Almost symmetric numerical semigroups

TL;DR

This paper investigates the properties of almost symmetric numerical semigroups and their semigroup rings, providing structural insights and simplified proofs for specific cases using row-factorization matrices.

Contribution

It introduces a characteristic property of the minimal free resolution for almost symmetric semigroup rings and offers a structure theorem for 4-generated cases using row-factorization matrices.

Findings

Characterization of minimal free resolution for almost symmetric semigroup rings

Structure theorem for 4-generated almost symmetric semigroups

Simplified proof of Komeda's theorem for pseudo-symmetric semigroups

Abstract

We study almost symmetric numerical semigroups and semigroup rings. We describe a characteristic property of the minimal free resolution of the semigroup ring of an almost symmetric numerical semigroup. For almost symmetric semigroups generated by $4$ elements we will give a structure theorem by using the \lq\lq row-factorization matrices", introduced by Moscariello. As a result, we give a simpler proof of Komeda's structure theorem of pseudo-symmetric numerical semigroups generated by $4$ elements. Row-factorization matrices are also used to study shifted families of numerical semigroups.