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 Dalmau's strategy.

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A k-edge operation ϕ on a finite set A is a k + 1-ary operation that satisfies the identities

Abstract. This paper studies the complexity of determining if a finite algebra generates a variety that satisfies various Maltsev conditions, such as congruence distributivity or modularity. For idempotent algebras we show that there are polynomial time algorithms to test for these conditions but that in general these problems are EXPTIME complete. In addition, we provide sharp bounds in terms of the size of two-generated free algebras on the number of terms needed to witness various Maltsev conditions, such as congruence distributivity.

Abstract. We describe an algebraic approach to the constraint satisfaction problem (CSP) and present recent results on the CSP that make use of, in an essential way, this algebraic framework.