Varieties of algebraic systems and propositional calculi
type: Item
instance of: scholarly article
title: Varieties of algebraic systems and propositional calculi
zbMATH Open document ID: 0599.08010
DOI: 10.1007/BF01979870
author: A. A. Shum
published in: Algebra and Logic
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/187275
review text: The author studies properties of varieties of algebraic systems defined in predicate calculi without equality. Such varieties are defined by means of systems of quasi-identities having the form \(f_ 1\&f_ 2\&...\&f_ n\to f\) where \(f,f_ 1,...,f_ n\) are identities. The first two sections of the paper contain necessary notations and definitions as well as some technical results. In the third section the theorem of G. Birkhoff concerning the structure and generation of varieties of universal algebras is generalized (Theorem 3.2) for the varieties of algebraic systems studied in the paper. An analogous generalization of Jónsson's theorem is given in the fifth section (Theorem 5.1). Some useful corollaries (5.2-5.7) are decuded from this theorem. Subdirect products of algebraic systems in those varieties that are used in the generalization of Jónsson's theorem are investigated in the fourth section. The author proves (Theorem 4.1) that in such varieties any model of the given base may be represented by a subdirect product of subdirectly irreducible models of the same base. In the sixth section, lattices of congruences and homomorphisms of algebraic systems are considered. In the last (seventh) section some classes of propositional calculi satisfying the conditions from the generalization of Jónsson's theorem are described. These classes include intuitionistic propositional calculus with arbitrary additional connectives and various nonnormal modal propositional calculi.
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 03B05
Mathematics Subject Classification ID: 03B20
Mathematics Subject Classification ID: 08B25
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 03B45
zbMATH DE Number: 3966164
zbMATH Keywords: varieties of algebraic systems
zbMATH Keywords: predicate calculi without equality
zbMATH Keywords: systems of quasi-identities
zbMATH Keywords: Jónsson's theorem
zbMATH Keywords: Subdirect products
zbMATH Keywords: subdirectly irreducible models
zbMATH Keywords: lattices of congruences
MaRDI profile type: Publication
cites work: Algebras Whose Congruence Lattices are Distributive.
cites work: Q3853703
cites work: Some structure results for propositional calculi
cites work: Q3963056
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cites work: Reduced direct products
cites work: The separation theorem of intuitionist propositional calculus
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description: scientific article; zbMATH DE number 3966164