Archive for Mathematical Logic

, Volume 55, Issue 5–6, pp 687–704 | Cite as

The strength of the Grätzer-Schmidt theorem

  • Katie Brodhead
  • Mushfeq Khan
  • Bjørn Kjos-Hanssen
  • William A. Lampe
  • Paul Kim Long V. Nguyen
  • Richard A. Shore


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.



Lattice theory Computability theory Universal algebra Congruence lattices 

Mathematics Subject Classification

03F35 03D28 06B15 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Florida A&M UniversityTallahasseeUSA
  2. 2.University of Hawaii at ManoaHonoluluUSA
  3. 3.University of Hawaii Leeward Community CollegePearl CityUSA
  4. 4.Cornell UniversityIthacaUSA

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