## 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 $$(X,e)$$, 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
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### References:

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