meaning of partially ordered set
1. partially ordered set A set with a partial ordering. partial ordering A relation R is a partial ordering if it is a pre-order i. e. it is reflexive x R x and transitive x R y R z => x R z and it is also antisymmetric 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 domain theory, if D is a set of values including the undefined value bottom 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 fx <= gx for all x in D. No f x is more defined than g x. A lattice is a partial ordering where all finite subsets have a least upper bound and a greatest lower bound. "<=" is written in LaTeX as sqsubseteq.
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