Lobachevskii Journal of Mathematics

, Volume 38, Issue 4, pp 600–614

Effective categoricity for distributive lattices and Heyting algebras

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Abstract

We study complexity of isomorphisms between computable copies of lattices and Heyting algebras. For a computable ordinal α, the Δα0dimension of a computable structure S is the number of computable copies of S, up to Δα0 computable isomorphism. The results of Goncharov, Harizanov, Knight, McCoy, Miller, Solomon, and Hirschfeldt, Khoussainov, Shore, Slinko imply that for every computable successor ordinal α and every non-zero natural number n, there exists a computable non-distributive lattice with Δα0 dimension n. In this paper, we prove that for every computable successor ordinal α ≥ 4 and every natural number n > 0, there is a computable distributive lattice with Δα0 dimension n. For a computable successor ordinal α ≥ 2, we build a computable distributive lattice M such that the categoricity spectrum of M is equal to the set of all PA degrees over Ø(α). We also obtain similar results for Heyting algebras.

Keywords and phrases

Distributive lattice Heyting algebra computable categoricity computable dimension categoricity spectrum degree of categoricity 

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© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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