Representations of additive relation algebras by modules
type: Item
instance of: scholarly article
title: Representations of additive relation algebras by modules
zbMATH Open document ID: 0596.06009
DOI: 10.1016/0022-4049(86)90061-7
author: B. George
published in: Journal of Pure and Applied Algebra
publication date: 1986-01-01T00:00:00Z
review text: An additive relation algebra is a bounded modular lattice together with a monoid multiplication, a commutative monoid addition and an involution for multiplication, all satisfying natural coherence conditions. If R is a ring with unit and M an R-module, Rel(M), the submodules of \(M\times M\), form an additive relation algebra. The elementary theory of this variety of algebras is developed. Further, the main result is a representation theorem for additive relation algebras which are complemented lattices as subalgebras of Rel(R), where R is an appropriate von Neumann ring. The proof is obtained combining ''projective generator'' methods from abelian category theory with the techniques developed by von Neumann for representation of complemented modular lattices.
Mathematics Subject Classification ID: 06C20
Mathematics Subject Classification ID: 06B20
Mathematics Subject Classification ID: 08A05
Mathematics Subject Classification ID: 08A30
zbMATH DE Number: 3959500
zbMATH Keywords: projective generator
zbMATH Keywords: injective cogenerator
zbMATH Keywords: bounded modular lattice
zbMATH Keywords: representation theorem
zbMATH Keywords: additive relation algebras
zbMATH Keywords: complemented lattices
zbMATH Keywords: von Neumann ring
reviewed by: Grigore Călugăreanu
MaRDI profile type: Publication
full work available at URL: 0022-4049(86)90061-7
OpenAlex ID: W1982391102
cites work: Q4056148
cites work: Q5331549
cites work: Addition von Korrespondenzen in abelschen Kategorien
cites work: Abelsche und exakte Kategorien, Korrespondenzen
cites work: The Coordinatization of Arguesian Lattices
cites work: Q5572358
cites work: On the Arithmetic of Projective Coordinate Systems
cites work: Q5586465
cites work: On the Addition of Relations in an Abelian Category
cites work: A duality principle for lattices and categories of modules
cites work: Q5829441
cites work: Representation of Modular Lattices and Of Relation Algebras
cites work: The class of Arguesian lattices is self-dual
cites work: AN ALGEBRA OF ADDITIVE RELATIONS
cites work: Q5508993
cites work: Korrespondenzen in abelschen Kategorien
cites work: Q5626909
cites work: Q4092160
cites work: Q3882609
cites work: Q5557964
description: scientific article; zbMATH DE number 3959500