Presentations for wreath products involving symmetric inverse monoids and categories

TL;DR

This paper develops presentations by generators and relations for wreath products involving symmetric inverse monoids, their singular ideals, and categories, which are important in algebra and science.

Contribution

It provides explicit presentations for wreath products of an arbitrary monoid with symmetric inverse monoids, singular ideals, and categories.

Findings

Presentations for $M\wr\mathcal I_n$, $M\wr\operatorname{Sing}(\mathcal I_n)$, and $M\wr\mathcal I$ are derived.

Facilitates algebraic analysis of wreath products in various mathematical contexts.

Enhances understanding of algebraic structures involving symmetric inverse monoids.

Abstract

Wreath products involving symmetric inverse monoids/semigroups/categories arise in many areas of algebra and science, and presentations by generators and relations are crucial tools in such studies. The current paper finds such presentations for $M\wr\mathcal I_n$, $M\wr\operatorname{Sing}(\mathcal I_n)$ and $M\wr\mathcal I$. Here $M$ is an arbitrary monoid, $\mathcal I_n$ is the symmetric inverse monoid, $\operatorname{Sing}(\mathcal I_n)$ its singular ideal, and $\mathcal I$ is the symmetric inverse category.