Nonhomogeneous patterns on numerical semigroups

TL;DR

This paper investigates conditions under which nonhomogeneous linear patterns are admissible on numerical semigroups, extending previous homogeneous pattern analysis and exploring the structure of semigroups admitting such patterns.

Contribution

It introduces criteria for admissibility of eventually non-homogeneous patterns and characterizes strongly admissible patterns, including their associated semigroup varieties and minimal generators.

Findings

Conditions for admissibility of non-homogeneous patterns

Representation of semigroup sets as m-varieties in a tree structure

Characterization of strongly admissible patterns with finite trees

Abstract

Patterns on numerical semigroups are multivariate linear polynomials, and they are said to be admissible if there exists a numerical semigroup such that evaluated at any nonincreasing sequence of elements of the semigroup gives integers belonging to the semigroup. In a first approach, only homogeneous patterns where analized. In this contribution we study conditions for an eventually non-homogeneous pattern to be admissible, and particularize this study to the case the independent term of the pattern is a multiple of the multiplicity of the semigroup. Moreover, for the so called strongly admissible patterns, the set of numerical semigroups admitting these patterns with fixed multiplicity $m$ form an $m$-variety, which allows us to represent this set in a tree and to describe minimal sets of generators of the semigroups in the variety with respect to the pattern. Furthermore, we characterize strongly admissible patterns having a finite associated tree.