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 $H(S)=\mathbb {N}^d \backslash S$ 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.