On atoms of the set of generalized numerical semigroups with fixed corner element

TL;DR

This paper explores the structure of atomic generalized numerical semigroups (GNS), introducing corner special gaps and characterizing atomic GNSs through maximal properties, with implications for irreducibility of Frobenius GNSs.

Contribution

It introduces the concept of corner special gaps and characterizes atomic GNSs using maximal properties, extending the understanding of atomic structures in numerical semigroups.

Findings

Characterization of atomic GNS via corner special gaps.

Conditions for Frobenius GNSs to be atoms or non-irreducible.

Necessary and sufficient conditions for maximal Frobenius GNSs to be irreducible.

Abstract

We study the so-called atomic GNS, which naturally extends the concept of atomic numerical semigroup. We introduce the notion of corner special gap and we characterize the class of atomic GNS in terms of the cardinality of the set of corner special gaps and also in terms of a maximal property. Using this maximal property we present some properties concerning irreducibility of Frobenius GNSs. In particular, we provide sufficient conditions for certain Frobenius GNSs to be an atom non-irreducible (ANI). Furthermore, we given necessary and sufficient conditions so that the maximal elements of a set of Frobenius GNSs with two fixes gaps to be all irreducible or not.