Ralph McKenzie

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 S P fin ( A ) \sf {SP}_\textsf {fin}(\mathbf {A}) of subalgebras of finite Cartesian powers of a finite universal algebra A \mathbf {A} . One particular strategy, advanced by Dalmau in his doctoral thesis (2000), has confirmed the conjecture for a certain class of finite algebras A \mathbf {A} which, among other things, have the property that the number of subalgebras of A n \mathbf {A}^n is bounded by an exponential polynomial. In this paper we characterize the finite algebras A \mathbf {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 \mathbf {A}^n and some related functions on the one hand, and some standard algebraic properties of A \mathbf {A} on the other hand. The theory developed here was applied to the Constraint Satisfaction Problem Dichotomy Conjecture, completing Dalmau’s strategy.

We give a new proof that a finitely generated congruence-distributive variety has finitely determined syntactic congruences (or, equivalently, term finite principal congruences), and show that the same does not hold for finitely generated congruence-permutable varieties, even under the additional assumption that the variety is residually very finite.

: Assume that all algebras are atomless. (1) Spind(A × B) = Spind(A) ∪ Spind(B). (2) Spind(A_i). Now suppose that κ and λ are infinite cardinals, with κ uncountable and regular and with κ < λ. (3) There is an atomless Boolean algebra A such that u(A) = κ and i(A) = λ. (4) If λ is also regular, then there is an atomless Boolean algebra A such that t(A) = s(A) = κ and α (A) = λ. All results are in ZFC, and answer some problems posed in Monk [01] and Monk [∞].