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AAA68: Workshop on General Algebra (68. Arbeitstagung Allgemeine Algebra)
June 10-13, 2004
Technische Universität Dresden
Dresden, Germany

Organizers
Reinhard Pöschel, Bernhard Ganter

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Groups, trees and quasigroup identities
by
Aleksandar Krapez
Matematicki institut SANU, Beograd, Serbia and Montenegro

Groups, trees and quasigroup identities


A. Krapez
Matematicki institut SANU,
Beograd,
Serbia and Montenegro


Abstract

A balanced identity s = t (in the language using multiplication only) is height preserving if the height of every variable is equal in s and t . It is a level identity if the height of any two variables is equal in both s and t . Commutativity and mediality (bisymmetry) are examples of level identities. The identity x ·yz = zy ·x is height preserving but not level. Associativity is not even height preserving.

Every height preserving quasigroup identity is either Belousov (i.e. a consequence of commutativity) or non-Belousov (i.e. implying group isotopy). The lattice of Belousov varieties is described in the article: A. Krapez, M. A. Taylor: Irreducible Belousov equations on quasigroups, Czechoslovak Math. J. 43 (118), (1993). All quasigroups satisfying non-Belousov identities are T-quasigroups and therefore isotopic to abelian groups.

Level identities can be understood as actions of permutations on the trees of terms of (the left and the right hand sides of) identities. Belousov identities then correspond to automorphisms of full binary trees of certain height. Varieties of quasigroups satisfying level (consequently: any height preserving) identities of height up to h can be related to special subgroups of the group of all permutations S2h .

Date received: May 19, 2004

Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # canq-74.