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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.

##### MSC:
 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
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