Joel Berman

We present an upper bound for the cardinality of any n -generated algebra in a locally finite variety ???? of algebras. This upper bound depends only on some fundamental numerical invariants of the n -generated subdirectly irreducible algebras in ???? . A theorem characterizing those varieties that contain algebras whose cardinalities achieve the upper bound is proved. Several explicit methods for computing the exact values of these invariants are described. The final section contains detailed concrete examples illustrating applications of the characterization theorem and of the various methods for computing the upper bound.

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.