Nonhomogeneous Patterns on Numerical Semigroups

Abstract

Patterns on numerical semigroups are multivariate linear polynomials, and they are said to admit a numerical semigroup if evaluating the pattern at any nonincreasing sequence of elements of the semigroup gives integers belonging to the semigroup. In a first approach, only homogeneous patterns were analyzed. In this contribution we study conditions for a nonhomogeneous pattern to admit a nontrivial numerical semigroup, 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 forms 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.

Patterns of ideals of numerical semigroups

“…Indeed, a homogeneous linear pattern as defined in [4] is according to our definition still a pattern admitted by a numerical semigroup. However, a non-homogeneous linear pattern as defined in [5] is now a pattern admitted by the maximal ideal of some numerical semigroup.…”

“…We start with necessary conditions for linear patterns to be endopatterns and surjective endopatterns of numerical semigroups. We will repeatedly make use of the following result, first proved in [4] for homogeneous linear patterns and in [5] for non-homogeneous linear patterns. Here we prove the result for ideals of numerical semigroups, making use of Abel's partial summation formula (this proof is due to Christian Gottlieb).…”

Sets of positive integers closed under product and the number of decimal digits

“…By using the terminology of [1] an LD-semigroup is a numerical semigroup fulfills a nonhomogeneous pattern x 1 + x 2 − 1. As a consequence of [1, Example 6.4] LD-semigroups can be characterized by the fact that the minimum element in each interval of nongaps is a minimal generator.…”

The numerical semigroup of the integers which are bounded by a submonoid of N2

“…Remark 2.5 Let us remember that the gaps of a numerical semigroup S are the elements of the set N \ S. As a consequence of Proposition 2.3 (Item 3), we have that a numerical A-semigroups S can be characterized as a numerical semigroup satisfying that, if s ∈ S is a maximum or a minimum of an interval of non-gaps, then s is a minimal generator of S or s = 0 (see [1]). …”

“…First of all, we will show a characterization of them, via the concept of nonhomogeneous pattern (see [1]), and how we can obtain all of them.…”