, Volume 29, Issue 2, pp 345-359

First online:

Finitely Related Clones and Algebras with Cube Terms

  • Petar MarkovićAffiliated withDepartment of Mathematics, University of Novi Sad
  • , Miklós MarótiAffiliated withDepartment of Mathematics, University of Szeged
  • , Ralph McKenzieAffiliated withDepartment of Mathematics, Vanderbilt University Email author 

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Aichinger et al. (2011) have proved that every finite algebra with a cube-term (equivalently, with a parallelogram-term; equivalently, having few subpowers) is finitely related. Thus finite algebras with cube terms are inherently finitely related—every expansion of the algebra by adding more operations is finitely related. In this paper, we show that conversely, if A is a finite idempotent algebra and every idempotent expansion of A is finitely related, then A has a cube-term. We present further characterizations of the class of finite idempotent algebras having cube-terms, one of which yields, for idempotent algebras with finitely many basic operations and a fixed finite universe A, a polynomial-time algorithm for determining if the algebra has a cube-term. We also determine the maximal non-finitely related idempotent clones over A. The number of these clones is finite.


Finitely related clones Cube terms Algebras with few subpowers Valeriote’s conjecture