Free dcpo-algebras via directed spaces

06/29/2023

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by   Yuxu Chen, et al.

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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 Σ 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.