Numerical Semigroups with unique Ap\'{e}ry expansions

TL;DR

This paper investigates two specific classes of numerical semigroups generated by sums of arithmetic and geometric progressions, focusing on their unique Apéry set expansions in embedding dimension 4.

Contribution

It provides a comprehensive analysis of these classes, highlighting their unique Apéry set expansion properties in a specific embedding dimension.

Findings

Characterization of semigroups generated by arithmetic progressions

Characterization of semigroups generated by geometric progressions

Identification of unique Apéry set expansions in these classes

Abstract

In this paper, we carry out a fairly comprehensive study of two special classes of numerical semigroups, one generated by the sequence of partial sums of an arithmetic progression and the other one generated by the partial sums of a geometric progression, in embedding dimension $4$. Both these classes have the common feature that they have unique expansions of the Ap\'{e}ry set elements.