Functionally dense relation algebras (Q1762473)

A theorem of Maddux and Tarski says that every functionally dense relation algebra is representable. The original proof uses, at Tarski's suggestion, the completion of a relation algebra (as well as the canonical (or perfect) extension, which Jónsson and Tarski had already used in proofs of special cases of that theorem). Tarski's Lemma (Lemma 4.1 of the paper under review) says that if a relation algebra is functionally dense (every nonzero element contains a nonzero functional element) then its completion is ``same-domain functionally dense'' (every nonzero element contains a nonzero functional element with the same domain). A second lemma (which is now the first part of the proof of Theorem~4.3) is that same-domain functional density of \(\mathfrak{A}\) entails representability, proved by mapping each element \(x\) of \(\mathfrak{A}\) to the set of pairs \((u,v)\) of atoms in the canonical extension of \(\mathfrak{A}\) such that \(u\) and \(v\) are included in functional elements of \(\mathfrak{A}\) and \(u\mathrel{;} x\geq v\). An arbitrary functionally dense relation algebra thus gets represented as an algebra of relations on a set of functional atoms in the canonical extension of its completion. The details of this proof, with all the requisite background material, are presented along with many related results. For example, in \S3, the authors show that if a relation algebra is atomic and functionally dense then it is completely representable by observing that a proof of Jónsson and Tarski yields complete representability, not just representability. After proving representability for the class of functionally dense relation algebras, the authors analyze this class with decomposition theorems, characterizations, and numerous examples. In \S5, they show every simple functionally dense relation algebra is either atomic or atomless, and every complete functionally dense relation algebra is isomorphic to a direct product of two algebras -- one of them is a complete functionally dense relation algebra that is atomless and has no simple factors, and the other is a direct product of simple complete functionally dense relation algebras (each either atomic or atomless). Three kinds of algebras are mentioned in this last theorem. One is characterized completely in \S6: a simple relation algebra is complete and atomic with functional atoms if and only if it is isomorphic to a matrix algebra over some group (a relation algebra whose elements are \(\kappa\)-by-\(\kappa\) matrices of sets of elements of the group). In later sections, constructions are given for classes of examples of the other two types, while their characterizations are left as open problems.