Adjoining units to residuated Boolean algebras (original) (raw)

We consider a variety ~K of r-algebras, -residuated Boolean algebras, and ask under what conditions a member A of "f-can be embedded in a member A' 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 variety ~, as long as ~K is closed under canonical extensions. This is so because if any extension of A has a unit, then the canonical extension has a unit. The second surprise is that, for varieties "K closed under canonical extensions, the members for which this extension has a unit form a subvariety with a very simple equational basis relatively to ~/~. Applied to the variety of all relation algebras, this latter result solves a problem of long standing due to A. Tarski. This problem was solved independently by H. Andr6ka and I. N6meti.