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Continuity in quantitative domains. (English) Zbl 1080.06007
Summary: Based on the notion of an $L$-fuzzy partially ordered set [see {\it L. Fan}, {\it Q.-Y. Zhang}, {\it W.-Y. Xiang} and {\it C. Y. Zheng}, “An $L$-fuzzy approach to quantitative domain. I. Generalized ordered set valued in frame and adjunction theory”, Fuzzy Syst. Math. 14, 6--7 (2000)] and by introducing the concepts of an $L$-fuzzy directed set and the join of an $L$-fuzzy set w.r.t. the $L$-fuzzy partial order, $L$-fuzzy domains are defined and the generalized Scott topology on an $L$-fuzzy domain is built. This approach is similar to Flagg’s logic approach to quantitative domain theory [{\it B. Flagg}, {\it P. Sünderhauf}, and {\it K. Wagner}, A logical approach to quantitative domain theory, Preprint (1996), submitted for publication]. In addition, the concepts of stratified approximation and a basis for an $L$-fuzzy domain are proposed, and a notion of a continuous $L$-fuzzy domain is developed. It is proved that if $L$ is a completely distributive lattice in which 1 is $\vee$-irreducible and the well-below relation is multiplicative, then the stratified interpolation property holds in a continuous $L$-fuzzy domain $(X,e)$, and $\{\Uparrow_ax\mid 0\ne a\lll 1$, $x\in X\}$ is a base for the generalized Scott topology on $(X,e)$.

##### MSC:
 06B35 Continuous lattices and posets, applications 68Q55 Semantics
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##### References:
 [1] Abramsky, S.; Jung, A.: Domain theory. Handbook of logic in computer science 3, 1-168 (1995) [2] Fan, L.; Zhang, Q. -Y.; Xiang, W. -Y.; Zheng, C. -Y.: An L-fuzzy approach to quantitative $domain(I)$ (generalized ordered set valued in frame and adjunction theory). Fuzzy systems math. (the special of theory of fuzzy sets and application) 14, 6-7 (2000) [3] L. Fan, Research of some problems in domain theory, Ph.D. Thesis, Capital Normal University, 2001 (in Chinese). [4] B. Flagg, P. Sünderhauf, K. Wagner, A logical approach to quantitative domain theory, Preprint, Elsevier, 1996, submitted. [5] U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Set Series, vol. 3, Kluwer Academic Publishers, Dordrecht, 1999. [6] Rutten, J. J. M.M.: Elements of generalized ultrametric domain theory. Theoret. comput. Sci. 170, 349-381 (1996) · Zbl 0874.68189 [7] Shi, F. -G.: Theory of L$\beta$- nested sets and L$\alpha$-nested sets and applications. Fuzzy systems math. 9, 65-72 (1995) · Zbl 1266.03063 [8] K.R. Wagner, Solving recursive domain equations with enriched categories, Ph.D. Thesis, School of Computer Science, Carnegie Mellon University, Pittsburgh, 1994. [9] Zheng, C. -Y.; Fan, L.; Cui, H.: Frame and continuous lattices. (2000)