Irreducible generalized numerical semigroups and uniqueness of the Frobenius element

Let Nd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {N}^{d}$$\end{document} be the d-dimensional monoid of non-negative integers. A generalized numerical semigroup is a submonoid S⊆Nd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ S\subseteq \mathbb {N}^d$$\end{document} such that H(S)=Nd\S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(S)=\mathbb {N}^d \backslash S$$\end{document} 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 Nd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {N}}^d$$\end{document}, 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.