We develop the theories of the strong commutator, the rectangular commutator, the strong rectangular commutator, as well as a solvability theory for the nonmodular TC commutator. These theories are… (More)

We describe a new order-theoretic property of the commutator for finite algebras. As a corollary we show that any right nilpotent congruence on a finite algebra is left nilpotent. The result is false… (More)

The aim of this paper is twofold. First some machinery is established to reveal the structure of abelian congruences. Then we describe all minimal, locally finite, locally solvable varieties. For… (More)

We clarify the relationship between the linear commutator and the ordinary commutator by showing that in any variety satisfying a nontrivial idempotent Mal’cev condition the linear commutator is… (More)

We show that the class of locally finite varieties omitting type 1 has the following properties. This class (1) is definable by an idempotent, linear, strong Mal’cev condition in a language with one… (More)

We examine idempotent, entropic algebras (modes) which have a semilattice term. We are able to show that any variety of semilattice modes has the congruence extension property and is residually… (More)

We construct two minimal clones on any finite set such that the join of the two clones contains all operations. Dually, we exhibit two maximal clones on any finite set with at least three elements… (More)

We show that a minimal clone has a nontrivial abelian representation if and only if it is isomorphic to a minimal subclone of a finite cyclic group. As an application, we show that a minimal clone… (More)