Affine convex body semigroups

TL;DR

This paper introduces convex body semigroups generated by convex bodies in R^k, generalizing numerical semigroups, and provides properties, characterizations, algorithms, and implementations for these semigroups.

Contribution

It presents the concept of convex body semigroups, explores their properties, and develops algorithms for their minimal generating systems, extending previous numerical semigroup theory.

Findings

Characterization of affine convex body semigroups from circles and polygons in R^2

Algorithms for computing minimal generators of these semigroups

Implementation of some algorithms for practical computation

Abstract

In this paper we present a new kind of semigroups called convex body semigroups which are generated by convex bodies of R^k. They generalize to arbitrary dimension the concept of proportionally modular numerical semigroup of [7]. Several properties of these semigroups are proven. Affine convex body semigroups obtained from circles and polygons of R^2 are characterized. The algorithms for computing minimal system of generators of these semigroups are given. We provide the implementation of some of them.