A formula for the conductor of a semimodule of a numerical semigroup with two generators
aa r X i v : . [ m a t h . A C ] J u l A FORMULA FOR THE CONDUCTOR OF A SEMIMODULE OF ANUMERICAL SEMIGROUP WITH TWO GENERATORS
PATRICIO ALMIR ´ON AND JULIO-JOS´E MOYANO-FERN ´ANDEZ
Dedicated to the memory of Fernando Eduardo Torres Orihuela
Abstract.
We provide an expression for the conductor c (∆) of a semimodule ∆ of anumerical semigroup Γ with two generators in terms of the syzygy module of ∆ and thegenerators of the semigroup. In particular, we deduce that the difference between theconductor of the semimodule and the conductor of the semigroup is an element of Γ, aswell as a formula for c (∆) in terms of the dual semimodule of ∆. Introduction
A classical problem in the combinatorics of natural numbers is to find a closed expressionfor the largest natural number that is not representable as a nonnegative linear combina-tion of some relatively prime numbers, called the Frobenius number. This problem canbe encoded in terms of getting a formula for the conductor of a numerical semigroup.A numerical semigroup Γ is an additive sub-semigroup of the monoid ( N , +) such thatthe greatest common divisor of all its elements is equal to 1. The complement N \ Γ istherefore finite, and its elements are called gaps of Γ. Moreover, Γ is finitely generatedand it is not difficult to find a minimal system of generators of Γ, see. e.g. Rosales andGarc´ıa S´anchez [6].The number c (Γ) = max( N \ Γ) + 1 is called the conductor of Γ; in particular c (Γ) − c (Γ) for an arbitrary number ofminimal generators of Γ is NP-hard (see Ram´ırez Alfons´ın [5] for a good account of this),but there are some special cases in which a closed formula is available. For example, aclosed formula is known for the case of two generators: if Γ = α N + β N := h α, β i , then c (Γ) = αβ − α − β + 1. However, for a numerical semigroup with more than two generatorsit is not possible in general to obtain a closed polynomial formula for its conductor interms of the minimal set of generators (see Curtis [2]).We are interested in subsets of N which have an additive structure over Γ (in analogy withthe structure of module over a ring): a Γ-semimodule is a non-empty subset ∆ of N with∆ + Γ ⊆ ∆. A system of generators of ∆ is a subset E of ∆ such that ∆ = S x ∈E ( x + Γ); itis called minimal if no proper subset of E generates ∆. Notice that, since ∆ \ Γ is finite,
Mathematics Subject Classification.
Primary: 20M14; Secondary: 05A19.
Key words and phrases.
Numerical semigroup, Frobenius problem, Γ-semimodule, syzygy.The first author was partially supported by Spanish Goverment, Ministerios de Ciencia e Innovaci´ony de Universidades MTM2016-76868-C2-1-P. The second author was partially supported by the SpanishGovernment, Ministerios de Ciencia e Innovaci´on y de Universidades, grant PGC2018-096446-B-C22, aswell as by Universitat Jaume I, grant UJI-B2018-10. every Γ-semimodule is finitely generated and has a conductor c (∆) = max( N \ ∆) + 1 . Motivated by the problem of obtaining a closed formula for the conductor of Γ, it isnatural to ask for a closed formula for the conductor of a Γ-semimodule. The purpose ofthis note is to give a formula for c (∆) in the case Γ = h α, β i in terms of the generators ofthe semimodule of syzygies of ∆, see [3], as well as in terms of the generators of the dualof this semimodule, see [4].2. Semimodules over a numerical semigroup
Let Γ be a numerical semigroup. This section is devoted to collect the main propertiesconcerning Γ-semimodules. The reader is referred to [6] or [5] for specific material aboutnumerical semigroups.Every Γ-semimodule ∆ has a unique minimal system of generators (see e.g. [3, Lemma2.1]). Two Γ-semimodules ∆ and ∆ ′ are called isomorphic if there is an integer n suchthat x x + n is a bijection from ∆ to ∆ ′ ; we write then ∆ ∼ = ∆ ′ . For every Γ-semimodule ∆ there is a unique semimodule ∆ ′ ∼ = ∆ containing 0; such a semimoduleis called normalized. Moreover, the minimal system of generators { x = 0 , . . . , x n } of anormalized Γ-semimodule is a Γ-lean set, i.e. it satisfies that | x i − x j | / ∈ Γ for any 0 ≤ i < j ≤ n, and conversely, every Γ-lean set of N minimally generates a normalized Γ-semimodule.Hence there is a bijection between the set of isomorphism classes of Γ-semimodules andthe set of Γ-lean sets of N . See Sect. 2 in [3] for the proofs of those statements.There is another kind of system of generators—not minimal—for a semimodule ∆ of Γrelative to s ∈ Γ \ { } : this is the set of the s smallest elements in ∆ in each of the s classes modulo s , namely the set ∆ \ ( s + ∆), and is called the Ap´ery set of ∆ with respectto s ; we write Ap(∆ , s ).A formula for the conductor in terms of Ap(∆ , s ) for s ∈ Γ \ { } is easily deduced. Proposition 2.1.
Let ∆ be a Γ -semimodule. For any s ∈ Γ we have that c (∆) = max ≤ N Ap(∆ , s ) . Proof.
The equality follows as in the case ∆ = Γ, see e.g. Lemma 3 in Brauer and Shockley[1]. (cid:3)
In this paper we will consider numerical semigroups with two generators, say Γ = h α, β i ,with α, β ∈ N with α < β and gcd( α, β ) = 1. As mentioned above, the conductor of Γcan be expressed as c = c ( h α, β i ) = ( α − β − h α, β i are also easy todescribe: they admit a unique representation αβ − aα − bβ , where a ∈ ]0 , β − ∩ N and b ∈ ]0 , α − ∩ N . This writing yields a map from the set of gaps of h α, β i to N given by αβ − aα − bβ ( a, b ) , which allows us to identify a gap with a lattice point in the lattice L = N ; since the gapsare positive numbers, the point lies inside the triangle with vertices (0 , , (0 , α ) , ( β, FORMULA FOR THE CONDUCTOR OF A SEMIMODULE OF SOME NUMERICAL SEMIGROUPS3
23 18 13 8 316 11 6 19 42
Figure 2.1.
Lattice path for the h , i -lean set { , , , } .In the following we will use the notation e = αβ − a ( e ) α − b ( e ) β for a gap e of the semigroup h α, β i ; if the gap is subscripted as e i then we write a i = a ( e i )and b i = b ( e ).Let us denote by ≤ the total ordering in N , sometimes we will denote it by ≤ N to remarkthat it is the natural ordering. In addition, we define the following partial ordering (cid:22) onthe set of gaps: Definition 2.2.
Given two gaps e , e of h α, β i , we define e (cid:22) e : ⇐⇒ a ≤ a ∧ b ≥ b and e ≺ e : ⇐⇒ a < a ∧ b > b . Let E = { , e , . . . , e n } ⊆ N with gaps e i = αβ − a i α − b i β of h α, β, i for every i = 1 , . . . , n such that a < a < · · · < a n . Corollary 3.3 in [3] ensures that E is h α, β i -lean if and onlyif b > b > · · · > b n .This simple fact leads to an identification (cf. [3, Lemma 3.4]) between an h α, β i -lean setand a lattice path with steps downwards and to the right from (0 , α ) to ( β,
0) not crossingthe line joining these two points, where the lattice points identified with the gaps in E mark the turns from the x -direction to the y -direction; these turns will be called ES-turnsfor abbreviation. Figure 2 shows the lattice path corresponding to the h , i -lean set { , , , } .Let g = 0 , g , . . . , g n be the minimal system of generators of a h α, β i -semimodule ∆.From now on, we will assume that the indexing in the minimal set of generators of ∆ issuch that g = 0 (cid:22) g (cid:22) · · · (cid:22) g n . In [3] it was introduced the notion of syzygy of ∆ asthe h α, β i -semimoduleSyz(∆) := [ i,j ∈{ ,...n } ,i = j (cid:16) (Γ + g i ) ∩ (Γ + g j ) (cid:17) . The semimodule of syzygies of the semimodule ∆ minimally generated by { g = 0 , g , . . . , g n } can be characterized as follows (see [3, Theorem 4.2]): PATRICIO ALMIR ´ON AND JULIO-JOS´E MOYANO-FERN ´ANDEZ
Definition 2.3.
Syz(∆) = [ ≤ k
Let ∆ be a Γ -semimodule with associated lean set [ I, J ] , with I = { g =0 , g , . . . , g n } and J = { h , . . . , h n } . Then, for any h ∈ J we have h − α − β / ∈ ∆ . Proof.
Consider h ∈ J such that that g i ≺ h ≺ g i +1 . Let us denote ( a j , b j ) resp.( a j +1 , b j +1 ) the coordinates of g j resp. g j +1 in the lattice L ; then the element h is repre-sented in the lattice path as ( a j , b j +1 ), see Definition 2.3. By contradiction, assume that h − α − β ∈ ∆; then there exists a gap g ∈ I together with two integers ν , ν ∈ N suchthat h − α − β = ν α + ν β + g. Since h − α − β / ∈ Γ, we may write h − α − β = αβ − ( a j + 1) α − ( b j +1 + 1) β. The writing of g as g = αβ − aα − bβ is unique whenever ( a, b ) ∈ L , therefore a j + 1 = a − ν , b j +1 + 1 = b − ν . These equalities yield the conditions a j < a, and , b j +1 > b . As the set I is ordered, thismeans that ( a, b ) = ( a j +1 , b j ), which contradicts the minimality of the set of generators,since g j ≺ g ≺ g j +1 with g j +1 = g = g j . (cid:3) A formula for the conductor of an h α, β i -semimodule In this section we are going to provide a formula for the conductor of a Γ-semimodulewith any number of generators in terms of the generators of Γ and a special syzygy ofthe Γ-semimodule. In particular, we will obtain some relations between the conductorof Γ and the conductor of the Γ-semimodule. Finally, we will provide a formula for theconductor of the Γ-semimodule in terms of its dual.
FORMULA FOR THE CONDUCTOR OF A SEMIMODULE OF SOME NUMERICAL SEMIGROUPS5
Theorem 3.1.
Let ∆ be a Γ -semimodule with associated lean set [ I, J ] as above, and let M := max ≤ N { h ∈ J } denote the biggest (with respect to the order of the natural numbers)minimal generator of Syz(∆) . Then c (∆) = M − α − β + 1 . In particular, if ( m , m ) are the coordinates of the point representing M in the lattice L ,we have c (∆) = c (Γ) − m α − m β. Proof.
Let us first prove that c (∆) − c (∆) − / ∈ ∆ and (ii) that if ℓ / ∈ ∆ with ℓ = c (∆) − ℓ < c (∆) − . The statement (i) is clear by Lemma 2.4, since M ∈ J . To see (ii), consider ℓ / ∈ ∆ with ℓ = c (∆) −
1; since ℓ / ∈ ∆, it holds in particular ℓ / ∈ Γ. Therefore, we canassociate to ℓ a point ( a, b ) in the lattice L . Moreover, ℓ is upon and not contained on thelattice path associated to I . This means that there exists some j ∈ J with coordinates( j , j ) in the lattice path such that a ≥ j and b ≥ j with some of the inequalities beingstrict, otherwise ℓ would be an element of ∆ since the elements represented by latticepoints on and under the lattice path belong to ∆. Thus, from the representation of ℓ and j as gaps it is easily checked that ℓ ≤ N j. Hence, since M = max ≤ N { h ∈ J } and M ∈ J ,we have that M ≥ N ℓ for any ℓ / ∈ ∆, which proves (ii).Finally, since M can be represented as a lattice point ( m , m ) ∈ L , we have c (∆) = M − α − β + 1 = αβ − m α − m β − α − β + 1 = c (Γ) − m α − m β. (cid:3) The value M can be easily characterized in terms of the Ap´ery set of ∆ with respect to α + β : Proposition 3.2.
Let M := max ≤ N { h ∈ J } be the biggest minimal generator of the syzygymodule with respect to the natural ordering of N as above, then M = max ≤ N Ap(∆ , α + β ) . Proof.
This is a consequence of Proposition 2.1 for n = α + β ∈ h α, β i . (cid:3) Corollary 3.3.
Let ∆ be a Γ semimodule. Then c (Γ) − c (∆) ∈ Γ . We conclude this paper rewriting the formula of Theorem 3.1 in terms of the dual Γ–semimodule of ∆, ∆ ∗ := { z ∈ Z | z + ∆ ⊂ Γ } , see [4]. An important fact about the dual semimodule is that the minimal set of generatorsof Syz(∆) is in bijection with the minimal set of generators of ∆ ∗ : Lemma 3.4 ([4], Lemma 6.1) . The minimal sets of generators of ∆ ∗ and Syz(∆) are incorrespondence via the map x αβ − x. In particular, this bijection together with Theorem 3.1 allows to computes the conductorof the semimodule ∆ in terms of its dual in a natural way:
PATRICIO ALMIR ´ON AND JULIO-JOS´E MOYANO-FERN ´ANDEZ
Corollary 3.5.
Let ∆ be a Γ -semimodule, and let ∆ ∗ be its dual, minimally generated by x , . . . , x n . Then c (∆) = min ≤ N { x , . . . , x n } − α − β + 1 . Proof.
By Theorem 3.1 we have that c (∆) = max ≤ N { h ∈ J } − α − β + 1 , where J is a minimal set of generators of Syz(∆) . Lemma 3.4 yields the equality min ≤ N { x i } = αβ − max ≤ N { h ∈ J } , which allows us to conclude. (cid:3) References
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Instituto de Matemtica interdisciplinar (IMI) y departamento de ´Algebra, Geometr´ıay Topolog´ıa, Facultad de Ciencias Matem´aticas, Universidad Complutense de Madrid,28040, Madrid, Spain.
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