Relativizations of relation algebras by the diversity
type: Item
instance of: scholarly article
title: Relativizations of relation algebras by the diversity
zbMATH Open document ID: 0859.03033
DOI: 10.1007/BF01197179
published in: Algebra Universalis
publication date: 1997-04-07T00:00:00Z
review text: The relative subalgebra \({\mathbf A}_{0a}\) of a Boolean algebra \({\mathbf A}_0 = (A,+, \cdot, ^-,0,1)\) by an element \(a\in A\) is defined by \({\mathbf A}_{0a} = (Aa,+, \cdot, ^{-a}, 0, a)\), where \(Aa= \{x\in A:x \leq a\}\) and \(x^{-a} = x^- \cdot a\) for \(x\in A\). The relative subalgebra \({\mathbf A} a\) of an augmented Boolean algebra \({\mathbf A} = ({\mathbf A}_0, F)\) is defined by \({\mathbf A} a= ({\mathbf A}_{0a},Fa)\), where \(Fa= \{f_a:f \in F\}\) and for each \(f\in F\), \(f_a\) is defined by \(f_a(x_1, \dots, x_n) =f(x_1, \dots, x_n) \cdot a\) for \(x_1, \dots, x_n\in Aa\). Now let \({\mathbf A} = ({\mathbf A}_0, ;,^\smile, 1')\) be a relation algebra; the element \(0'= 1'{}^-\) is called the diversity element of \({\mathbf A}\). The author proves that the relative subalgebras \({\mathbf A} 0'\) form a finitely based variety. It is also proved that every relation algebra \({\mathbf A}\) is determined by \({\mathbf A} 0'\), up to a direct factor that is Boolean (the relative product coincides with the meet). An open problem is pointed out.
Mathematics Subject Classification ID: 03G15
Mathematics Subject Classification ID: 08B99
zbMATH DE Number: 908010
zbMATH Keywords: relative subalgebra
zbMATH Keywords: relation algebra
zbMATH Keywords: diversity element
zbMATH Keywords: finitely based variety
zbMATH Keywords: relative product
zbMATH Keywords: meet
author: J. A. Rafter
reviewed by: Sergiu Rudeanu
MaRDI profile type: Publication
cites work: On taking subalgebras of relativized relation algebras
cites work: Varieties of relation algebras
cites work: Boolean Algebras with Operators. Part I
cites work: Relation algebras as residuated Boolean algebras
cites work: Relation algebras and projective geometries
cites work: Some Varieties Containing Relation Algebras
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description: scientific article; zbMATH DE number 908010