Linear identities in complex algebras of subalgebras
by
Agata Pilitowska
Warsaw University of Technology, Faculty of Mathematics and Information Science, Warsaw, Poland
Coauthors: Kira Adaricheva, Harold Washington College, Chicago, US

For an algebra G we define the complex algebra of subsets of G as the algebra of the same type as G whose carrier consists of all non-empty subsets of the underlying set of G. If instead of all subsets of G we consider only non-void subalgebras of G, so we receive the complex algebra of subalgebras SG. Evidently, the algebra SG is not always defined for an arbitrary algebra G. We describe the necessary and sufficient condition (called complex condition) for varieties V to guarantee for any algebra from V, the complex algebra of subalgebras is defined.

It was proved by G.Gratzer and H. Lakser that the variety VUS(G) generated by all algebras of subsets for the algebras from V(G) coincide with V(G) iff G is defined by linear identities. The question whether, if the variety VS(G) generated by all complex algebras of subalgebras of the algebras from V(G) is defined and VS(G)=V(G) then G is defined by linear or idempotent identities, is still open. We show that the premise of this conjecture can not be replaced with the weaker condition that SG is defined and V(SG)=V(G).

Date received: May 19, 2004