Adjoining units to residuated Boolean algebras

Abstract

We consider a varietyV ofr-algebras, — residuated Boolean algebras, — and ask under what conditions a memberA ofV can be embedded in a memberA' having a unit element. The answer, although quite simple, is somewhat surprising for two reasons. First, to a large extent the answer is independent of the varietyV, as long asV is closed under canonical extensions. This is so because if any extension ofA has a unit, then the canonical extension has a unit. The second surprise is that, for varietiesV closed under canonical extensions, the members for which this extension has a unit form a subvariety with a very simple equational basis relatively toV. Applied to the variety of all relation algebras, this latter result solves a problem of long standing due toA. Tarski. This problem was solved independently by H. Andréka and I. Németi.