1. In [7, Lemma 14] it is shown that thisconjecture holds true under several restrictions on S . We notice that the restrictions in part (a) and(b) of Lemma 14 from [7] cannot both hold at the same time, hence the statement of [7, Lemma 14]is void, in the sense that it does not find cyclotomic numerical semigroups satisfying Conjecture 7.1.We conclude this section by improving [7, Lemma 14]. Proposition 7.8. Let S be a cyclotomic numerical semigroup with cyclotomic exponent sequence e . Let j ∈ N with e j < and j < min { d ∈ N : e d > } . Then j is a minimal generator of S . As aconsequence, if max { d ∈ N : e d < } < min { d ∈ N : e d > } , then Conjecture 7.1 holds for S .Proof. Let j ∈ N with e j < j < min { d ∈ N : e d > } . In view of Theorem 1.1, either j is aminimal generator or d ( j ) ≥ 2. In the latter case, by Theorem 5.3, there is α ∈ Minimals ≤ S E ( S )with α ≤ S j , and that e α > 0. However, this implies thatmin { d ∈ N : e d > } ≤ α < j < min { d ∈ N : e d > } , a contradiction. We conclude that j must be a minimal generator of S . (cid:3) Open questions. Regarding cyclotomic exponent sequences of arbitrary numerical semi-groups, it would be interesting to study the values of these sequences at those Betti elementsthat are not in U(Betti( S )), where our current techniques fail to yield any result.Coming back to cyclotomic numerical semigroups, by Definition 1.1 a numerical semigroup iscyclotomic if and only if its cyclotomic exponent sequence has finitely many non-zero terms. ByProposition 2.4 this is equivalent with P S being a product of cyclotomic polynomials. Question 7.9. Is there a weaker condition than the cyclotomic exponent sequence having finitesupport that would ensure that P S is a product of cyclotomic polynomials? A possible way to weaken the condition would be, for instance, to require that the exponentsequence has infinitely many zeros.We point out that Conjecture 1.4 remains open, and it seems likely that further tools are neededin order to tackle it. One could start by showing that if S is a numerical semigroup such that |E ( S ) | ≤ 2, then S is a complete intersection. Here we have managed to address the case |E ( S ) | = 1in Theorem 1.3, but our techniques are not enough to analyze the case when |E ( S ) | = 2 and thetwo elements in E ( S ) are incomparable with respect to ≤ S . As seen in this section, Conjectures 7.1and 7.2 are equivalent with Conjecture 1.4. It is thus well possible that at least one of the two isconsiderably easier than Conjecture 1.4, and thus they warrant individual investigation. Acknowledgments Part of the work on this paper was done during an internship in the Fall of 2016 carried out bythe third author at the Max Planck Institute for Mathematics in Bonn and during a one-week visitin April 2019. He would like to thank the fourth author for the invitation and the institute stafffor their hospitality and support. The project was completed during a stay of the first author atthe same institute. Substantial progress on this paper was made in February 2017, when the firstand the fourth author were invited by the second and third author for one week to the Universityof Granada. They are grateful for the hospitality, for the inspiring and cheerful atmosphere and,last but not least, for the excellent tapas and wine! References [1] A. Assi and P. A. Garc´ıa-S´anchez, Numerical semigroups and applications, RSME Springer Series , Springer,2016.[2] A. Assi, P. A. Garc´ıa-S´anchez and I. Ojeda, Frobenius vectors, Hilbert series and gluings of affine semigroups, J.Commut. 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Max-Planck-Institut f¨ur Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany Email address : [email protected] Departamento de ´Algebra, Universidad de Granada, E-18071 Granada, Espa˜na Email address : [email protected] Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford,OX1 3QD, UK. Email address : [email protected] Max-Planck-Institut f¨ur Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany Email address ::