Continuity in quantitative domains

Abstract

Based on the notion of an L -fuzzy partially ordered set constructed in (An L-fuzzy approach to quantitative domain (I)—generalized ordered set valued in frame and adjunction theory, Fuzzy Systems and Mathematics 14 (2000) 6–7), 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 (A Logical Approach to Quantitative Domain Theory, Elsevier, Amsterdam, 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 x ∣ 0 ≠ a ⋘ 1 , x ∈ X } is a base for the generalized Scott topology on ( X , e ) .