Numerical semigroups on compound sequences

TL;DR

This paper introduces compound sequences as a generalization of geometric sequences for numerical semigroups, and computes key invariants such as Frobenius number, Apéry sets, and Betti elements, providing bounds on delta and tame degrees.

Contribution

It generalizes geometric sequences to compound sequences in numerical semigroups and calculates various algebraic and combinatorial invariants for these structures.

Findings

Computed Frobenius numbers and Apéry sets for compound sequence semigroups

Determined Betti elements and catenary degrees

Established bounds on delta set and tame degree

Abstract

We generalize the geometric sequence $\{a^p, a^{p-1}b, a^{p-2}b^2,...,b^p\}$ to allow the $p$ copies of $a$ (resp. $b$) to all be different. We call the sequence $\{a_1a_2a_3\cdots a_p, b_1a_2a_3\cdots a_p, b_1b_2a_3\cdots a_p,\ldots, b_1b_2b_3\cdots b_p\}$ a \emph{compound sequence}. We consider numerical semigroups whose minimal set of generators form a compound sequence, and compute various semigroup and arithmetical invariants, including the Frobenius number, Ap\'ery sets, Betti elements, and catenary degree. We compute bounds on the delta set and the tame degree.