A categorical equivalence between Q-domains and interpolative generalized Q-closure spaces

With a commutative unital quantale $Q$ as the truth value table, this study focuses on representations of $Q$-domains by means of generalized $Q$-closure spaces. First, the notions of interpolative generalized $Q$-closure spaces and directed closed sets are introduced. It is proved that for an interpolative generalized $Q$-closure space (resp., a $Q$-closure space), the collection of directed closed sets with respect to the inclusion $Q$-order forms a continuous $Q$-dcpo (resp., an algebraic $Q$-dcpo). Conversely, it is shown that every continuous $Q$-dcpo (resp., algebraic $Q$-dcpo) can be reconstructed from an interpolative generalized $Q$-closure space (resp., a $Q$-closure space). Second, the notion of dense subspaces of generalized $Q$-closure spaces is provided. By means of dense subspaces, an alternative representation for algebraic $Q$-dcpos is given. Furthermore, the notion of approximable $Q$-relations between interpolative generalized $Q$-closure spaces is introduced. Consequently, a categorical equivalence is established between the category of interpolative generalized $Q$-closure spaces (resp., $Q$-closure spaces) with approximable $Q$-relations and that of continuous $Q$-dcpos (resp., algebraic $Q$-dcpos) with Scott continuous mappings.