Betti elements and catenary degree of telescopic numerical semigroup families

TL;DR

This paper investigates the catenary degree of telescopic numerical semigroups with embedding dimension three by analyzing Betti elements, deriving formulas for Frobenius numbers and genus, and computing catenary degrees.

Contribution

It introduces formulas for Betti elements in specific telescopic numerical semigroup families and uses them to determine Frobenius numbers, genus, and catenary degrees.

Findings

Formulas for Betti elements in the studied families

Explicit calculations of Frobenius numbers and genus

Determination of catenary degrees for these semigroups

Abstract

The catenary degree is an invariant that measures the distance between factorizations of elements within a numerical semigroup. In general, all possible catenary degrees of the elements of the numerical semigroups occur as the catenary degree of one of its Betti elements. In this study, Betti elements of some telescopic numerical semigroup families with embedding dimension three were found and formulated. Then, with the help of these formulas, Frobenius numbers and genus of these families were obtained. Also, the catenary degrees of telescopic numerical semigroups were found with the help of factorizations of Betti elements of these semigroups