Weakly diagonal algebras and arithmetical affine complete varieties
by
Kalle Kaarli
University of Tartu

We call an algebra bare if it has no proper subalgebras. An algebra A is called weakly diagonal if every subuniverse of A² contains the graph of an automorphism of A. Clearly every weakly diagonal algebra is bare but the converse is not true. It is easily seen that every finitely generated variety contains a largest bare member. Our main results rest on the following somewhat surprising result whose proof is quite easy.

Theorem Let V be a variety generated by a finite algebra A. Then A is weakly diagonal iff it is a largest bare member of V.

This result is applied to categorical equivalence problem of finitely generated arithmetical affine complete varieties and to the problem of existence of a principal Pixley functions in such varieties. In particular it is proved that there is a 1-1 correspondence between categorical equivalence classes of finitely generated arithmetical affine complete varieties and certain finite factorisable monoids whose structure is quite well understood.

Some of the results of this work were obtained jointly with A. Pixley.

Date received: July 7, 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 Conferences Inc. Document # caoc-15.