Keith A. Kearnes

AN ORDER-THEORETIC PROPERTY OF THE COMMUTATOR

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 algebra whose congruence lattice contains a 0, 1-sublattice isomorphic to M3 is left nilpotent and any homomorphic image of a finite abelian algebra is left and right nilpotent.

Commutator theory for relatively modular quasivarieties

We develop a commutator theory for relatively modular quasivarieties that extends the theory for modular varieties. We characterize relatively modular quasivarieties, prove that they have an almost-equational axiomatization and we investigate the lattice of subquasivarieties. We derive the result that every finitely generated, relatively modular quasivariety of semigroups is finitely based

Minimal sets and varieties

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 generated by nilpotent algebras that have a trivial locally strongly solvable subvariety are congruence permutable.

Semilattice modes I: the associated semiring

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 small. We refine the proof of residual smallness by showing that any variety of semilattice modes of finite type is residually countable. To each variety of semilattice modes we associate a commutative semiring satisfying 1 +r=1 whose structure determines many of the properties of the variety. This semiring is used to describe subdirectly irreducible members, clones, subvariety lattices, and free spectra of varieties of semilattice modes.

CLONES OF ALGEBRAS WITH PARALLELOGRAM TERMS

We describe a manageable set of relations that generates the finitary relational clone of an algebra with a parallelogram term. This result applies to any algebra with a Maltsev term and to any algebra with a near unanimity term. One consequence of the main result is that on any finite set and for any finite k there are only finitely many clones of algebras with a k-ary parallelogram term which generate residually small varieties.

Residually finite, congruence meet-semidistributive varieties of finite type have a finite residual bound

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.

Inherently nonfinitely based solvable algebras

We prove that an inherently nonfinitely based algebra cannot generate an abelian variety. On the other hand, we show by example that it is possible for an inherently nonfinitely based algebra to generate a strongly solvable variety.

A geometric consequence of residual smallness

Abstract 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 types 2, 3 or 4 are residually small if and only if they are congruence modular. As a second application of the construction we characterize residually small locally finite abelian equational classes.

Cardinalities of Residue Fields of Noetherian Integral Domains

We determine the relationship between the cardinality of a Noetherian integral domain and the cardinality of a residue field. One consequence of the main result is that it is provable in Zermelo–Fraenkel Set Theory with Choice (ZFC) that there is a Noetherian domain of cardinality ℵ1 with a finite residue field, but the statement “There is a Noetherian domain of cardinality ℵ2 with a finite residue field” is equivalent to the negation of the Continuum Hypothesis.

Congruence join semidistributivity is equivalent to a congruence identity

Abstract. We show that a locally finite variety is congruence join semidistributive if and only if it satisfies a congruence identity that is strong enough to force join semidistributivity in any lattice.