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.
the set of indices of computable lattices that are complete is \(\Pi ^1_1\)-complete;
-
2.
the set of indices of computable lattices that are algebraic is \(\Pi ^1_1\)-complete;
-
3.
the set of compact elements of a computable lattice is \(\Pi ^{1}_{1}\) and can be \(\Pi ^1_1\)-complete; and
-
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.
Similar content being viewed by others
References
Brodhead, P., Kjos-Hanssen, B.: The strength of the Grätzer-Schmidt theorem. In: Mathematical Theory and Computational Practice, pp. 59–67 (2009)
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)
Grätzer, G., Schmidt, E.T.: Characterizations of congruence lattices of abstract algebras. Acta Sci. Math. (Szeged) 24, 34–59 (1963)
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 edn
Kjos-Hanssen, B.: Local initial segments of the Turing degrees. Bull. Symb. Log. 9(1), 26–36 (2003)
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)
Pavel, P.: A new proof of the congruence lattice representation theorem. Algebra Universalis 6(3), 269–275 (1976)
Sacks, G.E.: Higher Recursion Theory, Perspectives in Mathematical Logic. Springer, Berlin (1990)
Simpson, S.G.: Subsystems of Second Order Arithmetic, Second, Perspectives in Logic. Cambridge University Press, Cambridge; Association for Symbolic Logic, Poughkeepsie, NY (2009)
Soare, R.I.: Recursively Enumerable Sets and Degrees, Perspectives in Mathematical Logic. Springer, Berlin (1987). A study of computable functions and computably generated sets
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-016-0488-5