Free dcpo-algebras via directed spaces

TL;DR

This paper establishes a topological framework connecting directed spaces, dcpo-algebras, and powerdomains, revealing that free dcpo-algebras over Scott spaces are characterized by D-completions, thus unifying domain-theoretic and topological approaches.

Contribution

It demonstrates that D-completions of free algebras over Scott spaces are precisely the free dcpo-algebras over dcpos, linking directed powerspaces and powerdomains.

Findings

D-completion of free algebras over Scott spaces equals free dcpo-algebras

Provides a topological representation of powerdomains

Establishes a connection between directed spaces and domain theory

Abstract

Directed spaces are natural topological extensions of dcpos in domain theory and form a cartesian closed category. We will show that the D-completion of free algebras over a Scott space $\Sigma L$, on the context of directed spaces, are exactly the free dcpo-algebras over dcpo $L$, which reveals the close connection between directed powerspaces and powerdomains. By this result, we provide a topological representation of upper, lower and convex powerdomains of dcpos uniformly.