Ross Willard

Varieties with few subalgebras of powers

The Constraint Satisfaction Problem Dichotomy Conjecture of Feder and Vardi (1999) has in the last 10 years been profitably reformulated as a conjecture about the set SP fin (A) of subalgebras of finite Cartesian powers of a finite universal algebra A. One particular strategy, advanced by Dalmau in his doctoral thesis (2000), has confirmed the conjecture for a certain class of finite algebras A which, among other things, have the property that the number of subalgebras of A n is bounded by an exponential polynomial. In this paper we characterize the finite algebras A with this property, which we call having few subpowers, and develop a representation theory for the subpowers of algebras having few subpowers. Our characterization shows that algebras having few subpowers are the finite members of a newly discovered and surprisingly robust Maltsev class defined by the existence of a special term we call an edge term. We also prove some tight connections between the asymptotic behavior of the number of subalgebras of A n and some related functions on the one hand, and some standard algebraic properties of A on the other hand. The theory developed here was applied to the Constraint Satisfaction Problem Dichotomy Conjecture, completing Dalmaus strategy.

Essential arities of term operations in finite algebras

Abstract Given an algebra A , p n ( A ) denotes the number of distinct n -ary term operations t : A n → A of A which depend on all n variables. We solve some problems of Berman, Gratzer and Kisielewicz concerning the sequence 〈 p 0 ( A ), p 1 ( A ),…, p n ( A ),…〉 in case ¦A¦ is finite. Our methods yield new results about totally symmetric functions on a finite set.

Natural dualities for quasivarieties generated by a finite commutative ring

Abstract. Let R be a finite commutative ring with identity. If the Jacobson radical of R annihilates itself, then the quasivariety generated by R is dually equivalent to a category of structured Boolean spaces obtained in a natural way from R. If on the other hand the radical of R does not annihilate itself, then no such natural dual equivalence is possible. To illustrate the first result, a dual equivalence for the quasivariety generated by the ring

Testing expressibility is hard

\Bbb Z _{p^2}

An algebra that is dualizable but not fully dualizable

, where p is prime, is given.

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

We study the expressibility problem: given a finite constraint language Γ on a finite domain and another relation R, can Γ express R? We prove, by an explicit family of examples, that the standard witnesses to expressibility and inexpressibility (gadgets/formulas/conjunctive queries and polymorphisms respectively) may be required to be exponentially larger than the instances. We also show that the full expressibility problem is co-NEXPTIME-hard. Our proofs hinge on a novel interpretation of a tiling problem into the expressibility problem.

ON MCKENZIE'S METHOD

Abstract We give an example of a finite algebra which is dualizable but not fully dualizable in the sense of natural duality theory.

Near Unanimity Constraints Have Bounded Pathwidth Duality

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.

Equational complexity of the finite algebra membership problem

This is an expository account of R. McKenzies recent refutation of the RS conjecture.

SOME PROPERTIES OF FINITELY DECIDABLE VARIETIES

We show that if a finite relational structure has a near unanimity polymorphism, then the constraint satisfaction problem with that structure as its fixed template has bounded pathwidth duality, putting the problem in nondeterministic logspace. This generalizes the analogous result of Dalmau and Krokhin for majority polymorphisms and lends further support to a conjecture suggested by Larose and Tesson.