Representations of domains via closure spaces in the quantale-valued setting

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