Perfect numerical semigroups

Abstract

A numerical semigroup is perfect if it does not have isolated gaps. In this paper we will order the perfect numerical semigroups with a fixed multiplicity. This ordering allows us to give an algorithm procedure to obtain them. We also study the perfect monoid, which is a subset of N that can be expressed as an intersection of perfect numerical semigroups, and we present the perfect monoid generated by a subset of N. We give an algorithm to calculate it. We study the perfect closure of a numerical semigroup, as well as the perfect numerical semigroup with maximal embedding dimension, in particular Arf and saturated numerical semigroups.

Modularly equidistant numerical semigroups

“…This work is a generalization of the study of the parity numerical semigroups [3]. A numerical semigroup S is prefect if {x − 1, x + 1} ⊆ S then x ∈ S (see for instance [4,5]). Observe that every equidistant modularly numerical semigroup is a perfect numerical semigroup.…”

Perfect numerical semigroups with embedding dimension three

“…The perfect numerical semigroups were introduced in [3]. They are a family of numerical semigroups, whose name comes from topology, specifically the concept of a perfect set (set without isolated points).…”

“…They are a family of numerical semigroups, whose name comes from topology, specifically the concept of a perfect set (set without isolated points). The family of perfect semigroups is arranged in a tree, and this construction allows us to study certain aspects and properties of them (see [3]).…”

“…In [3], some properties of the perfect numerical semigroups have already been studied: construction of an algorithmic procedure that allows us to obtain all the perfect semigroups with a fixed multiplicity, perfect numerical semigroups with maximal embedding dimension, the perfect closure of a numerical semigroup, etc.…”

Parity numerical semigroups

“…Our main goal in this paper is to begin the study of the parity numerical semigroups. These semigroups are a distinguished class within the so-called perfect numerical semigroups introduced in [2]. Indeed, a numerical semigroup is perfect if {x − 1, x + 1} ⊆ S implies x ∈ S. It is clear then that every parity numerical semigroup is a perfect numerical semigroup.…”