On gluing semigroups in ${\mathbb{N}}^n$ and the consequences

A semigroup $⟨C⟩$ in ${\mathbb{N}}^n$ is a gluing of $\langle A\rangle$and$\langle B\rangle$ifitsfinitesetofgenerators$C$ splits into two parts, $C=k_{1}A\bigsqcup k_{2}B$ with $k_1,k_2\ge 1$, and the defining ideals of the corresponding semigroup rings satisfy that $I_C$ is generated by $I_A+I_B$ and one extra element. Two semigroups $⟨A⟩$ and $\langle B\rangle$canbegluedifthereexistpositiveintegers$k_1,k_2$ such that, for $C=k_{1}A\bigsqcup k_{2}B$, $⟨C⟩$ is a gluing of $⟨A⟩$ and $⟨B⟩$. Although any two numerical semigroups, namely semigroups in dimension $n=1$, can always be glued, it is no longer the case in higher dimensions. In this paper, we give necessary and sufficient conditions on $A$ and $B$ for the existence of a gluing of $⟨A⟩$ and $\langle B\rangle$, and give examples to illustrate why they are necessary. These generalize and explain the previous known results on existence of gluing. We also prove that the glued semigroup $⟨C⟩$ inherits the properties like Gorenstein or Cohen-Macaulay from the two parts $⟨A⟩$ and $⟨B⟩$.