Odd Right-End Numerical Semigroups

Abstract

An odd right-end semigroup (hereinafter Ore semigroup) is a numerical semigroup S verifying that x + 1 ∈ S for every x ∈ S\{0} such that x is even. The introduction and study of these semigroups is the purpose of the present work. In particular, we will give some algorithms which compute all Ore semigroups with a given genus, a fixed Frobenius number and aspecific multiplicity. We will see that if X is a set of positive integers, then there exists the smallest Ore semigroup, under the inclusion sets, that contains X. We will denote this semigroup by θ[X] and present an algorithm to calculate it. Finally, we will study the embedding dimension, the Frobenius number, and the genus of Ore semigroups of the form θ[{m}], where m is a positive integer. As a consequence of this study, we will prove that this kind of semigroup satisfies Wilf’s conjecture.