Skip to main content
×
×
Home

${\cal D}$ -MAXIMAL SETS

  • PETER A. CHOLAK (a1), PETER GERDES (a2) and KAREN LANGE (a3)
Abstract

Soare [20] proved that the maximal sets form an orbit in ${\cal E}$ . We consider here ${\cal D}$ -maximal sets, generalizations of maximal sets introduced by Herrmann and Kummer [12]. Some orbits of ${\cal D}$ -maximal sets are well understood, e.g., hemimaximal sets [8], but many are not. The goal of this paper is to define new invariants on computably enumerable sets and to use them to give a complete nontrivial classification of the ${\cal D}$ -maximal sets. Although these invariants help us to better understand the ${\cal D}$ -maximal sets, we use them to show that several classes of ${\cal D}$ -maximal sets break into infinitely many orbits.

Copyright
References
Hide All
[1]Cholak, Peter, Automorphisms of the lattice of recursively enumerable sets. Memoirs of the American Mathematical Society, vol. 113 (1995), no. 541.
[2]Cholak, Peter and Harrington, Leo A, Isomorphisms of splits of computably enumerable sets, this Journal, vol. 68 (2003), no. 3, pp. 10441064.
[3]Cholak, Peter and Harrington, Leo A. Extension theorems, orbits, and automorphisms of the computably enumerable sets. Transactions of the American Mathematical Society, vol. 360 (2008) no. 4, pp. 17591791.
[4]Cholak, Peter and Nies, André, Atomless r-maximal sets. Isreal Journal of Mathematics, vol. 113 (1999), pp. 305322.
[5]Cholak, Peter, Downey, Rod, and Herrmann, Eberhard, Some orbits for . Annals of Pure and Applied Logic, vol. 107 (2001), no. 1–3, pp. 193226.
[6]Cholak, Peter A., Downey, Rodney, and Harrington, Leo A., On the orbits of computably enumerable sets. Journal of the American Mathematical Society, vol. 21 (2008), no. 4, pp. 11051135.
[7]Dëgtev, A. N., Minimal 1-degrees, and truth-table reducibility. Sibirskii Matematicheskii Žhurnal, vol. 17 (1976), no. 5, pp. 10141022, 1196.
[8]Downey, R. G. and Stob, Michael, Automorphisms of the lattice of recursively enumerable sets: orbits. Advances in Mathematics, vol. 92 (1992), no. 2, pp. 237265.
[9]Harrington, Leo and Soare, Robert I, Post’s program and incomplete recursively enumerable sets. Proceedings of the National Academy of Sciences, U.S.A., vol. 88, no. 22, pp. 1024210246.
[10]Herrmann, E., Automorphisms of the lattice of recursively enumerable sets and hyperhypersimple sets. Proceedings of the fourth Easter conference on model theory (Gross Köris, 1986), Seminarberichte, vol. 86, pp. 69108, Humboldt University, Berlin, 1986.
[11]Herrmann, E., Automorphisms of the lattice of recursively enumerable sets and hyperhypersimple sets. Logic, methodology and philosophy of science, VIII (Moscow, 1987), Studies in Logic and the Foundations of Mathematics, vol. 126, pp. 179190, North-Holland, Amsterdam, 1989.
[12]Herrmann, Eberhard and Kummer, Martin, Diagonals and -maximal sets, this Journal, vol. 59 (1994), no. 1, pp. 6072. ISSN 0022-4812.
[13]Lachlan, Alistair H., The elementary theory of the lattice of recursively enumerable sets. Duke Mathematical Journal, vol. 35 (1968), pp. 123146.
[14]Lachlan, Alistair H., On the lattice of recursively enumerable sets. Transactions of the American Mathematical Society, vol. 130 (1968), pp. 137.
[15]Lerman, M. and Soare, R. I., A decidable fragment of the elementary theory of the lattice of recursively enumerable sets. Transactions of the American Mathematical Society, vol. 257 (1980), no. 1, pp. 137.
[16]Maass, W., On the orbit of hyperhypersimple sets, this Journal, vol. 49 (1984), pp. 5162.
[17]Maass, W. and Stob, M., The intervals of the lattice of recursively enumerable sets determined by major subsets. Annals of Pure and Applied Logic, vol. 24 (1983), pp. 189212.
[18]Martin,  Donald A., Classes of recursively enumerable sets and degrees of unsolvability. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 295310.
[19]Slaman, Theodore A. and Hugh Woodin, W., Slaman-Woodin conjecture. Personal Communication, 1989.
[20]Soare, Robert I., Automorphisms of the lattice of recursively enumerable sets I: maximal sets. Annals of Mathematics (2), vol. 100 (1974), pp. 80120.
[21]Soare, Robert I., Recursively Enumerable Sets and Degrees, Perspectives in Mathematical Logic, Omega Series, Springer-Verlag, Heidelberg, 1987.
[22]Stob, M., The Structure and Elementary Theory of the Recursive Enumerable Sets, PhD thesis, University of Chicago, Illinois 1979.
[23]Weber, Rebecca, Invariance in * and Π. Transactions of the American Mathematical Society, vol. 358 (2006), no. 7, pp. 30233059.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed