Varieties with few subalgebras of powers

Berman, Joel; Idziak, Paweł; Marković, Petar; McKenzie, Ralph; Valeriote, Matthew; Willard, Ross

doi:10.1090/S0002-9947-09-04874-0

Varieties with few subalgebras of powers
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by Joel Berman, Paweł Idziak, Petar Marković, Ralph McKenzie, Matthew Valeriote and Ross Willard

Trans. Amer. Math. Soc. 362 (2010), 1445-1473

DOI: https://doi.org/10.1090/S0002-9947-09-04874-0

Published electronically: October 19, 2009

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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 $\sf {SP}_\textsf {fin}(\mathbf {A})$ 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.

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