Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Clones from creatures


Authors: Martin Goldstern and Saharon Shelah
Journal: Trans. Amer. Math. Soc. 357 (2005), 3525-3551
MSC (2000): Primary 08A40; Secondary 03E50, 03E75
DOI: https://doi.org/10.1090/S0002-9947-04-03593-7
Published electronically: November 4, 2004
MathSciNet review: 2146637
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that (consistently) there is a clone $\mathcal{C}$ on a countable set such that the interval of clones above $\mathcal{C}$ is linearly ordered and has no coatoms.


References [Enhancements On Off] (What's this?)

  • 1. V. A. Buevich.
    A new version of the proof of completeness criterion for $k$-valued logic functions.
    Discrete Math. Appl., 6(5):505-530, 1996. MR 98g:03067
  • 2. G. P. Gavrilov.
    Certain conditions for completeness in countable-valued logic.
    Dokl. Akad. Nauk SSSR, 128:21-24, 1959.
    in Russian. MR 21:6327
  • 3. G.P. Gavrilov.
    Ueber funktionale Vollstaendigkeit in der abzaehlbar-wertigen Logik.
    Probl. Kibernetiki, 15:5-64, 1965. MR 37:5085
  • 4. Martin Goldstern and Saharon Shelah.
    Clones on regular cardinals.
    Fundamenta Mathematicae, 173:1-20, 2002.
    math.RA/0005273. MR 2003c:08006
  • 5. Martin Goldstern and Saharon Shelah.
    Large Intervals in the Clone Lattice.
    Algebra Universalis, 200x.
    math.RA/0208066.
  • 6. Lutz Heindorf.
    The maximal clones on countable sets that include all permutations.
    Algebra universalis, 48:209-222, 2002. MR 2003g:08005
  • 7. H. Machida and I. G. Rosenberg.
    A ``large'' essentially minimal clone over an infinite set.
    In Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989), volume 131 of Contemp. Math., pages 159-167, Providence, RI, 1992. Amer. Math. Soc. MR 93f:08005
  • 8. R. Pöschel and L. A. Kaluznin.
    Funktionen- und Relationenalgebren, volume 15 of Mathematische Monographien [Mathematical Monographs].
    VEB Deutscher Verlag der Wissenschaften, Berlin, 1979.
    Ein Kapitel der diskreten Mathematik. [A chapter in discrete mathematics].MR 81f:03075
  • 9. R. W. Quackenbush.
    A new proof of Rosenberg's primal algebra characterization theorem.
    Colloquia Mathematica Societatis János Bolyai, 28:603-634, 1971.MR 83f:08012
  • 10. I. G. Rosenberg.
    Über die funktionale Vollständigkeit in den mehrwertigen Logiken.
    Rozpravy Ceskoslovenské Akad. ved, Ser. Math. Nat. Sci., 80:3-93, 1970. MR 45:1732
  • 11. I. G. Rosenberg.
    Some maximal closed classes of operations on infinite sets.
    Math. Ann., 212:157-164, 1974/75. MR 50:4452
  • 12. I. G. Rosenberg.
    The set of maximal closed classes of operations on an infinite set ${A}$ has cardinality $2^{2^{\vert{A}\vert}}$.
    Arch. Math. (Basel), 27(6):561-568, 1976. MR 55:2711
  • 13. Ivo G. Rosenberg and Dietmar Schweigert.
    Locally maximal clones.
    Elektron. Informationsverarb. Kybernet., 18(7-8):389-401, 1982.MR 85b:08006
  • 14. Andrzej Roslanowski and Saharon Shelah.
    Norms on possibilities I: forcing with trees and creatures.
    Memoirs of the American Mathematical Society, 141(671), 1999.
    math.LO/9807172. MR 2000c:03036
  • 15. Saharon Shelah.
    On cardinal invariants of the continuum.
    In Axiomatic set theory (Boulder, Colo., 1983), volume 31 of Contemp. Mathematics, pages 183-207. Amer. Math. Soc., Providence, RI, 1984.
    Proceedings of the Conference in Set Theory, Boulder, June 1983; eds. J. Baumgartner, D. Martin, and S. Shelah. MR 86b:03064
  • 16. Ágnes Szendrei.
    Clones in universal algebra.
    Presses de l'Université de Montréal, Montreal, Que., 1986. MR 87m:08005
  • 17. Eric K. van Douwen.
    The integers and topology.
    In K. Kunen and J. E. Vaughan, editors, Handbook of Set-Theoretic Topology, pages 111-167. Elsevier Science Publishers, 1984. MR 87f:54008
  • 18. Jerry E. Vaughan.
    Small uncountable cardinals and topology, pages 195-218.
    North-Holland, Amsterdam, 1990.
    With an appendix by S. Shelah. MR 92c:54001

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 08A40, 03E50, 03E75

Retrieve articles in all journals with MSC (2000): 08A40, 03E50, 03E75


Additional Information

Martin Goldstern
Affiliation: Institute of Discrete Mathematics and Geometry, Vienna University of Technology, A-1040 Vienna, Austria
Email: goldstern@tuwien.ac.at

Saharon Shelah
Affiliation: Institute of Mathematics, Hebrew University of Jerusalem, 91904 Jerusalem, Israel – and – Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854
Email: shelah@math.huji.ac.il

DOI: https://doi.org/10.1090/S0002-9947-04-03593-7
Keywords: Precomplete clones, maximal clones, clones on infinite sets, creature forcing, continuum hypothesis
Received by editor(s): March 7, 2003
Received by editor(s) in revised form: December 2, 2003
Published electronically: November 4, 2004
Additional Notes: The first author is grateful to the Department of Mathematics, Rutgers University, New Jersey, for their hospitality during a visit in September 2002
The second author’s research was supported by the US-Israel Binational Science Foundation. Publication 808.
A preprint of this paper is available at http://www.arXiv.org/math.RA/0212379/
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society