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Elementary theories of free topo-Boolean and pseudo-Boolean algebras. (English. Russian original) Zbl 0593.03041
Math. Notes 37, 435-438 (1985); translation from Mat. Zametki 37, No. 6, 797-802 (1985).
As is well known there exists a dual isomorphism between the lattice of extensions of the modal logic S4 (the intuitionistic logic Int) and the lattice of varieties of topo-Boolean (pseudo-Boolean) algebras assigning to a logic \(\lambda\) the variety algebras var(\(\lambda)\). (The algebraic analog of ”topo-Boolean algebra” is ”interior algebra” or ”closure algebra” of McKinsey-Tarski.) In this note the elementary theories of free algebras of rank \(\omega\) (denoted: \({\mathcal F}_{\omega}(\lambda))\) from the variety var(\(\lambda)\), where \(\lambda\) \(\supseteq S4\) or \(\lambda\) \(\supseteq Int\), are investigated. We introduce some class of (\(\infty,2)\)-logics \(\lambda\), where \(\lambda\) \(\supseteq S4\) or \(\lambda\) \(\supseteq Int\). Main result of this note is: The free algebra \({\mathcal F}_{\omega}(\lambda)\) from the variety var(\(\lambda)\), where \(\lambda\) is (\(\infty,2)\)-logic and \(\lambda\) \(\supseteq S4\) or \(\lambda\) \(\supseteq Int\), has hereditarily undecidable elementary theory.
The classes of modal (\(\infty,2)\)-logics and of intermediate (\(\infty,2)\)-logics are extremely wide. For example, the first one contains all modal logics \(\lambda\) such that S4\(\subseteq \lambda \subseteq S4+\sigma_ 2\) or S4.1\(\subseteq \lambda \subseteq Grz+\sigma_ 2\), the second one includes all intermediate logics \(\lambda\) such that Int\(\subseteq \lambda \subseteq Int+I_ 2\). In particular, the free topo-Boolean algebra \({\mathcal F}_{\omega}(S4)\) and the free pseudo-Boolean algebra \({\mathcal F}_{\omega}(Int)\) have undecidable elementary theories.

03G10 Logical aspects of lattices and related structures
03B45 Modal logic (including the logic of norms)
03B55 Intermediate logics
03D35 Undecidability and degrees of sets of sentences
Full Text: DOI
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