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| | Hi there. Let me begin by introducing the writer, her title is Myrtle Cleary. Puerto Rico is exactly where he's always been residing but she requirements to move simply because of her family. Bookkeeping is what I do. Doing ceramics is what her family and her enjoy.<br><br>Here is my homepage - at home std testing ([http://drupal.12thirty4.com/gmaps/node/4991 Look At This]) |
| The powerset algebra of the set <math>\{1,2,3,4\}</math> with the upset <math>\uparrow\{1\}</math> colored green.]]
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| In [[mathematics]], an '''upper set''' (also called an '''upward [[Closure (mathematics)#Closed sets|closed]]''' set or just an '''upset''') of a [[partially ordered set]] (''X'',≤) is a subset ''U'' with the property that, if ''x'' is in ''U'' and ''x''≤''y'', then ''y'' is in ''U''.
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| The [[duality (order theory)|dual]] notion is '''lower set''' (alternatively, '''down set''', '''decreasing set''', '''initial segment'''; the set is '''downward closed'''), which is a subset ''L'' with the property that, if ''x'' is in ''L'' and ''y''≤''x'', then ''y'' is in ''L''.
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| == Properties ==
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| *Every partially ordered set is an upper set of itself.
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| *The [[intersection (set theory)|intersection]] and the [[union (set theory)|union]] of upper sets is again an upper set.
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| *The [[complement (set theory)|complement]] of any upper set is a lower set, and vice versa.
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| *Given a partially ordered set (''X'',≤), the family of lower sets of ''X'' ordered with the [[inclusion (set theory)|inclusion]] relation is a [[complete lattice]], the '''down-set lattice''' O(''X'').
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| *Given an arbitrary subset ''Y'' of an ordered set ''X'', the smallest upper set containing ''Y'' is denoted using an up arrow as ↑''Y''.
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| **Dually, the smallest lower set containing ''Y'' is denoted using a down arrow as ↓''Y''.
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| *A lower set is called '''principal''' if it is of the form ↓{''x''} where ''x'' is an element of ''X''.
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| *Every lower set ''Y'' of a finite ordered set ''X'' is equal to the smallest lower set containing all [[maximal element]]s of ''Y'': ''Y'' = ↓Max(''Y'') where Max(''Y'') denotes the set containing the maximal elements of ''Y''.
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| *A [[directed set|directed]] lower set is called an [[order ideal]].
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| *The [[minimal element]]s of any upper set form an [[antichain]].
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| **Conversely any antichain ''A'' determines an upper set {''x'': for some ''y'' in ''A'', ''x'' ≥ ''y''}. For partial orders satisfying the [[descending chain condition]] this correspondence between antichains and upper sets is 1-1, but for more general partial orders this is not true.
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| ==Ordinal numbers==
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| An [[ordinal number]] is usually identified with the set of all smaller ordinal numbers. Thus each ordinal number forms a lower set in the class of all ordinal numbers, which are totally ordered by set inclusion.
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| ==See also==
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| * [[Cofinal set]] – a subset ''U'' of a partially ordered set (''P'',≤) that contains for every element ''x'' of ''P'' an element ''y'' such that ''x'' ≤ ''y''
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| ==References==
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| *Blanck, J. (2000) "Domain representations of topological spaces". Theoretical Computer Science, '''247''', 229–255.
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| *Hoffman, K. H. (2001), [http://www.mathematik.tu-darmstadt.de:8080/Math-Net/Lehrveranstaltungen/Lehrmaterial/SS2003/Topology/separation.pdf ''The low separation axioms (T<sub>0</sub>) and (T<sub>1</sub>)'']
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| * {{cite book
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| | author = Davey, B.A., and Priestley, H. A.
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| | year = 2002
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| | title = Introduction to Lattices and Order
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| | edition = 2nd
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| | publisher = Cambridge University Press
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| | isbn = 0-521-78451-4
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| }}
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| [[Category:Order theory]]
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| [[ru:Частично упорядоченное множество#Верхнее и нижнее множество]]
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