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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 introduceirreducible generalized numerical semigroups and characterize them in terms ofthe cardinality of a special subset of $H(S)$. In particular, we describerelaxed monomial orders on $\mathbb N^d$, define the Frobenius element of $S$with respect to a given relaxed monomial order, and show that the Frobeniuselement of $S$ is independent of the order if the generalized numericalsemigroup is irreducible.

Irreducible Generalized Numerical Semigroups and uniqueness of the Frobenius element