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

A semigroup $$\langle C\rangle $$ in $$\mathbb {N}^n$$ is a gluing of $$\langle A\rangle $$ and $$\langle B\rangle $$ if its finite set of generators C splits into two parts, $$C=k_1A\sqcup k_2B$$ 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 $$\langle A\rangle $$ and $$\langle B\rangle $$ can be glued if there exist positive integers $$k_1,k_2$$ such that for $$C=k_1A\sqcup k_2B$$ , $$\langle C\rangle $$ is a gluing of $$\langle A\rangle $$ and $$\langle B\rangle $$ . 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 $$\langle A\rangle $$ 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 $$\langle C\rangle $$ inherits the properties like Gorenstein or Cohen–Macaulay from the two parts $$\langle A\rangle $$ and $$\langle B\rangle $$ .