# Characterizing affine $\mathcal{C}$-semigroups

## TL;DR

This paper characterizes affine -semigroups within finitely generated cones, providing algorithms for their identification, gap computation, and decomposition, along with bounds on their embedding dimension.

## Contribution

It offers a characterization of -semigroups from minimal generators, an algorithm to verify and compute gaps, and a method to decompose them into irreducible components.

## Key findings

- Characterization of -semigroups from minimal generating sets
- Algorithm to check if a semigroup is a -semigroup and to compute gaps
- Lower bounds for the embedding dimension of -semigroups

## Abstract

Let $\mathcal C \subset \mathbb N^p$ be a finitely generated integer cone and $S\subset \mathcal C$ be an affine semigroup such that the real cones generated by $\mathcal C$ and by $S$ are equal. The semigroup $S$ is called $\mathcal C$-semigroup if $\mathcal C\setminus S$ is a finite set. In this paper, we characterize the $\mathcal C$-semigroups from their minimal generating sets, and we give an algorithm to check if $S$ is a $\mathcal C$-semigroup and to compute its set of gaps. We also study the embedding dimension of $\mathcal C$-semigroups obtaining a lower bound for it, and introduce some families of $\mathcal C$-semigroups whose embedding dimension reaches our bound. In the last section, we present a method to obtain a decomposition of a $\mathcal C$-semigroup into irreducible $\mathcal C$-semigroups.