Geodesic
On the number of finite algebraic structures
Journal of the European Mathematical Society, Tome 16 (2014) no. 8, pp. 1673-1686

Voir la notice de l'article provenant de la source EMS Press

We prove that every clone of operations on a finite set A, if it contains a Malcev operation, is finitely related – i.e., identical with the clone of all operations respecting R for some finitary relation R over A. It follows that for a fixed finite set A, the set of all such Malcev clones is countable. This completes the solution of a problem that was first formulated in 1980, or earlier: how many Malcev clones can finite sets support? More generally, we prove that every finite algebra with few subpowers has a finitely related clone of term operations. Hence modulo term equivalence and a renaming of the elements, there are only countably many finite algebras with few subpowers, and thus only countably many finite algebras with a Malcev term.
DOI : 10.4171/jems/472
Classification : 08-XX, 00-XX
Keywords: Malcev conditions, few subpowers, term equivalence, clones, relations 
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Erhard Aichinger; Peter Mayr; Ralph McKenzie. On the number of finite algebraic structures. Journal of the European Mathematical Society, Tome 16 (2014) no. 8, pp. 1673-1686. doi: 10.4171/jems/472

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