On cross-sections of partial wreath product of inverse semigroups

TL;DR

This paper classifies certain cross-sections of the partial wreath product of inverse semigroups, providing new descriptions and counts for these structures, with applications to automorphisms of finite regular rooted trees.

Contribution

It introduces a classification of $ $- and $ $-cross-sections for the partial wreath product of inverse semigroups, extending understanding of their structure and enumeration.

Findings

Classified $ $- and $ $-cross-sections of the partial wreath product

Described $ $- and $ $-cross-sections of automorphisms of finite regular rooted trees

Computed the number of distinct cross-sections in the semigroup

Abstract

We classify $\mathcal{R}$- and $\mathcal{L}$-cross-sections of partial wreath product of inverse semigroups. As a corollary, we get the description of $\mathcal{R}$- and $\mathcal{L}$-cross-sections of the semigroupof partial automorphisms of finite regular rooted tree and compute also the number of different $\mathcal{R}$- ($\mathcal{L}$-) cross-sections in this semigroup.