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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 $\vee$-irreducible and the well-below relation is multiplicative, then the stratified interpolation property holds in a continuous $L$-fuzzy domain $\left(X,e\right)$, and $\left\{{⇑}_{a}x\mid 0\ne a⋘1$, $x\in X\right\}$ is a base for the generalized Scott topology on $\left(X,e\right)$.

##### MSC:
 06B35 Continuous lattices and posets, applications 68Q55 Semantics