Decision problems for equational theories of relation algebras

Abstract

The foundation of an algebraic theory of binary relations was laid by C. S. Peirce, building on earlier work of Boole and De Morgan. The basic universe of discourse of this theory is a collection of binary relations over some set, and the basic operations on these relations are those of forming unions, complements, relative products (i.e., compositions), and converses (i.e., inverses). There is also a distinguished relation, the identity relation. Other operations and distinguished relations studied by Peirce are definable in terms of the ones just mentioned. Such an algebra of relations is called a set relation algebra. A modern development of this theory as a theory of abstract relation algebras, axiomatized by a finite set of equations, was undertaken by Tarski and his students and colleagues, beginning around 1940. In 1942, Tarski proved that all of classical mathematics could be developed within the framework of the equational theory of relation algebras. Indeed, he created a general method for interpreting into the equational theory of relation algebras first-order theories that are strong enough to form a basis for the development of mathematics, in particular, set theories and number theories. As a consequence, he established that the equational theories of relation algebras and of set relation algebras are undecidable (see [10] and [11], and see [2] or [11], in particular Chapter 8, for unexplained terminology). They were the first known examples of undecidable equational theories. As was pointed out in [11], Tarski’s proof actually shows more. Namely, any class of relation algebras that contains the full set relation algebra on some infinite set (i.e., the set relation algebra whose universe consists of all binary relations on the infinite set) or, equivalently, that contains all set relation algebras on infinite sets must have an undecidable equational theory.