, Volume 29, Issue 2, pp 345359
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 cubeterm (equivalently, with a parallelogramterm; 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 cubeterm. We present further characterizations of the class of finite idempotent algebras having cubeterms, one of which yields, for idempotent algebras with finitely many basic operations and a fixed finite universe A, a polynomialtime algorithm for determining if the algebra has a cubeterm. We also determine the maximal nonfinitely related idempotent clones over A. The number of these clones is finite.
Keywords
Finitely related clones Cube terms Algebras with few subpowers Valeriote’s conjecture Title
 Finitely Related Clones and Algebras with Cube Terms
 Journal

Order
Volume 29, Issue 2 , pp 345359
 Cover Date
 201207
 DOI
 10.1007/s1108301192322
 Print ISSN
 01678094
 Online ISSN
 15729273
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

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

 Petar Marković ^{(1)}
 Miklós Maróti ^{(2)}
 Ralph McKenzie ^{(3)}
 Author Affiliations

 1. Department of Mathematics, University of Novi Sad, Novi Sad, Serbia
 2. Department of Mathematics, University of Szeged, Szeged, Hungary
 3. Department of Mathematics, Vanderbilt University, Nashville, TN, USA