# 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

## Abstract

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.

## 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