Atlas: On n-permutable and distriburive at 0 varieties by Ivan Chajda

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AAA60: Workshop on General Algebra (60. Arbeitstagung Allgemeine Algebra)
June 22-25, 2000
Dresden University of Technology
Dresden, Germany

Organizers
Reinhard Pöschel, Manfred Droste, Bernhard Ganter

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On n-permutable and distriburive at 0 varieties
by
Ivan Chajda
Palacky University Olomouc, Czech Republic

An algebra A with a constant 0 is called n-permutable at 0 if for any \theta, \phi in Con A it holds

[0]_{\theta\bullet\phi\bullet\theta\bullet ...} = [0]_{\phi\bullet\theta\bullet\phi\bullet ...}

where the symbol "\bullet" denotes relational products and there is exactly n-factors on every side of the foregoing equality. A variety V with 0 is n-permutable at 0 if each A in V has this property.

We are able to characterize varieties of 3-permutable at 0 varieties by a Mal'cev type condition but we do not know how to solve it for n >= 4. On the other hand, we can do it in the case if is simultaneously distributive at 0, i.e. if for each A in V and any \theta, \phi, \psi in Con A the condition

[0]_{\phi \cap (\theta \/ \psi)}=[0]_{(\phi \cap \theta) \/ (\phi \cap \psi)}

is satisfied. The corresponding Mal'cev type conditions will be explicitly listed.

Date received: April 27, 2000