Proportionally modular affine semigroups

Abstract

This work introduces a new kind of semigroup of Np called proportionally modular affine semigroup. These semigroups are defined by modular Diophantine inequalities and they are a generalization of proportionally modular numerical semigroups. We give an algorithm to compute their minimal generating sets. We also specialize on the case p = 2. For this case, we provide a faster algorithm to compute their minimal system of generators, prove they are Cohen-Macaulay and Buchsbaum, and determinate their (minimal) Frobenius vectors. Besides, Gorenstein proportionally modular affine semigroups are characterized.