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

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.