Elementary theories of free topo-Boolean and pseudo-Boolean algebras
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.
scientific article; zbMATH DE number 3952753