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The Hoare and Symth power domain constructors commute under composition. (English) Zbl 0699.06008
Summary: This paper examines the composition of the Smyth and Hoare power domains \((P_ S\) and \(P_ H)\) applied over the class of consistently complete algebraic partial orders represented by Scott’s category of information systems. It is proposed that both orders of applications yield equivalent domains. This is motivated by interpreting data elements in the Smyth as disjunction of propositions and those in the Hoare as conjunctions (as suggested by Scott), giving the two conjunctive-disjunctive forms for each of the double power domains. Then, using elementary category theory, it is shown that the images of any object under \(P_ S\circ P_ H\) and under \(P_ H\circ P_ S\) are isomorphic, and hence that both constructors are equivalent as functors.

MSC:
06B35 Continuous lattices and posets, applications
68P05 Data structures
68Q55 Semantics in the theory of computing
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