On gluing semigroups in Nn and the consequences

A semigroup ⟨ C ⟩ in N n is a gluing of ⟨ A ⟩ and ⟨ B ⟩ if its finite set of generators C splits into two parts, C = k 1 A ⊔ k 2 B with k 1 , k 2 ≥ 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 ⟨ B ⟩ can be glued if there exist positive integers k 1 , k 2 such that for C = k 1 A ⊔ 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 ⟨ B ⟩ , 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 ⟩ .