An extension of properties of symmetric group to monoids and a pretorsion theory in the category of mappings

TL;DR

This paper explores how properties of the symmetric group extend to the full transformation monoid, introducing a pretorsion theory where bijections are identified as torsion objects.

Contribution

It extends elementary properties of symmetric groups to monoids and establishes a pretorsion theory in the category of mappings with bijections as torsion objects.

Findings

Symmetric group properties extend to the transformation monoid.

Bijections are characterized as torsion objects within a pretorsion theory.

A new categorical framework for understanding mappings and their torsion elements.

Abstract

Several elementary properties of the symmetric group $S_n$ extend in a nice way to the full transformation monoid $M_n$ of all maps of the set $X:=\{1,2,3,\dots,n\}$ into itself. The group $S_n$ turns out to be in some sense the torsion part of the monoid $M_n$. More precisely, there is a pretorsion theory in the category of all maps $f\colon X\to X$, $X$ an arbitrary finite non-empty set, in which bijections are exactly the torsion objects.