On split extensions of preordered groups

TL;DR

This paper explores the structure of split extensions in preordered groups, focusing on how lexicographic order influences their compatibility, and establishes key conditions and properties for these extensions.

Contribution

It provides necessary and sufficient conditions for compatible orders in semidirect products and characterizes classes of split extensions with classifiers.

Findings

Lexicographic order determines compatibility in semidirect products.

Split Short Five Lemma holds for stably strong split extensions.

Identifies classes of split extensions with classifiers.

Abstract

We investigate the behaviour of split extensions in the category OrdGrp of (pre)\-ordered groups. Namely we show that the lexicographic order plays a key role on the existence of compatible orders for semidirect products, establishing necessary and sufficient conditions for such existence; we prove that the Split Short Five Lemma holds for stably strong split extensions, and identify classes of split extensions which admit a classifier.