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| partial ordering A R is a partial ordering if it is a (i.e. it is (x R x) and (x R y R z => x R z)) and it is also (x R y R x => x = y). The ordering is partial, rather than total, because there may exist elements x and y for which neither x R y nor y R x. In , if D is a set of values including the undefined value () then we can define a partial ordering relation <= on D by x <= y if x = bottom or x = y. The constructed set D x D contains the very undefined element, (bottom, bottom) and the not so undefined elements, (x, bottom) and (bottom, x). The partial ordering on D x D is then (x1,y1) <= (x2,y2) if x1 <= x2 and y1 <= y2. The partial ordering on D -> D is defined by f <= g if f(x) <= g(x) for all x in D. (No f x is more defined than g x.) A is a partial ordering where all finite subsets have a and a . ("<=" is written in as {\sqsubseteq}). (1995-02-03) |
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