Monotone Relations, Fixed Points and Recursive Definitions

  • Janusz CzelakowskiEmail author
Part of the Trends in Logic book series (TREN, volume 28)


The paper is concerned with reflexive points of relations. The significance of reflexive points in the context of indeterminate recursion principles is shown.


fixed-point monotone relation chain-σ-continuous relation definability by arithmetic recursion 


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  1. [Berman and Blok, 1989]
    Berman, J., Blok, W., [1989], ‘Generalizations of Tarski’s fixed-point theorem for order varieties of complete meet semilattices’, Order, 5(4): 381–392. zbMATHCrossRefMathSciNetGoogle Scholar
  2. [Cai and Paige, 1992]
    Cai, J., Paige, R., [1992], ‘Languages polynomial in the input plus output’, in Second International Conference on Algebraic Methodology and Software Technology (AMAST 91), Springer Verlag, London, pp. 287–300. Google Scholar
  3. [Chang and Keisler, 1973]
    Chang, C.C., Keisler, H.J., [1973], Model Theory, North-Holland and American Elsevier, Amsterdam–London–New York. zbMATHGoogle Scholar
  4. [Czelakowski, 2006]
    Czelakowski, J., [2006], ‘Fixed-points for relations and the back and forth method’, Bulletin of the Section of Logic, 35(2/3): 63–71. zbMATHMathSciNetGoogle Scholar
  5. [Davey and Priestley, 2002]
    Davey, B.A., Priestley, H., [2002] Introduction to Lattices and Order, 2nd ed., Cambridge University Press, Cambridge. zbMATHGoogle Scholar
  6. [Desharnais and Möller, 2005]
    Desharnais, J., Möller, B., [2005], ‘Least reflexive points of relations’, Higher-Order and Symbolic Computation, 18: 51–77. zbMATHCrossRefGoogle Scholar
  7. [Dugundji and Grana, 1982]
    Dugundji, J., Granas, A., [1982], Fixed Point Theory, Monografie Matematyczne, vol. 61, PWN, Warsaw. zbMATHGoogle Scholar
  8. [Fujimoto, 1984]
    Fujimoto, T., [1984], ‘An extension of Tarski’s fixed point theorem and its application to isotone complementarity problems’, Mathematical Programming, 28: 116–118. zbMATHCrossRefMathSciNetGoogle Scholar
  9. [Goebel and Kirk, 1990]
    Goebel, K., Kirk, W.A., [1990], Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge. zbMATHGoogle Scholar
  10. [Gunter and Scott, 1990]
    Gunter, C.A., Scott, D.S., [1990], ‘Semantic domains’, in Van Leeuwen, J. (Managing Editor), Handbook of Theoretical Computer Science, The MIT Press/Elsevier, Amsterdam, New York-Oxford-Tokyo/Cambridge, Massachusetts, pp. 634–674. Google Scholar
  11. [Kirk and Sims, 2001]
    Kirk, W.A., Sims, B. (eds.), [2001], Handbook of Metric Fixed Point Theory, Kluwer, Dordrecht, Boston–London. zbMATHGoogle Scholar
  12. [Kleene, 1952]
    Kleene, S.C., [1952], Introduction to Metamathematics, Van Nostrand. Google Scholar
  13. [Kunen, 1999]
    Kunen, K., [1999], Set Theory. An Introduction to Independence Proofs, Elsevier, Amsterdam–Lausanne–New York. Google Scholar
  14. [Markowsky, 1976]
    Markowsky, G., [1976], ‘Chain-complete posets and directed sets with applications’, Algebra Universalis, 6: 53–68. zbMATHCrossRefMathSciNetGoogle Scholar
  15. [Moschovakis, 1994]
    Moschovakis, Y.N., [1994], Notes on Set Theory, Springer-Verlag, New York–Berlin. zbMATHGoogle Scholar
  16. [Tarski, 1955]
    Tarski, A., [1955], ‘A lattice-theoretical fixpoint theorem and its applications’, Pacific Journal of Mathematics, 5: 285–309. zbMATHMathSciNetGoogle Scholar
  17. [Wright, Wagner and Thatcher, 1978]
    Wright, J., Wagner, E., Thatcher, J., [1978], ‘A uniform approach to inductive posets and inductive closure’, Theoretical Computer Science, 7: 57–77. zbMATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsOpole UniversityOpolePoland
  2. 2.Institute of Mathematics and PhysicsOpole University of TechnologyOpolePoland

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