Functionally dense relation algebras

Abstract

We give a new proof of a theorem due to Maddux and Tarski that every functionally dense relation algebra is representable. Our proof is very close in spirit to the original proof of the theorem of Jónsson and Tarski that atomic relation algebras with functional atoms are representable. We prove that a simple, functionally dense relation algebra is either atomic or atomless, and that every functionally dense relation algebra is essentially isomorphic to a direct product ${\mathfrak{B} \times \mathfrak{C}}$ , where ${\mathfrak{B}}$ is a direct product of simple, functionally dense relation algebras each of which is either atomic or atomless, and ${\mathfrak{C}}$ is a functionally dense relation algebra that is atomless and has no simple factors at all. We give several new structural descriptions of all atomic relation algebras with functional atoms. For example, each such algebra is essentially isomorphic to an algebra of matrices with entries from the complex algebra of some group. Finally, we construct examples of functionally dense relation algebras that are atomless and simple, and examples of functionally dense relation algebras that are atomless and have no simple factors at all.