, Volume 29, Issue 2, pp 345–359

Finitely Related Clones and Algebras with Cube Terms



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 


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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Petar Marković
    • 1
  • Miklós Maróti
    • 2
  • Ralph McKenzie
    • 3
  1. 1.Department of MathematicsUniversity of Novi SadNovi SadSerbia
  2. 2.Department of MathematicsUniversity of SzegedSzegedHungary
  3. 3.Department of MathematicsVanderbilt UniversityNashvilleUSA

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