Generalized diagonal algebras and rectangular algebras
by
Alena Vanžurová
Palacký University Olomouc

The theory of generalized diagonal algebras introduced by J. Plonka is closely related to rectangular algebras studied by R.Pöschel and M.Reichel.

The variety of rectangular algebras, RA, can be given by identities: idempotent law, diagonal law and commutability law. Rectangular algebras can be characterized either as homomorphic images of finite products of projection algebras of a given type, or as homomorphic images of subalgebras of direct products of two-element projection algebras of a given type. Subdirectly irreducible rectangular algebras are precisely two-element projection algebras. The variety is solid and not-normally presented.

Let the variety of generalized diagonal algebras, GDA, be introduced by diagonality and commutability. It appears that GDA is a nilpotent shift of RA. All subdirectly irreducible algebras of a nilpotent shift of a variety are those of the original variety and two-element constant algebras. As a consequence we obtain

Theorem. All subdirectly irreducible algebras in GDA of a given finite type are two-element constant algeras and two-element projection algebras.

Most of the results known for GDA with a single n-ary operation remain valid for a single operation of infinite arity.

Date received: June 1, 2000

Copyright © 2000 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 # caee-89.