Generalized Scott Topology on Sets with Families of Pre-orders

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• 作者：Li-Gang Wu ; Lei Fan
• 关键词：54A10 ; 06B35
• 刊名：Electronic Notes in Theoretical Computer Science
• 出版年：18 February, 2014
• 年：2014
• 卷：301
• 期：Complete
• 页码：103-115
• 全文大小：337 K

文摘

[Fan L., W. Ji and W. L. Wang. 鈥淭he Information Order Approximation and Generalized Chains' Completion鈥? Beijing: Capital Normal University, Preprint, 2005. (in Chinese), Lei Fan] proposed a class of sets with families of pre-orders (-posets for short). They are not only a non-symmetric generalization of sfe [Monteiro L., Semantic Domains Based on Sets with Families of Equivalences. Electronic Notes in Theoretical Computer Science 11 (1998): 1-34, L.Monteiro] but also a special case of quasi-metric spaces (qms, [Smyth M. B., Quasi Uniformities: Reconciling Domains with Metric Spaces. Proceedings of the 3rd Workshop on Mathematical Foundations of Programming Language Semantics, APR. 8-10, 1987. Lecture Notes In Computer Science, Vol. 298 (1988): 236-253. Springer-Verlag, Berlin, M. B. Smyth]) and generalized ultrametric spaces (gums, [Rutten J. J. M. M., 鈥淓lements of Generalized Ultrametric Domain Theory鈥? Technical Report CS-R9507, CWI, Amsterdam, 1995, J. J. M. M. Rutten]). In this paper, we define a kind of generalized Scott topology on -posets and discuss some basic properties of the topology. Some relevant interesting examples are offered. It is worth pointing out that an -monotone functions is -continuous if and only if (iff for short) it's continuous with respect to (w.r.t for short) the generalized Scott topology.