C-semigroups with its induced order

TL;DR

This paper introduces a natural order on C-semigroups within polyhedral cones, enabling the generalization of numerical semigroup concepts to higher-dimensional settings.

Contribution

It proposes using the semigroup-induced order instead of arbitrary monomial orders, broadening the theoretical framework for C-semigroups.

Findings

Generalized definitions from numerical semigroup theory to C-semigroups.

Established properties of C-semigroups under the new order.

Provided a foundation for further research in polyhedral semigroup structures.

Abstract

Let $C\subset\mathbb{N}^p$ be an integer polyhedral cone. An affine semigroup $S\subset C$ is a $ C$-semigroup if $| C\setminus S|<+\infty$. This structure has always been studied using a monomial order. The main issue is that the choice of these orders is arbitrary. In the present work we choose the order given by the semigroup itself, which is a more natural order. This allows us to generalise some of the definitions and results known from numerical semigroup theory to $C$-semigroups.