András Simon COMPLEXITY OF EQUATIONAL THEORY OF RELATIONAL ALGEBRAS WITH PROJECTION ELEMENTS

In connection with a problem of L. Henkin and J.D. Monk we show that the variety generated by TPA’s – relation algebras (RA’s) expanded with concrete set theoretical projection functions – and the first–order theory of the class TPA are not axiomatizable by any decidable set of axioms. Indeed, we show that Eq(TPA) – all equations valid in TPA – is exactly on the Π1 level. The same applies if we replace TPA’a with the expansions of RA’s suggested by P.A.S. Veloso and A.M. Haeberer in [17] as a candidate for finitary algebraization of first–order logic. Finally, we introduce TPA−, the RA–reducts of TPA, and prove that Eq(TPA−) is recursively enumerable, but not finitely axiomatizable.