partially ordered set

A set with a 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 f(x) <= g(x) 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). (1995-02-03)