TY - JOUR

T1 - Effective categoricity for distributive lattices and Heyting algebras

AU - Bazhenov, N. A.

PY - 2017/7/1

Y1 - 2017/7/1

N2 - We study complexity of isomorphisms between computable copies of lattices and Heyting algebras. For a computable ordinal α, the Δα 0dimension of a computable structure S is the number of computable copies of S, up to Δα 0 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 Δα 0 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 Δα 0 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.

AB - We study complexity of isomorphisms between computable copies of lattices and Heyting algebras. For a computable ordinal α, the Δα 0dimension of a computable structure S is the number of computable copies of S, up to Δα 0 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 Δα 0 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 Δα 0 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.

KW - categoricity spectrum

KW - computable categoricity

KW - computable dimension

KW - degree of categoricity

KW - Distributive lattice

KW - Heyting algebra

KW - FIELDS

KW - STABILITY

KW - COMPUTABLE CATEGORICITY

KW - DEGREE SPECTRA

KW - RECURSIVE STRUCTURES

KW - DIMENSIONS

KW - SYSTEMS

KW - MODEL-THEORY

KW - AUTOSTABILITY

UR - http://www.scopus.com/inward/record.url?scp=85024493948&partnerID=8YFLogxK

U2 - 10.1134/S1995080217040035

DO - 10.1134/S1995080217040035

M3 - Article

AN - SCOPUS:85024493948

VL - 38

SP - 600

EP - 614

JO - Lobachevskii Journal of Mathematics

JF - Lobachevskii Journal of Mathematics

SN - 1995-0802

IS - 4

ER -