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arXiv · 1211.0895

Nonhomogeneous patterns on numerical semigroups

Maria Bras-Amorós·Pedro A. García-Sánchez·Albert Vico-Oton
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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.

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Maria Bras-Amorós, Pedro A. García-Sánchez, Albert Vico-Oton. 2012-11-05. Nonhomogeneous patterns on numerical semigroups. https://arxiv.org/abs/1211.0895

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