Varieties of algebraic systems and propositional calculi
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.
scientific article; zbMATH DE number 3966164
scientific article