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 Izv. Akad. Nauk SSSR Ser. Mat., 1977, Volume 41, Issue 5, Pages 963–986 (Mi izv1875)

On finite-dimensional superintuitionistic logics

S. K. Sobolev

Abstract: A pseudoboolean algebra $\mathfrak M$ is called $n$-dimensional if the lattice $(Z_2)^{n+1}$ is not embeddable in $\mathfrak M$ as a lattice, where $Z_2$ is the two-element lattice. A superintuitionistic logic is said to be $n$-dimensional if the formula $E_n(x_1,…,x_n)\leftrightharpoons\bigvee_{i=1}^{n+1}(x_i=\bigvee_{j\ne i}x_j)$ belongs to it. A logic is $n$-dimensional if and only if it is approximable by $n$-dimensional algebras. All finite-dimensional logics are complete relative to Kripke semantics. An example is given of a formula that generates a logic not approximable by finite-dimensional algebras. It is proved that for every $n$, every finitely axiomatizable $n$-dimensional logic containing the formula $H(x,y)\leftrightharpoons(((x\to y)\to x)\to x)\vee (((y\to x)\to y)\to y)$ is decidable (already for $n=2$ there exist among such logics non-finitely-approximable ones). The proof uses the theory of finite automata on $\omega$-sequences.
Bibliography: 10 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1977, 11:5, 909–935

Bibliographic databases:

UDC: 51.01.16
MSC: Primary 02E05, 02J05; Secondary 02F10

Citation: S. K. Sobolev, “On finite-dimensional superintuitionistic logics”, Izv. Akad. Nauk SSSR Ser. Mat., 41:5 (1977), 963–986; Math. USSR-Izv., 11:5 (1977), 909–935

Citation in format AMSBIB
\Bibitem{Sob77}
\by S.~K.~Sobolev
\paper On finite-dimensional superintuitionistic logics
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1977
\vol 41
\issue 5
\pages 963--986
\mathnet{http://mi.mathnet.ru/izv1875}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=491051}
\zmath{https://zbmath.org/?q=an:0368.02062|0388.03027}
\transl
\jour Math. USSR-Izv.
\yr 1977
\vol 11
\issue 5
\pages 909--935
\crossref{https://doi.org/10.1070/IM1977v011n05ABEH001751}