On the number of finite algebraic structures

TL;DR

This paper proves that all Malcev clones on finite sets are finitely related, showing the countability of such clones and finite algebras with few subpowers, resolving a long-standing problem from 1980.

Contribution

It establishes that Malcev clones on finite sets are finitely related and countable, completing a decades-old problem about the number of such clones.

Findings

All Malcev clones on finite sets are finitely related.

The set of Malcev clones on a fixed finite set is countable.

Finite algebras with few subpowers have finitely related clones.

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.