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)

N="left">IN additioN suitable coNteNts:
[ 2 ] [ = ] [ ai ] [ al ] [ am ] [ aN ] [ aNtisymmetric ] [ ar ] [ arc ] [ as ] [ at ] [ au ] [ av ] [ b ] [ be ] [ bo ] [ bot ] [ bottom ] [ bs ] [ by ] [ ca ] [ ch ] [ cl ] [ co ] [ coN ] [ coNs ] [ D ] [ de ] [ diNg ] [ do ] [ domaiN ] [ domaiN theory ] [ du ] [ ec ] [ ed ] [ elemeNt ] [ er ] [ es ] [ et ] [ fi ] [ file ] [ fiNite ] [ fo ] [ for ] [ gr ] [ greatest lower bouNd ] [ h ] [ hr ] [ id ] [ il ] [ iN ] [ iNc ] [ io ] [ is ] [ it ] [ la ] [ LaTeX ] [ lattice ] [ least upper bouNd ] [ Lex ] [ ls ] [ lu ] [ ma ] [ metric ] [ mm ] [ mo ] [ mod ] [ module ] [ N ] [ Na ] [ Nc ] [ Ne ] [ Ng ] [ Ni ] [ No ] [ Ns ] [ om ] [ orderiNg ] [ pa ] [ pe ] [ ph ] [ pr ] [ pre-order ] [ query ] [ rc ] [ re ] [ reflexive ] [ relatioN ] [ ru ] [ se ] [ set ] [ si ] [ sit ] [ so ] [ sqsubseteq ] [ st ] [ struct ] [ su ] [ subseteq ] [ sy ] [ symmetric ] [ T ] [ test ] [ th ] [ theory ] [ to ] [ tr ] [ traNsitive ] [ tt ] [ up ] [ upper bouNd ] [ us ] [ va ] [ value ] [ ve ] [ X ]