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 $\{{\Uparrow}_{a}x\mid 0\ne a\u22d81$, $x\in X\}$ is a base for the generalized Scott topology on $(X,e)$.

##### MSC:

06B35 | Continuous lattices and posets, applications |

68Q55 | Semantics |