Quantale-valued fuzzy Scott topology

The aim of this paper is to extend the truth value table of lattice-valued convergence spaces to a more general case and then to use it to introduce and study the quantale-valued fuzzy Scott topology in fuzzy domain theory. Let $(L,*,\varepsilon)$ be a commutative unital quantale and let $\otimes$ be a binary operation on $L$ which is distributive over nonempty subsets. The quadruple $(L,*,\otimes,\varepsilon)$ is called a generalized GL-monoid if $(L,*,\varepsilon)$ is a commutative unital quantale and the operation $*$ is $\otimes$-semi-distributive. For generalized GL-monoid $L$ as the truth value table, we systematically propose the stratified $L$-generalized convergence spaces based on stratified $L$-filters, which makes various existing lattice-valued convergence spaces as special cases. For $L$ being a commutative unital quantale, we define a fuzzy Scott convergence structure on $L$-fuzzy dcpos and use it to induce a stratified $L$-topology. This is the inducing way to the definition of quantale-valued fuzzy Scott topology, which seems an appropriate way by some results.