Gluing semigroups: when and how

Given two semigroups $$\langle A\rangle$$ and $$\langle B\rangle$$ in $${\mathbb {N}}^n$$ , we wonder when they can be glued, i.e., when there exists a semigroup $$\langle C\rangle$$ in $${\mathbb {N}}^n$$ such that the defining ideals of the corresponding semigroup rings satisfy that $$I_C=I_A+I_B+\langle \rho \rangle$$ for some binomial $$\rho$$ . If $$n\ge 2$$ and $$k[A]$$ and $$k[B]$$ are Cohen–Macaulay, we prove that in order to glue them, one of the two semigroups must be degenerate. Then we study the two most degenerate cases: when one of the semigroups is generated by one single element (simple split) and the case where it is generated by at least two elements and all the elements of the semigroup lie on a line. In both cases we characterize the semigroups that can be glued and say how to glue them. Further, in these cases, we conclude that the glued $$\langle C\rangle$$ is Cohen–Macaulay if and only if both $$\langle A\rangle$$ and $$\langle B\rangle$$ are also Cohen–Macaulay. As an application, we characterize precisely the Cohen–Macaulay semigroups that can be glued when $$n=2$$ .