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

TL;DR

This paper develops a framework connecting $L$-closure spaces and $L$-domains using quantale-valued logic, establishing categorical equivalences and new representations for continuous and algebraic $L$-dcpos.

Contribution

It introduces interpolative generalized $L$-closure spaces and $L$-closure spaces, proving their correspondence with continuous and algebraic $L$-dcpos, and establishes categorical equivalences.

Findings

Directed closed sets form continuous and algebraic $L$-dcpos.

Every continuous/algebraic $L$-dcpo can be reconstructed from an $L$-closure space.

Categorical equivalence between $L$-closure spaces and $L$-dcpos.

Abstract

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.