Representations of frame-valued algebraic dcpos and domains via closure spaces

In this paper, with a frame as the truth value table, we propose the concepts of fuzzy F-closure spaces and fuzzy IG-closure spaces. Based on these concepts, we provide representations of fuzzy algebraic dcpos and fuzzy domains, respectively. Furthermore, we introduce the notion of fuzzy F-relations, which accurately represent fuzzy Scott continuous maps between fuzzy algebraic dcpos. Consequently, we establish a categorical equivalence between fuzzy F-closure spaces and fuzzy algebraic dcpos. Moreover, we introduce the concept of approximable L -relations and demonstrate that the category of fuzzy IG-closure spaces is equivalent to that of fuzzy domains.