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arXiv · 1103.2265

On the number of finite algebraic structures

Erhard Aichinger·Peter Mayr·Ralph McKenzie
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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.

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Erhard Aichinger, Peter Mayr, Ralph McKenzie. 2011-05-30. On the number of finite algebraic structures. https://arxiv.org/abs/1103.2265

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