Relation algebras with \(n\)-dimensional relational bases
type: Item
instance of: scholarly article
title: Relation algebras with \(n\)-dimensional relational bases
zbMATH Open document ID: 0953.03071
DOI: 10.1016/S0168-0072(99)00022-6
published in: Annals of Pure and Applied Logic
publication date: 2001-01-29T00:00:00Z
review text: This paper provides an illuminating relative representation theory for relation algebras with \(n\)-dimensional bases, and confirms a conjecture, first published in 1983, that the class of \((n+1)\)-dimensional relation algebras is not finitely axiomatizable relative to the larger class of \(n\)-dimensional relation algebras. Every relation algebra \(A\) has a relative representation, that is, an isomorphic embedding of \(A\) into a relation algebra whose elements are subsets of some binary relation \(R\), whose Boolean operations are intersection, union, and complementation relative to \(R\), and whose relative operations are relativized to \(R\). For example, the relative product of two binary relations is the result of composing the two relations and intersecting the result with \(R\). The binary relation \(R\) can always be arranged to be symmetric and reflexive, but it can be an equivalence relation only if \(A\) is a representable relation algebra. An \(n\)-dimensional relation algebra \(A\) is, by definition, a subalgebra of an atomic relation algebra with an \(n\)-dimensional basis. An \(n\)-dimensional basis is a special set of \(n\times n\) matrices of atoms. Each \(n\times n\) matrix of atoms serves as an abstract algebraic analog of a sequence of length \(n\). To be a basis a set of matrices must be closed under various operations that correspond in certain ways to the operations of permuting the elements of a sequence, or replacing one element in a sequence with another. The authors show that if \(A\) is an \(n\)-dimensional relation algebra, then it has what they call an \(n\)-square relativized representation, one in which \(R\) is a kind of binary approximation to an \(n\)-ary relation. If the algebra \(A\) is also atomic, then it has an \(n\)-square relativized representation that is also complete, that is, it preserves arbitrary meets and joins. From these results the authors also show that the class of \(n\)-dimensional relation algebras has a rec…
Mathematics Subject Classification ID: 03G15
Mathematics Subject Classification ID: 03C25
Mathematics Subject Classification ID: 03C90
zbMATH DE Number: 1398643
zbMATH Keywords: finite axiomatizability
zbMATH Keywords: relative representation theory
zbMATH Keywords: \(n\)-dimensional relation algebras
zbMATH Keywords: \(n\)-dimensional basis
zbMATH Keywords: recursive equational axiomatization
author: Robin Hirsch
author: Ian Hodkinson
reviewed by: Roger Maddux
MaRDI profile type: Publication
Recommended article: Relation algebras of every dimension
Recommended article: RELATION ALGEBRAS AND R
Recommended article: Q4011721
Recommended article: Relativized relation algebras
Recommended article: Varieties of relation algebras
Recommended article: Q4362875
Recommended article: Relation algebras
Recommended article: The Structure of Relation Algebras Generated by Relativizations
Recommended article: Many-valued relation algebras
Recommended article: Weak representations of relation algebras and relational bases
description: scientific article; zbMATH DE number 1398643