You are here: irt.org | FOLDOC | 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)
Nearby terms: partial function « partial key « partially ordered set « partial ordering » Partial Response Maximum Likelihood » partition » partitioned data set
FOLDOC, Topics, A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z, ?, ALL