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

TL;DR

This paper characterizes when semigroups in higher dimensions can be glued from smaller semigroups, extending known results from numerical semigroups, and explores how properties like Gorenstein or Cohen-Macaulay are inherited.

Contribution

It provides necessary and sufficient conditions for gluing in higher dimensions and shows property inheritance in the resulting semigroup.

Findings

Necessary and sufficient conditions for gluing in higher dimensions

Examples illustrating the conditions

Inheritance of properties like Gorenstein and Cohen-Macaulay

Abstract

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 $k_1,k_2\geq 1$, 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 $\langle B\rangle$.