Gluing semigroups -- when and how

TL;DR

This paper investigates the conditions under which two semigroups in a0^n can be combined into a larger semigroup, especially focusing on Cohen-Macaulay cases and specific degenerate scenarios, providing characterizations and gluing methods.

Contribution

It characterizes when semigroups can be glued in a0^n, especially in degenerate cases, and relates Cohen-Macaulay properties of the original and glued semigroups.

Findings

Gluing requires one semigroup to be degenerate if both are Cohen-Macaulay.

Characterization of gluing in simple split and colinear cases.

Glued semigroup is Cohen-Macaulay iff both original semigroups are Cohen-Macaulay.

Abstract

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 $n\geq 2$ 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 $n=2$.