Internal structures in the category of right-preordered groups

TL;DR

This paper develops an algebraic framework for right-preordered groups, explores their internal structures, and establishes properties like S-protomodularity and action representability, including classifications of groupoids.

Contribution

It provides explicit axioms for right-preordered groups, analyzes their internal structures, and proves key properties such as S-protomodularity and action representability.

Findings

Established S-protomodularity of right-preordered groups with Schreier split epimorphisms.

Proved the category of right-preordered groups is action representable for certain split epimorphisms.

Characterized groupoids within Schreier internal categories.

Abstract

We give explicit axioms for the algebraic theory of the quasivarieties of right-preordered groups and preordered groups. We then look at lattices of effective equivalence relations, which turn out to be similar to the lattices of equivalence relations in the category of groups. Once this is established, we study internal structures in the category of right-preordered groups. We start with some general results and then prove the S-protomodularity of the category of right-preordered groups, when considering the class S of Schreier split epimorphisms. Following this, we investigate further and prove that the category of right-preordered groups turns out to be action representable when we restrict our attention to split epimorphisms in S. Relatively to this class of split epimorphisms, we define the notion of S-precrossed modules, and then of S-crossed modules; that correspond exactly to Schreier internal reflexive graphs and Schreier internal categories, respectively. Lastly, we characterize groupoids among Schreier internal categories and give some examples.