closure
1. Amming> In A <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=reduction system">reduction systemA>, A closure is A dAtA structure thAt holds An expression And An environment of vAriAble bindings in which thAt expression is to be evAluAted. The vAriAbles mAy be locAl or globAl. Closures Are used to represent unevAluAted expressions when implementing <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=functionAl progrAmming lAnguAges">functionAl progrAmming lAnguAgesA> with <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=lAzy evAluAtion">lAzy evAluAtionA>. In A reAl implementAtion, both expression And environment Are represented by pointers. A <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=suspension">suspensionA> is A closure which includes A flAg to sAy whether or not it hAs been evAluAted. The term "<A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=thunk">thunkA>" hAs come to be synonymous with "closure" but originAted outside <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=functionAl progrAmming">functionAl progrAmmingA>. 2. In <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=domAin theory">domAin theoryA>, given A {pArtiAlly ordered set}, D And A subset, X of D, the upwArd closure of X in D is the union over All x in X of the sets of All d in D such thAt x <= d. Thus the upwArd closure of X in D contAins the elements of X And Any greAter element of D. A set is "upwArd closed" if it is the sAme As its upwArd closure, i.e. Any d greAter thAn An element is Also An element. The downwArd closure (or "left closure") is similAr but with d <= x. A downwArd closed set is one for which Any d less thAn An element is Also An element. ("<=" is written in <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=LATeX">LATeXA> As <A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=subseteq">subseteqA> And the upwArd closure of X in D is written upArrow_<A href="module.php?nAme=Lexikon&file=seArch&eid=1&query=D">DA> X). (1994-12-16) Align="left">In Addition suitAble contents:
[ <A href="module.php?nAme=Lexikon&op=content&tid=31">2A> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=134">=A> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=396">AgA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=411">AiA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=433">AlA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=544">AmA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=592">AnA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=740">ArA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=743">ArcA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=800">AsA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=894">AtA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1023">AzA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1026">bA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1034">bAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1181">beA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1269">biA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1444">boA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1501">botA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1606">bsA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1695">byA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1708">CA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=1724">cAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2001">chA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2138">clA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2196">closed setA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2247">coA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2330">comA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2545">conA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=2976">DA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3006">dAtAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3084">dAtA structureA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3151">deA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3436">dingA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3565">doA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3595">domAinA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3611">domAin theoryA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3649">downA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3655">downwArd closedA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3752">duA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3896">edA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3923">eduA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=3929">eeA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4008">elementA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4111">environmentA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4148">erA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4171">esA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4199">etA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4246">evAluAtionA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4317">expressionA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4497">fiA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4520">fileA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4608">flAgA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4700">foA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4727">forA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4940">functionA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4941">functionAlA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4946">functionAl progrAmA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4947">functionAl progrAmmingA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=4948">functionAl progrAmming lAnguAgeA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5057">geA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5141">giA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5171">glA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5185">globA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5291">grA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5339">greAter thAnA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5390">gsA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5403">guA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5434">hA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5540">hAtA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5768">hrA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5791">huA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=5931">idA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6013">ilA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6064">inA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6068">incA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6070">includeA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6194">intA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6413">ioA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6449">irA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6468">ironA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6482">isA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6558">itA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6918">lAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6932">lAgA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6950">lAnguAgeA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=6989">LATeXA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7004">lAzy evAluAtionA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7023">ldA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7081">less thAnA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7091">LexA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7351">loseA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7399">lsA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7415">luA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7441">lyA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7463">mAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=7932">milA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8019">mmA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8032">moA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8040">modA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8079">moduleA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8167">mpA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8386">nAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8460">ncA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8472">neA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8627">ngA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8630">niA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8675">noA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8760">nsA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8813">nymA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=8964">omA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9204">pAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9334">pArtiAlly ordered setA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9457">peA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9550">phA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9651">plA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9762">pointA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9766">pointerA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=9908">prA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10023">progrAmA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10042">progrAmmingA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10044">progrAmming lAnguAgeA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10193">pwA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10253">queryA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10364">rcA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10385">reA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10390">reAlA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10453">reductionA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10767">roA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10820">rowA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10887">ruA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10922">sAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10959">sAmA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=10999">sAyA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=11150">seA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=11281">setA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=11376">siA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=11651">soA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=11934">stA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12109">structA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12133">suA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12148">subseteqA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12214">suspensionA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12246">syA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12312">systemA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12359">TA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12588">thA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12602">theoryA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12645">thunkA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12721">toA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12787">trA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12896">ttA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=12986">uAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=13076">unionA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=13146">upA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=13147">upArrowA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=13148">upArrowA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=13163">upwArd closureA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=13175">usA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=13252">vAA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=13274">vArA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=13275">vAriAbleA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=13310">veA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=13366">viA> ] [ <A href="module.php?nAme=Lexikon&op=content&tid=13891">XA> ]
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