Locally Boolean spectra
type: Item
instance of: scholarly article
title: Locally Boolean spectra
zbMATH Open document ID: 0821.08001
DOI: 10.1007/BF01190703
published in: Algebra Universalis
publication date: 1995-09-18T00:00:00Z
review text: This paper is a continuation of the author's earlier paper [ibid. 29, 232-272 (1992; Zbl 0772.08005)]. He uses the same approach to study now locally Boolean representability. The generality of the results obtained in both works put into perspective the power of universal algebra in the study of sheaf representations of algebras (in all cases the results are obtained by combining a minimal amount of particular algebra with strong general universal algebra theorems). The main results of this paper are found in Sections 6, 7 and 8. In section 6, the author defines locally Boolean spectra and locally projectives and proves two important theorems characterizing them. In section 7, the author introduces the SUP (spectra uniqueness property) and uses Beth's definability theorem to prove that for every \(\forall \exists\) class \({\mathcal M}\) with the SUP, the class of (locally) Boolean \({\mathcal M}\)-representable algebras is elementary. Also he proves that every class of finitely subdirectly irreducible algebras in a congruence distributive variety has the SUP and that the class of domains and the class of simple rings with identity (in the language of rings and rings with identity, respectively) have the SUP. In section 8 the author uses the characterization theorems proved in Section 6 to study the locally Boolean representation of semi-irreducible algebras by finitely subdirectly irreducibles.
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 03C20
Mathematics Subject Classification ID: 08B26
zbMATH DE Number: 759385
zbMATH Keywords: subdirect product
zbMATH Keywords: first-order axiomatizability
zbMATH Keywords: ordered groups
zbMATH Keywords: spectra uniqueness property
zbMATH Keywords: locally Boolean representability
zbMATH Keywords: sheaf representations of algebras
zbMATH Keywords: locally Boolean spectra
zbMATH Keywords: locally projectives
zbMATH Keywords: congruence distributive variety
zbMATH Keywords: domains
zbMATH Keywords: simple rings
zbMATH Keywords: semi-irreducible algebras
author: Diego J. Vaggione
reviewed by: Dimitru Buşneag
MaRDI profile type: Publication
cites work: Congruence relations on De Morgan algebras
cites work: Stone lattices
cites work: Finitely subdirectly irreducible algebras with pseudocomplementation
cites work: Q5789040
cites work: Equational compactness in quasi-primal varieties
cites work: Q3934450
cites work: Remarks on Boolean products
cites work: Sheaf Constructions and Their Elementary Properties
cites work: The lattice of global sections of sheaves of chains over Boolean spaces
cites work: Linear reasoning. A new form of the Herbrand-Gentzen theorem
cites work: Congruence distributive quasivarieties whose finitely subdirectly irreducible members form a universal class
cites work: The representation of biregular rings by sheaves
cites work: Q3996619
cites work: Principal Congruences of p-Algebras and Double p-Algebras
cites work: Representations of algebras by continuous sections
cites work: Q5622399
cites work: Subdirectly irreducible modular p-algebras
cites work: The structure of distributive double p-algebras. Regularity and congruences
cites work: Q5638369
cites work: Global subdirect products
cites work: Q5519166
cites work: On spectra, and the negative solution of the decision problem for identities having a finite nontrivial model
cites work: Relation of the Exact Transient Solution for a Line Source near an Interface between Two Fluids to Geometrical Acoustics
cites work: Q5560258
cites work: A General Theory of Spectra. I
cites work: Uniformity of congruences
cites work: Projectable and strongly projectable lattice-ordered groups
cites work: Sheaf representation and Chinese Remainder Theorems
cites work: Q3964534
full work available at URL: https://doi.org/10.1007/bf01190703
OpenAlex ID: W2081870297
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description: scientific article; zbMATH DE number 759385