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Zbl 1080.06007
Zhang, Qi-Ye; Fan, Lei
Continuity in quantitative domains. (English)
[J] Fuzzy Sets Syst. 154, No. 1, 118-131 (2005). ISSN 0165-0114

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 2000:: *06B35 Continuous lattices
68Q55 Semantics

Keywords: $L$-fuzzy domain; Generalized Scott topology; Stratified approximation relation; Continuous $L$-fuzzy domain; Stratified interpolation

Cited in: Zbl 1116.06006

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Zentralblatt MATH Berlin [Germany]