Keith A. Kearnes

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We develop the theories of the strong commutator, the rectangular commutator, the strong rectangular commutator, as well as a solvabil-ity theory for the nonmodular TC commutator. These theories are used to show that each of the following sets of statements are equivalent for a variety V of algebras. (I) (a) V satisfies a nontrivial congruence identity. (b)(More)
We describe a new way to construct large subdirectly irreducibles within an equational class of algebras. We use this construction to show that there are forbidden geometries of multitraces for finite algebras in residually small equational classes. The construction is first applied to show that minimal equational classes generated by simple algebras of(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 nite, locally solvable varieties. For locally solvable varieties, this solves problems 9 and 10 of Hobby and McKenzie, 6]. We generalize part of this result by proving that all locally nite varieties(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 for infinite algebras and the converse is false even for finite algebras. We show further that any solvable E-minimal algebra is left nilpotent, any finite(More)