On Free Numerical Semigroups and the Construction of Minimal Telescopic Sequences

TL;DR

This paper investigates free numerical semigroups generated by telescopic sequences, demonstrating how to construct minimal telescopic sequences for any such semigroup and exploring operations on these sequences.

Contribution

It introduces methods to derive minimal telescopic sequences from any telescopic sequence, enhancing understanding of free numerical semigroups and their generators.

Findings

Existence of minimal telescopic sequences for any telescopic sequence.

Construction of minimal telescopic sequences for given free numerical semigroups.

Analysis of operations and constructions on telescopic sequences.

Abstract

A free numerical semigroup is a submonoid of the non-negative integers with finite complement that is additively generated by the terms in a telescopic sequence with gcd 1. However, such a sequence need not be minimal, which is to say that some proper subsequence may generate the same numerical semigroup, and that subsequence need not be telescopic. In this paper, we will see that for a telescopic sequence with any gcd, there is a minimal telescopic sequence which generates the same submonoid. In particular, given a free numerical semigroup we can construct a telescopic generating sequence which is minimal. In the process, we will examine some operations on and constructions of telescopic sequences in general.