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On the number of finite algebraic structures

Erhard AichingerPeter MayrR. McKenzie — 2014

Journal of the European Mathematical Society

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...

Quasiequational theories of flat algebras

Jaroslav JežekM. MarótiR. McKenzie — 2005

Czechoslovak Mathematical Journal

We prove that finite flat digraph algebras and, more generally, finite compatible flat algebras satisfying a certain condition are finitely q -based (possess a finite basis for their quasiequations). We also exhibit an example of a twelve-element compatible flat algebra that is not finitely q -based.

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