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On the number of finite algebraic structures

Erhard Aichinger, Peter Mayr, R. 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 Ježek, M. Maróti, R. 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.

Rees ideal algebras

Ivan Chajda (1997)

Mathematica Bohemica

We describe algebras and varieties for which every ideal is a kernel of a one-block congruence.

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