Irreducible generalized numerical semigroups and uniqueness of the Frobenius element

Let N^d be the d-dimensional monoid of non-negative integers. A generalized numerical semigroup is a submonoid S ⊆ N^d such that H(S) = N^d 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 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.