Elementary generation and canonicity for varieties of Boolean algebras with operators
type: Item
instance of: scholarly article
title: Elementary generation and canonicity for varieties of Boolean algebras with operators
zbMATH Open document ID: 0854.06020
DOI: 10.1007/BF01181878
author: Robert Goldblatt
published in: Algebra Universalis
publication date: 1997-01-19T00:00:00Z
review text: Every Boolean algebra \({\mathfrak A}\) with (finitely additive) operators is a (certain special kind of) subalgebra of a complete atomic Boolean algebra with operators called the canonical embedding algebra or the perfect extension of \({\mathfrak A}\). A variety \(V\) of Boolean algebras with operators is said to be canonical if \(V\) contains the perfect extensions of its members. Jónsson and Tarski proved that a variety \(V\) of Boolean algebras with operators is canonical if \(V\) has an axiomatization consisting exclusively of equations in which no variable appears in the scope of an odd number of complementations. The canonical embedding algebra can be obtained by first constructing a relational structure, called the canonical structure of \({\mathfrak A}\), by defining from each \(n\)-ary operator a corresponding \(n + 1\)-ary relation on the ultrafilters of \({\mathfrak A}\), and then taking the complex algebra of that structure. The complex algebra of a relational structure \({\mathfrak S}\) is the complete atomic Boolean algebra of subsets of the universe of \({\mathfrak S}\) augmented with an \(n\)-ary operator for every \(n + 1\)-ary relation in \({\mathfrak S}\). For example, the unary operation on sets that corresponds to a binary relation \(R\) in \({\mathfrak S}\) is the operation of forming the image of a set under \(R\). Operators obtained in this way distribute over arbitrary unions. It frequently happens that a variety \(V\) of Boolean algebras with operators can be characterized as the class of algebras isomorphic to subalgebras of complex algebras of structures in some interesting elementary class \(K\). The construction of complete atomic Boolean algebras with operators from relational structures induces a duality between algebraic and model-theoretic notions. For example, the algebraic notions of subalgebra, homomorphism, and direct product correspond to bounded morphisms, inner substructures, and disjoint unions. This duality is the guiding…
Mathematics Subject Classification ID: 06E25
Mathematics Subject Classification ID: 03G15
zbMATH DE Number: 836834
zbMATH Keywords: Boolean algebra with operators
zbMATH Keywords: canonical embedding algebra
zbMATH Keywords: perfect extension
zbMATH Keywords: canonical structure
zbMATH Keywords: complex algebra of a relational structure
zbMATH Keywords: duality
zbMATH Keywords: model-theoretic closure operators
zbMATH Keywords: representable cylindric algebras
zbMATH Keywords: canonical variety
zbMATH Keywords: cylindric-relativized set algebras
zbMATH Keywords: elementary class
reviewed by: Roger Maddux
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01181878
OpenAlex ID: W1968888224
description: scientific article; zbMATH DE number 836834