Irreducible Generalized Numerical Semigroups and uniqueness of the Frobenius element

Abstract

Let ${\mathbb{N}}^d$ be the $d$-dimensional monoid of non-negative integers. A generalized numerical semigroup is a submonoid $S\subseteq {\mathbb{N}}^d$ such that $H(S)={\mathbb{N}}^d\setminus S$ is a finite set. We introduce irreducible generalized numerical semigroups and characterize them in terms of the cardinality of a special subset of $H(S)$. In particular, we describe relaxed monomial orders on ${\mathbb{N}}^d$, define the Frobenius element of $S$ with respect to a given relaxed monomial order, and show that the Frobenius element of $S$ is independent of the order if the generalized numerical semigroup is irreducible.

15 pages