Categorical Representations of Continuous Domains and Continuous L-Domains Based on Closure Spaces

TL;DR

This paper introduces a new approach using closure spaces to characterize continuous domains and related structures, establishing categorical equivalences and representations for various types of domains.

Contribution

It presents F-augmented generalized closure spaces that generate continuous domains and identifies approximable mappings as Scott-continuous functions, expanding the theoretical framework.

Findings

F-augmented generalized closure spaces generate exactly continuous domains

Establishment of a category equivalent to continuous domains with Scott-continuous functions

Representation of continuous L-domains and bounded complete domains using subclasses

Abstract

Closure space has proven to be a useful tool to restructure lattices and various order structures.This paper aims to provide a novel approach to characterizing some important kinds of continuous domains by means of closure spaces. By introducing an additional map into a given closure space, the notion of F-augmented generalized closure space is presented. It is shown that F-augmented generalized closure spaces generate exactly continuous domains. Moreover, the notion of approximable mapping is identified to represent Scott-continuous functions between continuous domains. These results produce a category equivalent to that of continuous domains with Scottcontinuous functions. At the same time, two subclasses of F-augmented generalized closure spaces are considered which are representations of continuous L-domains and continuous bounded complete domains, respectively.