Varieties with few subalgebras of powers

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