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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
Article
  • 51 Downloads

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.

     

Keywords

Lattice theory Computability theory Universal algebra Congruence lattices 

Mathematics Subject Classification

03F35 03D28 06B15 

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References

  1. 1.
    Brodhead, P., Kjos-Hanssen, B.: The strength of the Grätzer-Schmidt theorem. In: Mathematical Theory and Computational Practice, pp. 59–67 (2009)Google Scholar
  2. 2.
    Cholak, P.A., Jockusch, C.G., Slaman, T.A.: On the strength of Ramsey’s theorem for pairs. J. Symb. Log. 66(1), 1–55 (2001)Google Scholar
  3. 3.
    Grätzer, G., Schmidt, E.T.: Characterizations of congruence lattices of abstract algebras. Acta Sci. Math. (Szeged) 24, 34–59 (1963)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Grätzer, G.: General Lattice Theory. Birkhäuser, Basel (2003). With appendices by Davey, B.A., Freese, R., Ganter, B., Greferath, M., Jipsen, P., Priestley, H.A., Rose, H., Schmidt, E.T., Schmidt, S.E., Wehrung, F., Wille, R., Reprint of the 1998, 2nd ednGoogle Scholar
  5. 5.
    Kjos-Hanssen, B.: Local initial segments of the Turing degrees. Bull. Symb. Log. 9(1), 26–36 (2003)Google Scholar
  6. 6.
    Odifreddi, P.: Classical recursion theory, Studies in Logic and the Foundations of Mathematics, vol. 125. North-Holland Publishing Co., Amsterdam. The theory of functions and sets of natural numbers. With a foreword by Sacks, G.E. (1989)Google Scholar
  7. 7.
    Pavel, P.: A new proof of the congruence lattice representation theorem. Algebra Universalis 6(3), 269–275 (1976)Google Scholar
  8. 8.
    Sacks, G.E.: Higher Recursion Theory, Perspectives in Mathematical Logic. Springer, Berlin (1990)CrossRefGoogle Scholar
  9. 9.
    Simpson, S.G.: Subsystems of Second Order Arithmetic, Second, Perspectives in Logic. Cambridge University Press, Cambridge; Association for Symbolic Logic, Poughkeepsie, NY (2009)Google Scholar
  10. 10.
    Soare, R.I.: Recursively Enumerable Sets and Degrees, Perspectives in Mathematical Logic. Springer, Berlin (1987). A study of computable functions and computably generated setsGoogle Scholar

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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