Relevant logic and relation algebras

4 pages•Published: July 28, 2014

Abstract

In 2007, Maddux observed that certain classes of representable relation
algebras (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 as
well. Later that year I proved a modest positive result in a similar direction,
namely that weakly associative relation algebras, a class (much) larger than
RRA, is sound and complete for positive relevant logic B. In 2008, Mikulas proved a
negative result: that RRAs of transitive relations are not complete for R.
His proof is indirect: he shows that the quasivariety of appropriate
reducts of transitive RRAs is not finitely based. Later Maddux re-established
the 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 language
without Ackermann constant can be embedded into a square-increasing relation
algebra. Therefore, the variety of such algebras is sound and complete for R.

Keyphrases: de morgan monoids, relation algebras, relevant logics

In: Nikolaos Galatos, Alexander Kurz and Constantine Tsinakis (editors). TACL 2013. Sixth International Conference on Topology, Algebra and Categories in Logic, vol 25, pages 125-128.

Links:	https://easychair.org/publications/paper/c83
	https://doi.org/10.29007/8gj7