A pretorsion theory for right groups

TL;DR

This paper introduces a pretorsion theory for right groups, showing how they decompose into products of groups and right zero semigroups, and explores their categorical coproduct structure.

Contribution

It establishes a pretorsion theory in the category of right groups, characterizing their structure via congruences and coproduct decompositions.

Findings

Right groups can be decomposed into products of groups and right zero semigroups.

A pretorsion theory with right zero semigroups as torsion objects and groups as torsion-free objects is developed.

The coproduct structure of pointed right groups is characterized in the category of pointed right groups.

Abstract

Let $S$ be a right group. Then there exist two congruences $\sim$ and $\equiv$ on $S$ such that $S$ is the product of its quotient semigroups $S/{\sim}$ and $S/{\equiv}$, where $S/{\sim}$ is a group and $S/{\equiv}$ is a right zero semigroup. If $E$ is the set of all idempotents of $S$ and we fix an element $e_0\in E$, then the pointed right group $(S,e_0)$ is the coproduct of its pointed subsemigroups $(Se_0,e_0)$ and $(E,e_0)$ in the category of pointed right groups. In general, there is a pretorsion theory in the category of right groups in which the torsion objects are right zero semigroups and the torsion-free objects are groups.