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### Meaning of POWERDOMAIN

 Computing Dictionary Definition: The powerdomain of a domain D is a domain containing some of the subsets of D. Due to the asymmetry condition in the definition of a partial order (and therefore of a domain) the powerdomain cannot contain all the subsets of D. This is because there may be different sets X and Y such that X <= Y and Y <= X which, by the asymmetry condition would have to be considered equal.There are at least three possible orderings of the subsets of a powerdomain:Egli-Milner:``` X <= Y iff for all x in X, exists y in Y: x <= y and for all y in Y, exists x in X: x <= y```("The other domain always contains a related element").Hoare or Partial Correctness or Safety:` X <= Y iff for all x in X, exists y in Y: x <= y`("The bigger domain always contains a bigger element").Smyth or Total Correctness or Liveness:` X <= Y iff for all y in Y, exists x in X: x <= y`("The smaller domain always contains a smaller element").If a powerdomain represents the result of an abstract interpretation in which a bigger value is a safe approximation to a smaller value then the Hoare powerdomain is appropriate because the safe approximation Y to the powerdomain X contains a safe approximation to each point in X.("<=" is written in latex as \sqsubseteq).