Reduced type of certain numerical semigroup rings

TL;DR

This paper investigates the extremal behavior of the reduced type invariant in certain numerical semigroup rings, providing detailed descriptions of pseudo-Frobenius elements for specific classes and analyzing their impact on the rings' properties.

Contribution

It offers new characterizations of pseudo-Frobenius elements for Bresinsky's and duplicated numerical semigroups, advancing understanding of reduced type extremal behavior.

Findings

Complete descriptions of pseudo-Frobenius elements for Bresinsky's semigroups

Analysis of reduced type extremal behavior in specific semigroup rings

Insights into the relationship between pseudo-Frobenius elements and reduced type

Abstract

For a reduced one-dimensional complete local $k$-algebra $R$, Huneke et al. (Res. Math. Sci., 8(4), paper no. 60, 2021) introduced an important invariant, the reduced type. In this article, we study the extremal behavior of reduced type of some special numerical semigroup rings. For a numerical semigroup ring, the behavior of reduced type can be studied by analyzing the set of pseudo-Frobenius elements of the numerical semigroup. We give complete descriptions of pseudo-Frobenius elements of Bresinsky's numerical semigroups and duplication of numerical semigroups. Further, we explore the extremal behavior of reduced type for the associated semigroup rings.