Proportionally modular affine semigroups

TL;DR

This paper introduces proportionally modular affine semigroups, generalizing numerical semigroups, proves their finite generation, and provides algorithms for their minimal generators, especially for the case p=2, with properties like Cohen-Macaulay and Gorenstein characterized.

Contribution

It defines a new class of semigroups, proves their finite generation, and offers algorithms for computing minimal generators, including special properties for the case p=2.

Findings

Semigroups are finitely generated.

Algorithms for minimal generators are provided.

Special properties like Cohen-Macaulay and Gorenstein are characterized for p=2.

Abstract

This work introduces a new kind of semigroup of $\N^p$ called proportionally modular affine semigroup. These semigroups are defined by modular Diophantine inequalities and they are a generalization of proportionally modular numerical semigroups. We prove they are finitely generated and we give an algorithm to compute their minimal generating sets. We also specialise on the case $p=2$. For this case, we provide a faster algorithm to compute their minimal system of generators and we prove they are Cohen-Macaulay and Buchsbaum. Besides, the Gorenstein property is charactized, and their (minimal) Fr\"obenius vectors are determinated.