JournalsjemsVol. 16, No. 8pp. 1673–1686

On the number of finite algebraic structures

  • Erhard Aichinger

    Johannes Kepler University, Linz, Austria
  • Peter Mayr

    Johannes Kepler University, Linz, Austria
  • Ralph McKenzie

    Vanderbilt University, Nashville, USA
On the number of finite algebraic structures cover
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Abstract

We prove that every clone of operations on a finite set AA, if it contains a Malcev operation, is finitely related – i.e., identical with the clone of all operations respecting RR for some finitary relation RR over AA. It follows that for a fixed finite set AA, 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.

Cite this article

Erhard Aichinger, Peter Mayr, Ralph McKenzie, On the number of finite algebraic structures. J. Eur. Math. Soc. 16 (2014), no. 8, pp. 1673–1686

DOI 10.4171/JEMS/472