Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-18T19:01:46.053Z Has data issue: false hasContentIssue false
  • English
  • Français

Lattice-valued Scott topology on dcpos

Published online by Cambridge University Press:  18 May 2015

WEI YAO
WEI YAO*
Affiliation:
Department of Mathematics, Hebei University of Science and Technology, Shijizhuang 050018, P.R. China Email: yaowei0516@163.com
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies the fuzzy Scott topology on dcpos with a *-continuous semigroup (L, *) as the truth value table. It is shown that the fuzzy Scott topological space on a continuous dcpo is an ιL-sober space. The fuzzy Scott topology is completely distributive iff L is completely distributive and the underlying dcpo is continuous. For (L, *) being an integral quantale, semantics of L-possibility of computations is studied by means of a duality.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 27 , Special Issue 4: Symposium on Domain Theory (ISDT 2013) , May 2017 , pp. 516 - 529
Copyright
Copyright © Cambridge University Press 2015 

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.)

References

Chen, Y.X. and Wu, H.Y. (2008). Domain semantics of possibility computations. Information Sciences 178 2661–2679.Google Scholar
Gierz, G., Hofmann, H.H., Keimel, K., Lawson, J.D., Mislove, M. and Scott, D.S. (2003). Continuous Lattices and Domains, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Hájek, P. (1998). Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Dordrecht.Google Scholar
He, J.F., Seidel, K. and McIver, A.K. (1997). Probabilistic models for the guarded command language. Science of Computer Programming 28 171–192.Google Scholar
Hoare, C.A.R. (1985). Communicating Sequential Process, Prentice Hall International.Google Scholar
Hoffmann, R.-E. (1981). Continuous posets, prime spectra of completely distributive complete lattices, and Hausdorff compactifications. In: Continuous Lattices. Lecture Notes in Mathematics 871 159–208 Springer-Verlag (Berlin–Heidelberg–New York).Google Scholar
Höhle, U. and Šostak, A.P. (1999). Axiomatic foundations of fixed-basis fuzzy topology. Chapter 3 In: Höhle, U. and Rodabaugh, S.E. (eds.) Mathematics of Fuzzy Sets–-Logic, Topology, and Measure Theory, Kluwer Academic Publishers (Boston/Dordrecht/London) 123–272.Google Scholar
Höhle, U. and Kubiak, T. (2007). Many valued topologies and lower semicontinuity. Semigroup Forum 75 1–17.Google Scholar
Jones, C. (1990). Probabilistic Non-Determinism, PhD Thesis, University of Edinburgh, Edinburgh.Google Scholar
Kubiak, T. (1992). The topological modification of the L-fuzzy unit interval. In: Rodabaugh, S.E., Klement, E.P. and Höhle, U. (eds.) Applications of Category Theory to Fuzzy Subsets, Kluwer, Dordrecht 276–305.Google Scholar
Lawson, J.D. (1979). The duality of continuous posets. Houston Journal of Mathematics 5 357–386.Google Scholar
Liu, Y.M. and Luo, M.K. (1997). Fuzzy Topology, World Scientific, Singapore.Google Scholar
Lu, L.-X. (2012). Fuzzy Scott topology on directed complete posets. Computer Engineering and Applications 48 (25) 57–60. (In Chinese)Google Scholar
Plotkin, G.D. (1976). A powerdomains construction. SIAM Journal on Computing 5 452–487.Google Scholar
Plotkin, G.D. (1982). Probabilistic powerdomains. In: Proceedings CAAP 271–287.Google Scholar
Pultr, A. and Rodabaugh, S.E. (2003). Examples of diffirent sobrieties in fixed-basis topology. In: Rodabaugh, S.E. and Klement, E.P. (eds.) Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Trends in Logic volume 20, Kluwer Academic Publishers, Boston/Dordrecht/London 427–440.Google Scholar
Rosenthal, K. (1990). Quantales and Their Applications, Pitman Research Notes in Mathematics Series volume 234, Longman Scientific & Technical, Harlow.Google Scholar
Scott, D.S. (1970). Outline of a mathematical theory of computation. In: The 4th Annual Princeton Conference on Information Science and Systems 169–176.Google Scholar
Scott, D.S. (1972). Continuous lattices. In: Lawvere, E. (ed.) Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics volume 274, Springer-Verlag pp. 97–136.CrossRefGoogle Scholar
Smyth, M.B. (1978). Powerdomain. Journal of Computer and System Sciences 16 23–36.Google Scholar
Sugeno, M. (1974). Theory of Fuzzy Integrals and Its Applications, PhD Doctoral Dissertation, Tokyo Institute of Technology.Google Scholar
Tix, R., Keimel, K. and Plotkin, G.D. (2005). Semantic domains for combining probability and non-determinism. Electronic Notes in Theoretical Computer Science 129 1–104.Google Scholar
Warner, M.W. (1990). Fuzzy topology with respect to continuous lattice. Fuzzy Sets and Systems 35 85–91.Google Scholar
Wu, H.Y. and Chen, Y.X. (2012). Semantics of non-deterministic possibility computation. Fuzzy Sets and Systems 199 47–63.CrossRefGoogle Scholar
Yao, W. (2010). Quantitative domains via fuzzy sets: Part I: Continuity of fuzzy directed complete posets. Fuzzy Sets and Systems 161 973–987.CrossRefGoogle Scholar
Yao, W. (2012). A survey of fuzzifications of frames, the Papert–Papert–Isbell adjunction and sobriety. Fuzzy Sets and Systems 190 63–81.CrossRefGoogle Scholar
Yao, W., Li, Y.L. and Wu, G.N. (2013). Crisp posets can not be roughly considered as fuzzy posets. Journal of Weinan Normal University 28 5–7.Google Scholar
Yao, W. and Shi, F.G. (2011). Quantitative domains via fuzzy sets: Part II: Fuzzy Scott topologies on fuzzy dcpos. Fuzzy Sets and Systems 173 60–80.Google Scholar
Yao, W. and Zhao, B. (2014). A duality between fuzzy domains and strongly completely distributive L-ordered sets. Iranian Journal of Fuzzy Systems 11 23–43.Google Scholar