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Lower and upper power domain constructions commute on all cpos. (English) Zbl 0748.68038
Summary: The initial lower and upper power domain constructions \({\mathcal P}_ \vee\) and \({\mathcal P}_ \wedge\) commute under composition for all cpos (K. E. Flannery and J. J. Martin [J. Comput. Syst. Sci. 40(2), 125-135 (1990; Zbl 0699.06008)]). The common result \({\mathcal P}_ \vee({\mathcal P}_ \wedge X)\) and \({\mathcal P}_ \wedge({\mathcal P}_ \vee X)\) is the free frame over the cpo \(X\).
Our proof differs completely from the proof of K. E. Flannery and J. J. Martin [loc. cit.]. It uses algebraic methods and does not rely on any explicit representations of the power domains.

MSC:
68Q55 Semantics in the theory of computing
06B35 Continuous lattices and posets, applications
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References:
[1] Flannery, K.E.; Martin, J.J., The Hoare and smyth power domain constructors commute under composition, J. comput. system sci., 40, 125-135, (1990) · Zbl 0699.06008
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