Partial order: Difference between revisions

Browse history interactively ← Previous editContent deleted Content addedVisual WikitextInline

Revision as of 15:33, 16 November 2001 editAxelBoldt (talk \| contribs)Administrators44,502 editsNo edit summary← Previous edit		Latest revision as of 18:59, 2 September 2024 edit undoTule-hog (talk \| contribs)Extended confirmed users4,580 edits add rcat
(50 intermediate revisions by 24 users not shown)
Line 1:		Line 1:
			#REDIRECT ]
	A '''partial order''' <= on a ] ''X'' is a ] that is '''reflexive''', '''antisymmetric''' and '''transitive''', i.e., it holds for all ''a'', ''b'' and ''c'' in ''X'' that:

			{{Rcatsh\|
	* (reflexivity) ''a'' <= ''a''
			{{R to section}}

			{{R with Wikidata item}}
	* (antisymmetry) if ''a'' <= ''b'' and ''b'' <= ''a'' then ''a'' = ''b''
			{{R with history}}

			}}
	* (transitivity) if ''a'' <= ''b'' and ''b'' <= ''c'' then ''a'' <= ''c''



	A set with a partial order on it is called a '''partially ordered set''', or '''poset'''. We can then also define an irreflexive relation <, where ''a'' < ''b'' if and only if ''a'' <= ''b'' and ''a'' ≠ ''b''.



	Examples of partial orders include implications and inclusions ("is a subset of" and the more general "is a subobject of" in the sense of ]). Clearly, any ] is also a partially ordered set, for example <= on the real numbers.



	Finite posets are most easily visualized as ''Hasse diagrams'', that is, graphs where the vertices are the elements of the poset and the ordering relation is indicated by edges and the relative positioning of the vertices. The element ''x'' is smaller than ''y'' if and only if there exists a path from ''x'' to ''y'' always going upwards.



	New partially ordered sets can be constructed from other partially ordered sets by cartesian products, disjoint unions and so on.



	A poset where any two elements have both a greatest lower bound and a least upper bound forms an algebraic structure called a ]. Every poset (''X'',<=) has a unique dual poset (''X'',>=).



	A partially ordered set is ''complete'' if any increasing chain of elements has a least upper bound. Various types of complete partially ordered sets are used in, for example, ]. The most well-known type of complete partially ordered sets are the ]. These structures are important in that they constitute a ] and in that they provide a natural theory of approximations.

	That the class of Scott-Ershov domains is cartesian closed category enables the solution of so-called domain equations, e.g., ''D'' = , where the right-hand side denotes the space of all continuous functions on ''D''.



	Partially ordered sets can be given a ], for example, the Alexandrov topology, consisting of all upwards closed subsets. A subset ''U'' of a partially ordered set is ''upwards closed'' if ''x'' in ''U'' and ''x'' <= ''y'' implies that ''y'' belongs to ''U''. For special types of partially ordered sets other topologies may be more interesting. For example, the natural topology on Scott-Ershov domains is the Scott topology.



	----

	/Talk

			{{Authority control}}

Latest revision as of 18:59, 2 September 2024

Redirect to:

Partially ordered set#Partial order

This page is a redirect. The following categories are used to track and monitor this redirect:

To a section: This is a redirect from a topic that does not have its own page to a section of a page on the subject. For redirects to embedded anchors on a page, use {{R to anchor}} instead.

From a cross-project redirect: This is a redirect from a title linked to an item on Wikidata. The Wikidata item linked to this page is partial order (Q1069998).
- Use this template only on hard redirects – for soft redirects use {{Soft redirect with Wikidata item}}.

With history: This is a redirect from a page containing substantive page history. This page is kept as a redirect to preserve its former content and attributions. Please do not remove the tag that generates this text (unless the need to recreate content on this page has been demonstrated), nor delete this page.
- This template should not be used for redirects having some edit history but no meaningful content in their previous versions, nor for redirects created as a result of a page merge (use {{R from merge}} instead), nor for redirects from a title that forms a historic part of Misplaced Pages (use {{R with old history}} instead).

When appropriate, protection levels are automatically sensed, described and categorized.