Pretorsion theories, stable category and preordered sets

TL;DR

This paper introduces a pretorsion theory in the category of preordered sets, distinguishing torsion and torsion-free objects, and constructs a stable category by factoring out objects that are both.

Contribution

It defines a natural pretorsion theory in preordered sets and constructs a related stable category, advancing categorical understanding of preordered structures.

Findings

Partially ordered sets are torsion-free objects.

Sets with an equivalence relation are torsion objects.

A stable category is constructed by factoring out torsion and torsion-free objects.

Abstract

We show that in the category of preordered sets, there is a natural notion of pretorsion theory, in which the partially ordered sets are the torsion-free objects and the sets endowed with an equivalence relation are the torsion objects. Correspondingly, it is possible to construct a stable category factoring out the objects that are both torsion and torsion-free.