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Rieger-Nishimura lattices. (English. Russian original) Zbl 0412.03010

Sov. Math., Dokl. 19, 1014-1018 (1978); translation from Dokl. Akad. Nauk SSSR 241, 1288-1291 (1978).
The Rieger-Nishimura lattice is the free Brower algebra with 1 generator, that is the Lindenbaum algebra of intuitionistic propositional calculus in 1 variable. The author’s aim is to describe free algebras in \(m\) generators for S4 and some of its extensions. It is done via universal Kripke models in \(M\) variables, i.e. Kripke frames equipped with a valuation for the first \(m\) variables such that the derivability of a formula in \(m\) variables is equivalent to the truth in the model. Universal models constructed are the simplest possible ones in a certain sense. The form of the universal model for Grz (biggest extension of S4 having the same “superintuitionistic” fragment under Gödel-Tarski translation) in 1 variable allows to prove that the number of Grz nonequivalent formulas of degree \(n\) in 2 variable is not Kalmar elementary function of \(n\) (and the more so for S4). It is pointed out that the same method can be adapted for other extensions of S4 and to superintuitionistic logics. In the English translation p. 1024 1.4 “remains” should be changed to “was”; “is finitely approximable” (bottom page 101’7) to “has finite model property”.
[There are given several corrections: ibid. 19, vii (1978).]

MSC:

03B55 Intermediate logics
03B45 Modal logic (including the logic of norms)
03C90 Nonclassical models (Boolean-valued, sheaf, etc.)
03G25 Other algebras related to logic
03G10 Logical aspects of lattices and related structures
06D20 Heyting algebras (lattice-theoretic aspects)
06F30 Ordered topological structures
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