Weakly associative relation algebras with projections

Built on the foundations laid by Peirce, Schröder and others in the 19th century, the modern development of relation algebras started with the work of Tarski and his colleagues [21, 22]. They showed that relation algebras can capture strong rst-order theories like ZFC, and so their equational theory is undecidable. The less expressive classWA of weakly associative relation algebras were introduced by Maddux [7]. Németi [16] showed that WAs have a decidable universal theory. There has been extensive research on increasing the expressive power of WA by adding new operations [1, 4, 11, 13, 20]. Extensions of this kind usually also have decidable universal theories. Here we give an example——extending WAs with set-theoretic projection elements——where this is not the case. These ‘‘logical’’ connectives are set-theoretic counterparts of the axiomatic quasi-projections that have been investigated in the representation theory of relation algebras [6, 19, 22]. We prove that the quasi-equational theory of the extended class PWA is not recursively enumerable. By adding the difference operator one can turnWA and PWA to discriminator classes where each universal formula is equivalent to some equation. Hence our result implies that the projections turn the decidable equational theory of ‘‘WA ’’ to non-recursively enumerable.