The Zappa-Szep product of left-orderable groups

TL;DR

This paper extends the understanding of how complex group constructions, specifically the Zappa-Szep product, preserve left-orderability and bi-orderability, providing new conditions for these properties in such products.

Contribution

It introduces conditions under which the Zappa-Szep product of left-orderable and bi-orderable groups remains left-orderable or bi-orderable, expanding the class of groups known to have these orderings.

Findings

Zappa-Szep product of left-orderable groups is left-orderable under certain conditions.

Conditions for the existence of partial left and right invariant orderings in Zappa-Szep products.

Properties of bi-orderability in Zappa-Szep products of bi-orderable groups.

Abstract

It is well-known that the direct product of left-orderable groups is left-orderable and that, under a certain condition, the semi-direct product of left-orderable groups is left-orderable. We extend this result and show that, under a similar condition, the Zappa-Szep product of left-orderable groups is left-orderable. Moreover, we find conditions that ensure the existence of a partial left and right invariant ordering (bi-order) in the Zappa-Szep product of bi-orderable groups and prove some properties.