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Mardaev frames

Explore this construction
Motivation

The dual algebras were introduced to study pre-locally tabular superintuitionistic logics and to prove that there are continuum many of them.

Details of the construction

Begin with a maximal point and a three point antichain below it. We give the maximum the colour {p,q}, and {p}, {q}, and ∅ for each point on the antichain.

A Mardaev letter Mn+2 is a sequence of a three point antichain, followed by n many points. The comparisons are made as follows:

xk+1 < xk,   yk+1 < yk
xk+1 < yk−1,   yk+1 < xk−1

When stacking together different Mardaev letters, we can make a Mardaev word Mw. This is done by gluing together two Mardaev letters along their antichains.

Given an ω-word w∈Σω, for Σ a finite alphabet, this allows us to construct one Mardaev frame for each ω-word. Special Mardaev frames -- those corresponding to the algebras studied by Mardaev in his 1987 paper -- give rise to pre-locally tabular logics.

References

  1. S. I. Mardaev (1984)
    Number of prelocally table superintuitionistic propositional logics.
    Algebra and Logic 23(1), 56–66. Russian original: Algebra i Logika 23(1), 74–87.
    DOI
  2. Rodrigo Nicolau Almeida and Luca Carai (2026, forthcoming)
    Mardaev’s theorem via duality.
  3. Leo Esakia (2019)
    Heyting Algebras: Duality Theory.
    Trends in Logic, vol. 50. Edited by Guram Bezhanishvili and Wesley H. Holliday; translated by Anton Evseev. Springer, Cham.
    DOI
  4. Guram Bezhanishvili and Revaz Grigolia (2005)
    Locally finite varieties of Heyting algebras.
    Algebra Universalis 54(4), 465–473. Problem 2.4 asks whether every 2-finite variety of Heyting algebras is locally finite.
    DOI