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The strength of the Grätzer-Schmidt theorem

Abstract

The Grätzer-Schmidt theorem of lattice theory states that each algebraic lattice is isomorphic to the congruence lattice of an algebra. We study the reverse mathematics of this theorem. We also show that

  1. 1.

    the set of indices of computable lattices that are complete is \(\Pi ^1_1\)-complete;

  2. 2.

    the set of indices of computable lattices that are algebraic is \(\Pi ^1_1\)-complete;

  3. 3.

    the set of compact elements of a computable lattice is \(\Pi ^{1}_{1}\) and can be \(\Pi ^1_1\)-complete; and

  4. 4.

    the set of compact elements of a distributive computable lattice is \(\Pi ^{0}_{3}\), and there is an algebraic distributive computable lattice such that the set of its compact elements is \(\Pi ^0_3\)-complete.

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Correspondence to Bjørn Kjos-Hanssen.

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Brodhead, K., Khan, M., Kjos-Hanssen, B. et al. The strength of the Grätzer-Schmidt theorem. Arch. Math. Logic 55, 687–704 (2016). https://doi.org/10.1007/s00153-016-0488-5

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  • DOI: https://doi.org/10.1007/s00153-016-0488-5

Keywords

  • Lattice theory
  • Computability theory
  • Universal algebra
  • Congruence lattices

Mathematics Subject Classification

  • 03F35
  • 03D28
  • 06B15