On isomorphisms of $\mathcal{R}$- and $\mathcal{L}$-cross-sections of wreath products of finite inverse symmetric semigroups

TL;DR

This paper classifies and counts the isomorphism classes of certain cross-sections in wreath products of finite inverse symmetric semigroups, showing all isomorphisms are conjugacies.

Contribution

It provides a complete classification of $ $- and $ L$-cross-sections of these wreath products up to isomorphism, establishing conjugacy as the key equivalence.

Findings

All isomorphisms of cross-sections are conjugacies.

Explicit enumeration of non-isomorphic cross-sections.

Auxiliary result on isomorphisms of cross-sections of $ ext{IS}_n$.

Abstract

We classify $\mathcal{R}$- and $\mathcal{L}$-cross-sections of wreath products of finite inverse symmetric semigroups $\mathcal{IS}_m \wr_p \mathcal{IS}_n$ up to isomorphism. We show that every isomorphism of $\mathcal{R}$ ($\mathcal{L}$-) cross-sections of $\mathcal{IS}_m \wr_p \mathcal{IS}_n$ is a conjugacy. As an auxiliary result, we get that every isomorphism of $\mathcal{R}$- ($\mathcal{L}$-) cross-sections of $\mathcal{IS}_n$ is also a conjugacy. We also compute the number of non-isomorphic $\mathcal{R}$- ($\mathcal{L}$-) cross-sections of $\mathcal{IS}_m \wr_p \mathcal{IS}_n$.