Normal extensions and full restricted semidirect products of inverse semigroups

TL;DR

This paper characterizes normal extensions of inverse semigroups as full restricted semidirect products and extends classical embedding theorems by using a more general wreath product construction.

Contribution

It introduces a broader Kalouznin-Krasner theorem for inverse semigroups, utilizing a full restricted semidirect product instead of a lambda-semidirect product, and improves the understanding of kernel class structures.

Findings

Characterization of normal extensions as full restricted semidirect products.

A generalized Kalouznin-Krasner theorem for inverse semigroups.

Enhanced structure of kernel classes in wreath product constructions.

Abstract

We characterize the normal extensions of inverse semigroups isomorphic to full restricted semidirect products, and present a Kalouznin-Krasner theorem which holds for a wider class of normal extensions of inverse semigroups than that in the well-known embedding theorem due to Billhardt, and also strengthens that result in two respects. First, the wreath product construction applied in our result, and stemmming from Houghton's wreath product, is a full restricted semidirect product not merely a lambda-semidirect product. Second, the Kernel classes of our wreath product construction are direct products of some Kernel classes of the normal extension to be embedded rather than only inverse subsemigroups of the direct power of its whole Kernel.