Star operations on numerical semigroups: antichains and explicit results

Abstract

We introduce an order on the set of non-divisorial ideals of a numerical semigroup S, and link antichains of this order with the star operations on S; subsequently, we use this order to find estimates on the number of star operations on S. We then use them to find an asymptotic estimate on the number of nonsymmetric numerical semigroups with n or less star operations, and to determine these semigroups explicitly when n = 10.2010 Mathematics Subject Classification. 20M12,20M14.

Embedding the set of nondivisorial ideals of a numerical semigroup into ℕn

“…for every J ∈ F(S); moreover, every star operation on S is in the form * ∆ , for some antichain ∆ of (G 0 (S), ≤ * ) [17,Section 3].…”

“…If x < y and Q x = ∅, then also Q y = ∅, and , and thus it is the maximum of (G 0 (S), ≤ * ); it is called the canonical ideal of S. An atom of S (or of G 0 (S)) is an I ∈ G 0 (S) such that, whenever . Sufficient conditions for I ∈ G 0 (S) to be an atom are that [17,Proposition 4.8] and that I is an element of Q x such that |M x \ I| ≤ 1 [17,Proposition 5.3]. If every non-divisorial ideal I is an atom, then the number of star operations on S is equal to the number of antichains of (G 0 (S), ≤ * ), and conversely [17,Proposition 4.9].…”

“…A deeper examination of the questions tackled in [15] led in [16] and [17] to the introduction of a partial order on the set G 0 (S) of the classes of non-divisorial ideals of a numerical semigroup S: in particular, it was shown that there is a strong link between the set Star(S) of star operations on S and the antichains of G 0 (S) (an antichain of a partially ordered set P is a subset of P composed by pairwise incomparable elements). In particular, [16] provided a full analysis of the case when the multiplicity of the numerical semigroup is 3, while [17] presented some estimates on the number of star operations on a numerical semigroup and, consequently, on the cardinality of the set of semigroups with some fixed number of star operations.…”

“…, µ − 2}. The claim is proved.The description of G 0 (S) also allows to prove that |Star(S)| = 1 + ω(µ − 2) (see[17, Proposition 6.3]).…”