Commutator theory for loops

JOURNAL OF ALGEBRA

卷:399

Article

Stanovsky, David^1,2  Vojtechovsky, Petr¹ 

[1] Univ Denver, Dept Math, Denver, CO 80208 USA

[2] Charles Univ Prague, Fac Math & Phys, Dept Algebra, Prague 18675 8, Czech Republic

关键词: Commutator theory;    Congruence commutator;    Loop;    Commutator of normal subloops;    Commutator;    Associator;    Associator subloop;    Derived subloop;    Inner mapping;    Inner mapping group.;    Total inner mapping group;   

DOI  :  10.1016/j.jalgebra.2013.08.045

来源: Elsevier

【 摘 要 】

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. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free