Frobenius Allowable Gaps of Generalized Numerical Semigroups

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