The Representation of Relation Algebras, II

Abstract

This paper completes, in a sense, the investigation undertaken in our earlier paper RRA;1 in particular, we correct an error in that paper, which was pointed out by Professor Alfred Tarski and Mr. Dana Scott.2 Tarski3 has recently shown, as a consequence of a general theorem, that, contrary to the erroneous Theorem IV of RRA, there exists a set of axioms in the form of equations, which is necessary and sufficient for a relation algebra to be isomorphic to a proper algebra of relations. His theorem provides no means of constructing such a set of axioms. By a refinement of the method of RRA, an explicit set of equational axioms for representable relation algebras is here obtained. The chief new tool is a polarization operator which expresses that a property p of elements x of A, a Boolean algebra with operators,4 is divisible. Divisible properties are dual to local properties, and are such that 4 holds on x only if 0 holds on some member of every finite covering of x. By this device, assertions of the existence of certain maximal dual ideals in 2[ are expressed by universal sentences in the elementary language of W. A second crucial fact, implicit in much of our argument, is that the Stone-Jo6nsson-Tarski completion I of Wf is compact under the topology defined by 20 For I countable and simple, a countable Boolean representation of a Boolean algebra dense in ft is contructed in such a way that it yields a representation of 2[ as relation algebra. The condition of simplicity is removed by a device of Tarski, and that of countability by a theorem of Henkin.' In an Appendix we itemize the erroneous statements in RRA, and formulate a proper reinterpretation of the parts thus affected. In this connection we reproduce an example, due to Tarski,2 of a complete and atomistic algebra of sets in which not every element of a set in the algebra belongs to an atom.