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Free algebras over directed spaces

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.