Quantale-valued fuzzy Scott topology

Abstract

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