Free algebras over directed spaces

TL;DR

This paper establishes the existence of free algebras over directed spaces, characterizes c-spaces and b-spaces, and demonstrates that free algebra carriers preserve continuity and algebraicity using categorical methods.

Contribution

It proves the existence of free algebras over directed spaces and characterizes c-spaces and b-spaces as continuous and algebraic spaces, respectively, with preservation properties.

Findings

Free algebras over directed spaces exist by the Adjoint Functor Theorem.

c-spaces are characterized as continuous directed spaces.

The carrier spaces of free algebras retain continuity and algebraicity.

Abstract

Directed spaces are natural topological extensions of dcpos in domain theory and form a cartesian closed category. In order to model nondeterministic semantics, the power structures over directed spaces were defined through the form of free algebras. We show that free algebras over any directed space exist by the Adjoint Functor Theorem. An c-space (resp.\ b-space) can be characterized as a continuous (resp.\ algebraic) directed space. We show that continuous spaces are just all retracts of algebraic spaces by means of topological ideals, which are generalizations of the rounded ideals. Moreover, by categorical methods, we show that the carrier spaces of free algebras over continuous (resp.\ algebraic) spaces are still continuous (resp.\ algebraic) spaces.