en>CommonsDelinker: Replacing Directed_set,_but_no_join_semi-lattice.gif with File:Directed_set,_but_no_join_semi-lattice.png (by GifTagger because: Replacing GIF by exact PNG duplicate.).

2015-01-04T15:54:52Z

Replacing Directed_set,_but_no_join_semi-lattice.gif with File:Directed_set,_but_no_join_semi-lattice.png (by GifTagger because: Replacing GIF by exact PNG duplicate.).

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en>Jochen Burghardt: /* Contrast with semilattices */ suggest pic.

2013-09-21T23:04:36Z

Contrast with semilattices: suggest pic.

← Older revision		Revision as of 01:04, 22 September 2013
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	<br><br>~~You might~~ also ~~face it~~, ~~inflexible economy you could probably use some more income coming in~~ . ~~The internet~~ is ~~actually~~ an ~~appealing place~~ to ~~earn~~.<br><br>~~Finding a reliable dissertation Writing Services~~ can be ~~tricky. But~~, ~~don~~'~~t frustration. Here we offers some solid tips certain that that essential get cheated. To make certain you obtain a high quality dissertation help~~, ~~you require to find a reliable writing company which will not period money available~~. In this ~~way your money will~~ be ~~secure~~ and ~~trusted hands anyone will go for complete work without facing any nightmare~~.<br><br>~~Pack~~ a ~~strategy B. Insert system prepared for those elusive events when you~~ can'~~t ensure it for instance school gathering and occasions while busy doing your essay paper at enterprise. Talk to~~ a ~~close other relatives about picking your boy or girl up~~ and ~~watching them in~~ a ~~nip~~.<br><br>~~If your project~~ is ~~the press release type, you need to give facts to aid your writer in need to best website~~ that ~~need to~~. ~~Seek and choose~~ a ~~contract writer with experience inside this type of project. It~~ is a ~~big plus to guarantee that~~ a ~~truth~~ is ~~obtaining.~~<br><br>~~Stay~~ in ~~touch with~~ the ~~writer while they work~~ on ~~your project~~. It'~~s very essential~~ a ~~person need to stay~~ in ~~contact with~~ the ~~writer while they work round the project. Be around to answer any questions they will probably have.~~<br><br>~~First, choose~~ a ~~domain name that is short~~ and ~~memorable~~. ~~Jot down all the keywords your web visitors would almost certainly to use on search websites~~. ~~Then~~, ~~mix match these keywords~~ if ~~you come lets start work on a domain name that will perfectly satisfy your website~~'~~s theme~~. ~~Ensure~~ that ~~your domain name set up to spell~~ and ~~in order to pronounce.~~<br><br>~~So try~~ and ~~regarding this practice~~. ~~You don~~'~~t ever have anything~~ to ~~lose~~. ~~The website cost~~ is ~~cheap~~, ~~you~~'~~ll design~~ it ~~yourself~~, an ~~individual also can manage it~~ on ~~residence. Eventually~~, ~~could possibly grow for the point you~~ need an ~~employee~~. ~~Wouldn't that be great? And~~ it~~'s manage this~~ is to ~~make fast your money~~.<br><br>~~If you~~ have ~~almost any questions with regards to where by and tips on how to work with~~ [~~http://imnewshere~~.~~blogspot~~.~~in/2014/08/malaysia~~.~~html/ Resume]~~, ~~you~~'~~ll be able to e~~-~~mail us in our internet site~~.		In [[mathematics]], a '''directed set''' (or a '''directed preorder''' or a '''filtered set''') is a nonempty [[Set (mathematics)\|set]] ''A'' together with a [[reflexive relation\|reflexive]] and [[transitive relation\|transitive]] [[binary relation]] ≤ (that is, a [[preorder]]), with the additional property that every pair of elements has an [[upper bound]]:<ref>Kelley, p. 65.</ref> In other words, for any ''a'' and ''b'' in ''A'' there must exist a ''c'' in ''A'' with ''a'' ≤ ''c'' and ''b'' ≤ ''c''.

			Directed sets are a generalization of nonempty [[totally ordered set]]s, that is, all totally ordered sets are directed sets (contrast [[partially ordered sets\|''partially'' ordered sets]] which need not be directed). In [[topology]], directed sets are used to define [[net (topology)\|nets]], which generalize [[sequence]]s and unite the various notions of [[limit (mathematics)\|limit]] used in [[mathematical analysis\|analysis]]. Directed sets also give rise to [[direct limit]]s in [[abstract algebra]] and (more generally) [[category theory]].

			== Equivalent definition ==
			In addition to the definition above, there is an equivalent definition. A '''directed set''' is a set ''A'' with a preorder such that every finite subset of ''A'' has an upper bound. In this definition, we take the upper bound of the [[empty set\|empty subset]] to be any existing element of ''A'' and require that ''A'' be nonempty.

			== Examples ==
			Examples of directed sets include:
			* The set of [[natural number]]s '''N''' with the ordinary order ≤ is a directed set (and so is every [[total order\|totally ordered set]]).
			* Let '''D<sub>1</sub>''' and '''D<sub>2</sub>''' be directed sets. Then the Cartesian product set '''D<sub>1</sub> <math>\times</math> D<sub>2</sub>''' can be made into a directed set by defining (''n''<sub>1</sub>, ''n''<sub>2</sub>) ≤ (''m''<sub>1</sub>, ''m''<sub>2</sub>) if and only if ''n''<sub>1</sub> ≤ ''m''<sub>1</sub> and ''n''<sub>2</sub> ≤ ''m''<sub>2</sub>. In analogy to the [[product order]] this is the product direction on the Cartesian product.
			* It follows from previous example that the set '''N''' <math>\times</math> '''N''' of pairs of natural numbers can be made into a directed set by defining (''n''<sub>0</sub>, ''n''<sub>1</sub>) ≤ (''m''<sub>0</sub>, ''m''<sub>1</sub>) if and only if ''n''<sub>0</sub> ≤ ''m''<sub>0</sub> and ''n''<sub>1</sub> ≤ ''m''<sub>1</sub>.
			* If ''x''<sub>0</sub> is a [[real number]], we can turn the set '''R''' − {''x''<sub>0</sub>} into a directed set by writing ''a'' ≤ ''b'' if and only if <br>\|''a'' − ''x''<sub>0</sub>\| ≥ \|''b'' − ''x''<sub>0</sub>\|. We then say that the reals have been ''directed towards x<sub>0</sub>''. This is an example of a directed set that is not [[partial order\|ordered]] (neither totally nor partially).
			* A (trivial) example of a partially ordered set that is '''''not''''' directed is the set {''a'', ''b''}, in which the only order relations are ''a'' ≤ ''a'' and ''b'' ≤ ''b''. A less trivial example is like the previous example of the "reals directed towards x<sub>0</sub>" but in which the ordering rule only applies to pairs of elements on the same side of x<sub>0</sub>.
			* If ''T'' is a [[topological space]] and ''x''<sub>0</sub> is a point in ''T'', we turn the set of all [[topological neighbourhood\|neighbourhoods]] of ''x''<sub>0</sub> into a directed set by writing ''U'' ≤ ''V'' if and only if ''U'' contains ''V''.
			** For every ''U'': ''U'' ≤ ''U''; since ''U'' contains itself.
			** For every ''U'',''V'',''W'': if ''U'' ≤ ''V'' and ''V'' ≤ ''W'', then ''U'' ≤ ''W''; since if ''U'' contains ''V'' and ''V'' contains ''W'' then ''U'' contains ''W''.
			** For every ''U'', ''V'': there exists the set ''U'' <math>\cap</math> ''V'' such that ''U'' ≤ ''U'' <math>\cap</math> ''V'' and ''V'' ≤ ''U'' <math>\cap</math> ''V''; since both ''U'' and ''V'' contain ''U'' <math>\cap</math> ''V''.
			* In a [[poset]] ''P'', every lower closure of an element, i.e. every subset of the form {''a''\| ''a'' in ''P'', ''a'' ≤''x''} where ''x'' is a fixed element from ''P'', is directed.

			[[File:Directed set, but no join semi-lattice.gif\|thumb\|x100px\|Witness]]
			== Contrast with semilattices ==

			Directed sets are a more general concept than (join) semilattices: every [[semilattice\|join semilattice]] is a directed set, as the join or least upper bound of two elements is the desired ''c''. The converse does not hold however, witness the directed set {1000,0001,1101,1011,1111} ordered bitwise (e.g. 1000 ≤ 1011 holds, but 0001 ≤ 1000 does not, since in the last bit 1 > 0), where {1000,0001} has three upper bounds but no ''least'' upper bound, cf. picture.

			== Directed subsets ==
			The order relation in a directed sets is not required to be [[antisymmetric relation\|antisymmetric]], and therefore directed sets are not always [[partial order]]s. However, the term ''directed set'' is also used frequently in the context of posets. In this setting, a subset ''A'' of a partially ordered set (''P'',≤) is called a '''directed subset''' if it is a directed set according to the same partial order: in other words, it is not the [[empty set]], and every pair of elements has an upper bound. Here the order relation on the elements of ''A'' is inherited from ''P''; for this reason, reflexivity and transitivity need not be required explicitly.

			A directed subset of a poset is not required to be [[lower set\|downward closed]]; a subset of a poset is directed if and only if its downward closure is an [[ideal (order theory)\|ideal]]. While the definition of a directed set is for an "upward-directed" set (every pair of elements has an upper bound), it is also possible to define a downward-directed set in which every pair of elements has a common lower bound. A subset of a poset is downward-directed if and only if its upper closure is a [[filter (mathematics)\|filter]].

			Directed subsets are used in [[domain theory]], which studies [[directed complete partial order]]s.<ref>Gierz, p. 2.</ref> These are posets in which every upward-directed set is required to have a [[least upper bound]]. In this context, directed subsets again provide a generalization of convergent sequences.

			== See also ==
			* [[Filtered category]]
			* [[Centered set]]
			* [[Linked set]]

			==Notes==
			{{reflist}}

			== References ==

			* J. L. Kelley (1955), ''General Topology''.

			* Gierz, Hofmann, Keimel, ''et al.'' (2003), ''Continuous Lattices and Domains'', Cambridge University Press. ISBN 0-521-80338-1.

			[[Category:Mathematical relations]]
			[[Category:Order theory]]
			[[Category:General topology]]

71.46.49.198: Grammar clarity.

2012-08-20T22:13:42Z

Grammar clarity.

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Directed set - Revision history

en>CommonsDelinker: Replacing Directed_set,_but_no_join_semi-lattice.gif with File:Directed_set,_but_no_join_semi-lattice.png (by GifTagger because: Replacing GIF by exact PNG duplicate.).

en>Jochen Burghardt: /* Contrast with semilattices */ suggest pic.

71.46.49.198: Grammar clarity.