A geometrical characterization of proportionally modular affine semigroups

TL;DR

This paper provides a geometric framework for understanding proportionally modular affine semigroups, which are defined by modular Diophantine inequalities, and offers algorithms to identify such semigroups in the integer lattice.

Contribution

It introduces a novel geometric characterization of these semigroups and develops algorithms to verify if a given semigroup is proportionally modular.

Findings

Geometric characterization of proportionally modular affine semigroups

Algorithms for identifying such semigroups in  lattice

Application of geometric approach to semigroup verification

Abstract

A proportionally modular affine semigroup is the set of nonnegative integer solutions of a modular Diophantine inequality $f_1x_1+\cdots +f_nx_n \mod b \le g_1x_1+\cdots +g_nx_n$ where $g_1,\dots,g_n,$ $f_1,\ldots ,f_n\in \mathbb{Z}$ and $b\in\mathbb{N}$. In this work, a geometrical characterization of these semigroups is given. Moreover, some algorithms to check if a semigroup $S$ in $\mathbb{N}^n$, with $\mathbb{N}^n\setminus S$ a finite set, is a proportionally modular affine semigroup are provided by means of that geometrical approach.