Functionally dense relation algebras
type: Item
instance of: scholarly article
title: Functionally dense relation algebras
zbMATH Open document ID: 1272.03161
author: Steven R. Givant
author: Hajnal Andréka
published in: Algebra Universalis
publication date: 2012-11-27T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 03G15
zbMATH DE Number: 6110482
zbMATH Keywords: representable relation algebra
zbMATH Keywords: perfect extension
zbMATH Keywords: canonical extension
zbMATH Keywords: completion
zbMATH Keywords: group
zbMATH Keywords: Brandt groupoid
zbMATH Keywords: matrix algebra
reviewed by: Roger Maddux
MaRDI profile type: Publication
full work available at URL: s00012-012-0197-9
OpenAlex ID: W2030278250
cites work: Q5810602
cites work: The Structure of Relation Algebras Generated by Relativizations
cites work: Complete representations in algebraic logic
cites work: Relation algebras by games
cites work: Atom structures of cylindric algebras and relation algebras
cites work: Relation algebras and Schröder categories
cites work: Boolean Algebras with Operators. Part I
cites work: Some sufficient conditions for the representability of relation algebras
cites work: Completions of B<scp>OOLEAN</scp> Algebras with operators
DOI: 10.1007/S00012-012-0197-9
description: scientific article; zbMATH DE number 6110482