The bounded ideal monad on the category of quasi-metric spaces and its algebras

TL;DR

This paper introduces the bounded ideal monad for quasi-metric spaces, characterizes its algebras as metric local dcpos, and explores their properties within the categorical framework.

Contribution

It defines the bounded ideal monad on quasi-metric spaces and characterizes its algebras as metric local dcpos and local domains.

Findings

Algebras of the monad are standard quasi-metric spaces with formal balls forming local dcpos.

Continuous algebras correspond to standard quasi-metric spaces with formal balls forming local domains.

The work extends the categorical understanding of quasi-metric spaces and their domain-theoretic structures.

Abstract

The notion of bounded ideals is introduced for quasi-metric spaces. Such ideals give rise to a monad, the bounded ideal monad, on the category of quasi-metric spaces and non-expansive maps. Algebras of this monad are metric version of local dcpos of Mislove. It is shown that an algebra of the bounded ideal monad is a standard quasi-metric space of which the formal balls form a local dcpo; and that a continuous algebra is a standard quasi-metric space of which the formal balls form a local domain.