Commutator theory for loops

TL;DR

This paper extends commutator theory from congruence modular varieties to loops, linking generators of the commutator to inner mappings and proposing revised definitions for loop invariants.

Contribution

It develops a new commutator theory for loops based on congruence modular varieties and introduces revised definitions for associators and commutators linked to inner mappings.

Findings

Generators of the congruence commutator relate to inner mapping groups

Subloop generated by new associators is automatically normal

Proposes revised definitions linking loop invariants to inner mappings

Abstract

Using the Freese-McKenzie commutator theory for congruence modular varieties as the starting point, we develop commutator theory for the variety of loops. The fundamental theorem of congruence commutators for loops relates generators of the congruence commutator to generators of the total inner mapping group. We specialize the fundamental theorem into several varieties of loops, and also discuss the commutator of two normal subloops. Consequently, we argue that some standard definitions of loop theory, such as elementwise commutators and associators, should be revised and linked more closely to inner mappings. Using the new definitions, we prove several natural properties of loops that could not be so elegantly stated with the standard definitions of loop theory. For instance, we show that the subloop generated by the new associators defined here is automatically normal. We conclude with a preliminary discussion of abelianess and solvability in loops.