A computational approach to the study of finite-complement submonids of an affine cone

TL;DR

This paper introduces algorithms for analyzing finite-complement submonoids of affine cones, focusing on invariants like genus and Frobenius element, and explores their properties and applications to Wilf's conjecture.

Contribution

It develops new algorithms for computing specific invariants of $\

Findings

Introduction of $\

Application to Wilf's conjecture

Abstract

Let $\mathcal{C}\subseteq \mathbb{N}^p$ be an integer cone. A $\mathcal{C}$-semigroup $S\subseteq \mathcal{C}$ is an affine semigroup such that the set $\mathcal{C}\setminus S$ is finite. Such $\mathcal{C}$-semigroups are central to our study. We develop new algorithms for computing $\mathcal{C}$-semigroups with specified invariants, including genus, Frobenius element, and their combinations, among other invariants. To achieve this, we introduce a new class of $\mathcal{C}$-semigroups, termed $\mathcal{B}$-semigroups. By fixing the degree lexicographic order, we also research the embedding dimension for both ordinary and mult-embedded $\mathbb{N}^2$-semigroups. These results are applied to test some generalizations of Wilf's conjecture.