A characterization of primitive Boolean algebras
type: Item
instance of: scholarly article
title: A characterization of primitive Boolean algebras
zbMATH Open document ID: 0754.06009
DOI: 10.1007/BF01191085
author: Alain Touraille
published in: Algebra Universalis
publication date: 1992-06-27T00:00:00Z
review text: Heyting* algebras (i.e. Heyting algebras with a unary operation \(^*\) such that \(a\leq a^*\), \((a\land b)^*=a^*\land b^*\), \((a\to b)^*=a\lor(a\to b))\) were introduced and extensively studied by the author [ibid. 24, 21-31 (1987; Zbl 0644.06004)]. In this paper he proves that primitive Boolean algebras (in the sense of Hanf) can be characterized in terms of Heyting* algebras of ideals, i.e. Heyting algebras closed under the operation \(^*\) defined by \(I^*=\{a: a/I\) is a finite sum of atoms\}; namely, a denumerable Boolean algebra is primitive iff its smallest complete Heyting* algebra of ideals has the property that each element is the sum of completely join-irreducible elements.
Mathematics Subject Classification ID: 06E05
Mathematics Subject Classification ID: 06D20
zbMATH DE Number: 25316
zbMATH Keywords: lattice of ideals
zbMATH Keywords: Heyting* algebras
zbMATH Keywords: primitive Boolean algebras
zbMATH Keywords: join- irreducible elements
reviewed by: Leo Esakia
MaRDI profile type: Publication
cites work: Q4083730
cites work: Q4111555
cites work: Algèbres de Boole primitives
cites work: Primitive Boolean algebras: Hanf and Pierce reconciled
cites work: Compact zero-dimensional metric spaces of finite type
cites work: Heyting* algebras, topological Boolean algebras and P.O. systems
description: scientific article; zbMATH DE number 25316