Relevant logic and relation algebras

Abstract

In 2007, Maddux observed that certain classes of representable relationalgebras (RRAs) form sound semantics for some relevant logics. In particular,(a) RRAs of transitive relations are sound for R, and (b) RRAs of transitive,dense relations are sound for RM. He asked whether they were complete aswell. Later that year I proved a modest positive result in a similar direction,namely that weakly associative relation algebras, a class (much) larger thanRRA, is sound and complete for positive relevant logic B. In 2008, Mikulas proved anegative result: that RRAs of transitive relations are not complete for R.His proof is indirect: he shows that the quasivariety of appropriatereducts of transitive RRAs is not finitely based. Later Maddux re-establishedthe result in a more direct way. In 2010, Maddux proved a contrasting positive result: that transitive, dense RRAs are complete for RM. He found an embedding of Sugihara algebras into transitive, dense RRAs. I will show that if we give up the requirement of representability, the positive result holds for R as well. To be precise, the following theorem holds.Theorem. Every normal subdirectly irreducible De Morgan monoid in the languagewithout Ackermann constant can be embedded into a square-increasing relationalgebra. Therefore, the variety of such algebras is sound and complete for R.