Commutator Studies in Pursuit of Finite Basis Results

Several new results of a general algebraic scope are developed in an effort to build tools for use in finite basis proofs. Many recent finite basis theorems have involved assumption of a finite residual bound, with the broadest result concerning varieties with a difference term (Kearnes, Szendrei, and Willard (2013+)). However, in varieties with a difference term, the finite residual bound hypothesis is known to strongly limit the degree of nilpotence observable in a variety, while, on the other hand, there is another, older series of results in which nilpotence plays a key role (beginning with those of Lyndon (1952) and Oates and Powell (1964).) Thus, we have chosen to further study nilpotence, commutator theory, and related matters in fairly general settings. Among other results, we have been able to establish the following:• If variety V has a finite signature, is generated by a nilpotent algebra and possesses a finite 2-freely generated algebra, then for all large enough N, the variety based on the N-variable laws true in V is locally finite and has a finite bound on the index of the annihilator of any chief factor of its algebras.• If variety V has a finite signature, is congruence permutable, locally finite and generated by a supernilpotent algebra, then V is finitely based.We have also established several new results concerning the commutator in varieties with a difference term, including an order-theoretic property, a “homomorphism” property, a property concerning affine behavior, and new characterizations of nilpotence in such a setting—extending work of Smith (1976), Freese and McKenzie (1987), Lipparini (1994), and Kearnes (1995).