Frobenius allowable gaps of Generalized Numerical Semigroups

A generalized numerical semigroup is a submonoid S of Nd for which the complement Nd \ S is finite. The points in the complement Nd \ S are called gaps. A gap F is considered Frobenius allowable if there is some relaxed monomial ordering on Nd with respect to which F is the largest gap. We characterize the Frobenius allowable gaps of a generalized numerical semigroup. A generalized numerical semigroup that has only one maximal gap under the natural partial ordering of Nd is called a Frobenius generalized numerical semigroup. We show that Frobenius generalized numerical semigroups are precisely those whose Frobenius gap does not depend on the relaxed monomial ordering. We estimate the number of Frobenius generalized numerical semigroup with a given Frobenius gap F = (F(1), ... , F(d)) is an element of Nd and show that it is close to root 3(F(1)+/- 1)center dot center dot center dot(F(d)+/- 1) for large d. We define notions of quasiirreducibility and quasi-symmetry for generalized numerical semigroups. While in the case of d = 1 these notions coincide with irreducibility and symmetry, they are distinct in higher dimensions.