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.

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We give a characterization on the sets A ⊆ ℕd such that the monoid generated by A is a generalized numerical semigroup (GNS) in ℕd. Furthermore we give a procedure to compute the hole set ℕd \ S, where S is a GNS, if a finite set of generators of S is known.

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