A ref="module.php?name=Lexikon&file=search&eid=1&query=relation">relation R is a partial ordering if it is a ref="module.php?name=Lexikon&file=search&eid=1&query=pre-order">pre-order (i.e. it is ref="module.php?name=Lexikon&file=search&eid=1&query=reflexive">reflexive (x R x) and ref="module.php?name=Lexikon&file=search&eid=1&query=transitive">transitive (x R y R z => x R z)) and it is also ref="module.php?name=Lexikon&file=search&eid=1&query=antisymmetric">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 ref="module.php?name=Lexikon&file=search&eid=1&query=domain theory">domain theory, if D is a set of values including the undefined value (ref="module.php?name=Lexikon&file=search&eid=1&query=bottom">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 ref="module.php?name=Lexikon&file=search&eid=1&query=lattice">lattice is a partial ordering where all finite subsets have a ref="module.php?name=Lexikon&file=search&eid=1&query=least upper bound">least upper bound and a ref="module.php?name=Lexikon&file=search&eid=1&query=greatest lower bound">greatest lower bound. ("<=" is written in ref="module.php?name=Lexikon&file=search&eid=1&query=LaTeX">LaTeX as ref="module.php?name=Lexikon&file=search&eid=1&query=sqsubseteq">sqsubseteq). (1995-02-03)

In addition suitable contents:
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ref="module.php?name=Lexikon&op=content&tid=1503">bottom ] [ ref="module.php?name=Lexikon&op=content&tid=1606">bs ] [ ref="module.php?name=Lexikon&op=content&tid=1695">by ] [ ref="module.php?name=Lexikon&op=content&tid=1724">ca ] [ ref="module.php?name=Lexikon&op=content&tid=2001">ch ] [ ref="module.php?name=Lexikon&op=content&tid=2138">cl ] [ ref="module.php?name=Lexikon&op=content&tid=2247">co ] [ ref="module.php?name=Lexikon&op=content&tid=2545">con ] [ ref="module.php?name=Lexikon&op=content&tid=2606">cons ] [ ref="module.php?name=Lexikon&op=content&tid=2976">D ] [ ref="module.php?name=Lexikon&op=content&tid=3151">de ] [ ref="module.php?name=Lexikon&op=content&tid=3436">ding ] [ ref="module.php?name=Lexikon&op=content&tid=3565">do ] [ ref="module.php?name=Lexikon&op=content&tid=3595">domain ] [ ref="module.php?name=Lexikon&op=content&tid=3611">domain theory ] [ ref="module.php?name=Lexikon&op=content&tid=3752">du ] [ ref="module.php?name=Lexikon&op=content&tid=3865">ec ] [ ref="module.php?name=Lexikon&op=content&tid=3896">ed ] [ ref="module.php?name=Lexikon&op=content&tid=4008">element ] [ ref="module.php?name=Lexikon&op=content&tid=4148">er ] [ ref="module.php?name=Lexikon&op=content&tid=4171">es ] [ ref="module.php?name=Lexikon&op=content&tid=4199">et ] [ ref="module.php?name=Lexikon&op=content&tid=4497">fi ] [ ref="module.php?name=Lexikon&op=content&tid=4520">file ] [ ref="module.php?name=Lexikon&op=content&tid=4559">finite ] [ ref="module.php?name=Lexikon&op=content&tid=4700">fo ] [ ref="module.php?name=Lexikon&op=content&tid=4727">for ] [ ref="module.php?name=Lexikon&op=content&tid=5291">gr ] [ ref="module.php?name=Lexikon&op=content&tid=5341">greatest lower bound ] [ ref="module.php?name=Lexikon&op=content&tid=5434">h ] [ ref="module.php?name=Lexikon&op=content&tid=5768">hr ] [ ref="module.php?name=Lexikon&op=content&tid=5931">id ] [ ref="module.php?name=Lexikon&op=content&tid=6013">il ] [ ref="module.php?name=Lexikon&op=content&tid=6064">in ] [ ref="module.php?name=Lexikon&op=content&tid=6068">inc ] [ ref="module.php?name=Lexikon&op=content&tid=6413">io ] [ ref="module.php?name=Lexikon&op=content&tid=6482">is ] [ ref="module.php?name=Lexikon&op=content&tid=6558">it ] [ ref="module.php?name=Lexikon&op=content&tid=6918">la ] [ ref="module.php?name=Lexikon&op=content&tid=6989">LaTeX ] [ ref="module.php?name=Lexikon&op=content&tid=6991">lattice ] [ ref="module.php?name=Lexikon&op=content&tid=7048">least upper bound ] [ ref="module.php?name=Lexikon&op=content&tid=7091">Lex ] [ ref="module.php?name=Lexikon&op=content&tid=7399">ls ] [ ref="module.php?name=Lexikon&op=content&tid=7415">lu ] [ ref="module.php?name=Lexikon&op=content&tid=7463">ma ] [ ref="module.php?name=Lexikon&op=content&tid=7822">metric ] [ ref="module.php?name=Lexikon&op=content&tid=8019">mm ] [ ref="module.php?name=Lexikon&op=content&tid=8032">mo ] [ ref="module.php?name=Lexikon&op=content&tid=8040">mod ] [ ref="module.php?name=Lexikon&op=content&tid=8079">module ] [ ref="module.php?name=Lexikon&op=content&tid=8384">N ] [ ref="module.php?name=Lexikon&op=content&tid=8386">na ] [ ref="module.php?name=Lexikon&op=content&tid=8460">nc ] [ ref="module.php?name=Lexikon&op=content&tid=8472">ne ] [ ref="module.php?name=Lexikon&op=content&tid=8627">ng ] [ ref="module.php?name=Lexikon&op=content&tid=8630">ni ] [ ref="module.php?name=Lexikon&op=content&tid=8675">no ] [ ref="module.php?name=Lexikon&op=content&tid=8760">ns ] [ ref="module.php?name=Lexikon&op=content&tid=8964">om ] [ ref="module.php?name=Lexikon&op=content&tid=9115">ordering ] [ ref="module.php?name=Lexikon&op=content&tid=9204">pa ] [ ref="module.php?name=Lexikon&op=content&tid=9457">pe ] [ ref="module.php?name=Lexikon&op=content&tid=9550">ph ] [ ref="module.php?name=Lexikon&op=content&tid=9908">pr ] [ ref="module.php?name=Lexikon&op=content&tid=9929">pre-order ] [ ref="module.php?name=Lexikon&op=content&tid=10253">query ] [ ref="module.php?name=Lexikon&op=content&tid=10364">rc ] [ ref="module.php?name=Lexikon&op=content&tid=10385">re ] [ ref="module.php?name=Lexikon&op=content&tid=10476">reflexive ] [ ref="module.php?name=Lexikon&op=content&tid=10508">relation ] [ ref="module.php?name=Lexikon&op=content&tid=10887">ru ] [ ref="module.php?name=Lexikon&op=content&tid=11150">se ] [ ref="module.php?name=Lexikon&op=content&tid=11281">set ] [ ref="module.php?name=Lexikon&op=content&tid=11376">si ] [ ref="module.php?name=Lexikon&op=content&tid=11506">sit ] [ ref="module.php?name=Lexikon&op=content&tid=11651">so ] [ ref="module.php?name=Lexikon&op=content&tid=11900">sqsubseteq ] [ ref="module.php?name=Lexikon&op=content&tid=11934">st ] [ ref="module.php?name=Lexikon&op=content&tid=12109">struct ] [ ref="module.php?name=Lexikon&op=content&tid=12133">su ] [ ref="module.php?name=Lexikon&op=content&tid=12148">subseteq ] [ ref="module.php?name=Lexikon&op=content&tid=12246">sy ] [ ref="module.php?name=Lexikon&op=content&tid=12270">symmetric ] [ ref="module.php?name=Lexikon&op=content&tid=12359">T ] [ ref="module.php?name=Lexikon&op=content&tid=12557">test ] [ ref="module.php?name=Lexikon&op=content&tid=12588">th ] [ ref="module.php?name=Lexikon&op=content&tid=12602">theory ] [ ref="module.php?name=Lexikon&op=content&tid=12721">to ] [ ref="module.php?name=Lexikon&op=content&tid=12787">tr ] [ ref="module.php?name=Lexikon&op=content&tid=12819">transitive ] [ ref="module.php?name=Lexikon&op=content&tid=12896">tt ] [ ref="module.php?name=Lexikon&op=content&tid=13146">up ] [ ref="module.php?name=Lexikon&op=content&tid=13155">upper bound ] [ ref="module.php?name=Lexikon&op=content&tid=13175">us ] [ ref="module.php?name=Lexikon&op=content&tid=13252">va ] [ ref="module.php?name=Lexikon&op=content&tid=13260">value ] [ ref="module.php?name=Lexikon&op=content&tid=13310">ve ] [ ref="module.php?name=Lexikon&op=content&tid=13891">X ]