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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 an easy way to determine whether the realization of a set of idempotent identities guarantees congruence modularity or the satisfaction of a nontrivial congruence identity. Our results yield slight strengthenings of Day's Theorem and Gumm's Theorem, which each characterize congruence modularity.

- Keith A. Kearnes, Emil W. Kiss, Matthew Valeriote
- Ann. Pure Appl. Logic
- 1999

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)

- Keith A. Kearnes, Ágnes Szendrei
- IJAC
- 1998

We clarify the relationship between the linear commutator and the ordinary com-mutator by showing that in any variety satisfying a nontrivial idempotent Mal'cev condition the linear commutator is definable in terms of the centralizer relation. We derive from this that abelian algebras are quasi–affine in such varieties. We refine this by showing that if A… (More)

Nearly twenty years ago, two of the authors wrote a paper on congruence lattices of semilattices [9]. The problem of finding a really useful characterization of congruence lattices of finite semilattices seemed too hard for us, so we went on to other things. Thus when Steve Seif asked one of us at the October 1990 meeting of the AMS in Amherst what we had… (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)

- KEITH A. KEARNES
- 2007

We show that a residually finite, congruence meet-semidistributive variety of finite type is residually < N for some finite N. This solves Pixley's problem and a special case of the restricted Quackenbush problem.

- Keith A. Kearnes, Ágnes Szendrei
- IJAC
- 1997

We show that a locally finite variety which omits abelian types is self–rectangulating if and only if it has a compatible semilattice term operation. Such varieties must have type–set {5 }. These varieties are residually small and, when they are finitely generated, they have de-finable principal congruences. We show that idempotent varieties with a… (More)

- Keith A. Kearnes, Steven T. Tschantz
- IJAC
- 2007

We show that certain finite groups do not arise as the automorphism group of the square of a finite algebraic structure, nor as the automorphism group of a finite, 2-generated, free, algebraic structure.

- Keith A. Kearnes
- IJAC
- 1993

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)