Complexity of equational theory of relational algebras with standard projection elements

The class TPA of true pairing algebras is defined to be the class of relation algebras expanded with concrete set theoretical projection functions. The main results of the present paper is that neither the equational theory of TPA nor the first order theory of TPA are decidable. Moreover, we show that the set of all equations valid in TPA is exactly on the Π11 level. We consider the class TPA of the relation algebra reducts of TPA’s, as well. We prove that the equational theory of TPA is much simpler, namely, it is recursively enumerable. We also give motivation for our results and some connections to related work. ∗Research supported by Hungarian National Foundation for Scientific Research grant Numbers T030314, T034861, and T035192.