Varieties with few subalgebras of powers

Source abstract

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.