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

2019/02/14 by Facchini, Alberto, Zadeh, Leila Heidari
#18A99 #20M20 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1902.05507

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,…,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→ X, X an arbitrary finite non-empty set, in which bijections are exactly the torsion objects.

Related

• Building pretorsion theories from torsion theories
• Short presentations for transformation monoids
• Torsion groups and the Bienvenu–Geroldinger conjecture
• Monoids in the mapping class group
• Simplicity of augmentation submodules for transformation monoids