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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 L. Fan, Q.-Y. Zhang, W.-Y. Xiang and 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 [B. Flagg, P. Sünderhauf, and 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 -irreducible and the well-below relation is multiplicative, then the stratified interpolation property holds in a continuous L-fuzzy domain (X,e), and { a x0a1, xX} is a base for the generalized Scott topology on (X,e).

MSC:
06B35Continuous lattices and posets, applications
68Q55Semantics