Continuity in quantitative domains. (English) Zbl 1080.06007

Zhang, Qi-Ye; Fan, Lei

Fuzzy Sets Syst. 154, No. 1, 118-131 (2005).

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 \((X,e)\), and \(\{\Uparrow_ax\mid 0\neq a\lll 1\), \(x\in X\}\) is a base for the generalized Scott topology on \((X,e)\).

MSC:

06B35	Continuous lattices and posets, applications
68Q55	Semantics in the theory of computing

Keywords:

\(L\)-fuzzy domain; Generalized Scott topology; Stratified approximation relation; Continuous \(L\)-fuzzy domain; Stratified interpolation

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