Hostname: page-component-cb9f654ff-c75p9 Total loading time: 0 Render date: 2025-08-29T08:42:43.319Z Has data issue: false hasContentIssue false
  • English
  • Français

Fixed points of contractive maps on dcpo's

Published online by Cambridge University Press:  13 May 2013

E. COLEBUNDERS ,
S. DE WACHTER  and
R. LOWEN
E. COLEBUNDERS
Affiliation:
Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium Email: evacoleb@vub.ac.be; sdewacht@vub.ac.be
S. DE WACHTER
Affiliation:
Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium Email: evacoleb@vub.ac.be; sdewacht@vub.ac.be
R. LOWEN
Affiliation:
Department of Mathematics and Computer Science, Universiteit Antwerpen, Middelheimlaan 1, 2020 Antwerpen, Belgium Email: bob.lowen@ua.ac.be
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study approach structures on dcpo's. A dcpo (X, ≤) will be endowed with several other structures: the Scott topology; an approach structure generated by a collection of weightable quasi metrics on X; and a collection of weights corresponding to the quasi metrics. Understanding the interaction between these structures on X will eventually lead to some fixed-point theorems for the morphisms in the category of approach spaces, which are called contractions. Existing fixed-point theorems on both monotone and non-monotone maps are obtained as special cases.

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 24 , Issue 1 , February 2014 , e240102
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Adámek, J., Herrlich, H. and Strecker, G. E. (1990) Abstract and concrete categories, John Wiley and Sons.Google Scholar
Berckmoes, B., Lowen, R. and Van Casteren, J. (2011) Distances on probability measures and random variables. Journal of Mathematical Analysis and Applications 374 412–428.Google Scholar
Colebunders, E., De Wachter, S. and Lowen, B. (2011) Intrinsic approach spaces on domains. Topology and its Applications 158 2343–2355.CrossRefGoogle Scholar
Colebunders, E., De Wachter, S. and Schellekens, M. (2012) Complexity analysis via approach spaces. Applied Categorical Structures 20 1–18.Google Scholar
Dikranjan, D. and Tholen, W. (1995) Categorical structure of closure operators, Kluwer Academic Publishers.Google Scholar
Edalat, A. and Heckmann, R. (1998) A computational model for metric spaces. Theoretical Computer Science 193 53–73.CrossRefGoogle Scholar
García-Raffi, L. M., Romaguera, S. and Schellekens, M. P. (2008) Applications of the complexity space to the general probabilistic divide and conquer algorithms. Journal of Mathematical Analysis and Applications 348 (1)346–355.Google Scholar
Gierz, G., Hofmann, K., Keimel, K., Lawson, J., Mislove, M. and Scott, D. (2003) Continuous lattices and domains, Encyclopedia of Mathematics and its applications, 93, Cambridge University Press.Google Scholar
Gutierres, G. and Hofmann, D. (2012) Approaching metric domains. Applied Categorical Structures.CrossRefGoogle Scholar
Heckmann, R. (1999) Approximation of metric spaces by partial metric spaces. Applied Categorical Structures 7 71–83.Google Scholar
Künzi, H.-P. (2001) Nonsymmetric distances and their associated topologies: about the origins of basic ideas in the area of asymmetric topology. In: Aull, C. and Lowen, R. (eds.) Handbook of the History of General Topology 3, Kluwer Academic Publishers 853–968.Google Scholar
Künzi, H.-P. and Vajner, V. (1994) Weighted quasi metrics. Annals of the New York Academy of Sciences 728 64–77.Google Scholar
Lowen, R. (1997) Approach spaces, Oxford Mathematical Monographs, The Clarendon Press.Google Scholar
Martin, K. (2000a) A foundation for computation, Ph.D. thesis, Tulane University.Google Scholar
Martin, K. (2000b) The measurement process in domain theory. Springer-Verlag Lecture Notes in Computer Science 1853 116–126.Google Scholar
Matthews, S. (1994) Partial metric topology. In: Proceedings of the 8th Summer conference on General Topology and Applications. Annals of the New York Academy of Sciences 728 183–197.CrossRefGoogle Scholar
Oltra, S. and Valero, O. (2004) Banach's fixed point theorem for partial metric spaces. Rendiconti dell'Istituto di Matematica dell'Universit di Trieste 36 17–26.Google Scholar
Romaguera, S. and Schellekens, M. (1999) Quasi metric properties of complexity spaces. Topology and its Applications 98 311–322.Google Scholar
Romaguera, S., Schellekens, M. and Valero, O. (2011) Complexity spaces as quantitative domains of computation. Topology and its Applications 158 853–860.Google Scholar
Salbany, S. (1984) A bitopological view of topology and order: In: Bentley, H. L., Herrlich, H., Rajagopalan, M. and Wolff, H. (eds.) Categorical topology – Proceedings Conference Toledo, Ohio 1983, Sigma Series in Pure Mathematics 5, Heldermann 481–504.Google Scholar
Schellekens, M. P. (1995) The Smyth completion: A common foundation for denotational semantics and complexity analysis. Electronic Notes in Theoretical Computer Science 1, 535–556.Google Scholar
Schellekens, M. P. (2003) A characterization of partial metrizability: domains are quantifiable. Theoretical Computer Science 305 (1–3)409–432.Google Scholar
Scott, D. (1972) Continuous lattices. Springer-Verlag Lecture Notes in Computer Science 274 97–136.Google Scholar
Waszkiewicz, P. (2001) Distance and measurement in Domain Theory. Electronic Notes in Theoretical Computer Science 45 1–15.Google Scholar
Waszkiewicz, P. (2003) Quantitative continuous domains. Applied Categorical Structures 11 41–67.Google Scholar