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-{{Distinguish|Pre-order}}
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-{{About|binary relations|the graph vertex ordering|Depth-first search|other uses}}
-{{Redirect|Quasiorder|irreflexive transitive relations|strict order}}
-In [[mathematics]], especially in [[order theory]], a '''preorder''' or '''quasiorder''' is a [[binary relation]] that is [[reflexive relation|reflexive]] and [[transitive relation|transitive]]. All [[partial order]]s and [[equivalence relation]]s are preorders, but preorders are more general.
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-The name 'preorder' comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders, but not quite; they're neither necessarily [[anti-symmetric relation|anti-symmetric]] nor [[symmetric relation|symmetric]].  Because a preorder is a binary relation, the symbol ≤ can be used as the notational device for the relation.  However, because they are not necessarily anti-symmetric, some of the ordinary intuition associated to the symbol ≤ may not apply.  On the other hand, a pre-order can be used, in a straightforward fashion, to define a partial order and an equivalence relation.  Doing so, however, is not always useful or worthwhile, depending on the problem domain being studied.
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-In words, when ''a'' ≤ ''b'', one may say that ''b'' ''covers'' ''a'' or that ''b'' ''precedes'' ''a'', or that ''b'' ''reduces'' to ''a''.  Occasionally, the notation ← or <math>\lesssim</math> is used instead of ≤.
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-: if ''a'' ≤ ''b'' and ''b'' ≤ ''c'' then ''a'' ≤ ''c'' (transitivity)
+   <li>[http://www.hbsdstny.com/news/html/?27916.html http://www.hbsdstny.com/news/html/?27916.html]</li>
-A set that is equipped with a preorder is called a '''preordered set''' (or '''proset''').<ref>For "proset", see e.g. {{citation
- | last1 = Eklund | first1 = Patrik
+   <li>[http://team-impressive.de/index.php?site=news_comments&newsID=148 http://team-impressive.de/index.php?site=news_comments&newsID=148]</li>
- | last2 = Gähler | first2 = Werner
- | doi = 10.1002/mana.19901470123
+   <li>[http://enseignement-lsf.com/spip.php?article66#forum18059111 http://enseignement-lsf.com/spip.php?article66#forum18059111]</li>
- | journal = Mathematische Nachrichten
- | mr = 1127325
+   <li>[http://www.tc139.cn/news/html/?223433.html http://www.tc139.cn/news/html/?223433.html]</li>
- | pages = 219–233
- | title = Generalized Cauchy spaces
+   <li>[http://f1fanclub.com.au/activity/p/23426/ http://f1fanclub.com.au/activity/p/23426/]</li>
- | volume = 147
- | year = 1990}}.</ref>
+ </ul>
-If a preorder is also [[antisymmetric relation|antisymmetric]], that is, ''a'' ≤ ''b'' and ''b'' ≤ ''a'' implies ''a'' = ''b'', then it is a [[partially ordered set|partial order]].
-On the other hand, if it is [[symmetric relation|symmetric]], that is, if ''a'' ≤ ''b'' implies ''b'' ≤ ''a'', then it is an [[equivalence relation]].
-A preorder which is preserved in all contexts (i.e. respected by all functions on ''P'') is called a '''precongruence'''.
-A precongruence which is also [[symmetric relation|symmetric]] (i.e. is an [[equivalence relation]]) is a [[congruence relation]].
-Equivalently, a preordered set ''P'' can be defined as a [[category theory|category]] with [[object (category theory)|objects]] the elements of ''P'', and each [[hom-set]] having at most one element (one for objects which are related, zero otherwise).
-Alternately, a preordered set can be understood as an [[enriched category]], enriched over the category '''2''' = (0→1).
-A [[preordered class]] is a [[class (mathematics)|class]] equipped with a [[preorder]]. Every set is a class and so every preordered set is a preordered class. Preordered classes can be defined as [[thin category|thin categories]], i.e. as categories with at most one morphism from an object to another.
-==Examples==
-* The [[reachability]] relationship in any [[directed graph]] (possibly containing cycles) gives rise to a preorder, where ''x''&nbsp;≤&nbsp;''y'' in the preorder if and only if there is a path from ''x'' to ''y'' in the directed graph. Conversely, every preorder is the reachability relationship of a directed graph (for instance, the graph that has an edge from ''x'' to ''y'' for every pair (''x'', ''y'') with ''x''&nbsp;≤&nbsp;''y''). However, many different graphs may have the same reachability preorder as each other. In the same way, reachability of [[directed acyclic graph]]s, directed graphs with no cycles, gives rise to [[partially ordered set]]s (preorders satisfying an additional anti-symmetry property).
-* Every [[finite topological space]] gives rise to a preorder on its points, in which ''x'' ≤ ''y'' if and only if ''x'' belongs to every neighborhood of ''y'', and every finite preorder can be formed as the [[Specialization_(pre)order|specialization preorder]] of a topological space in this way. That is, there is a [[bijection|1-to-1 correspondence]] between finite topologies and finite preorders. However, the relation between infinite topological spaces and their specialization preorders is not 1-to-1.
-* A [[net (mathematics)|net]] is a [[directed set|directed]] preorder, that is, each pair of elements has an [[upper bound]].  The definition of convergence via nets is important in [[topology]], where preorders cannot be replaced by [[partially ordered set]]s without losing important features.
-* The relation defined by <math>x \le y</math> [[iff]] <math>f(x) \le f(y)</math>, where ''f'' is a function into some preorder.
-* The relation defined by <math>x \le y</math> [[iff]] there exists some [[injective function|injection]] from ''x'' to ''y''. Injection may be replaced by [[surjection]], or any type of structure-preserving function, such as [[ring homomorphism]], or [[permutation]].
-* The [[embedding]] relation for countable [[total order]]ings.
-* The [[graph-minor]] relation in [[graph theory]].
-* A [[category (mathematics)|category]] with at most one [[morphism]] from any object <math>x</math> to any other object <math>y</math> is a preorder. Such categories are called [[thin category|thin]].  In this sense, categories "generalize" preorders by allowing more than one relation between objects: each morphism is a distinct (named) preorder relation.
-In computer science, one can find examples of the following preorders.
-* [[Many-one reduction|Many-one]] and [[Turing reduction|Turing reductions]] are preorders on complexity classes.
-* The [[Subtype|subtyping]] relations are usually preorders.
-* [[Simulation preorder]]s are preorders (hence the name).
-* [[Reduction relation]]s in [[abstract rewriting system]]s.
-Example of a [[Strict weak ordering#Total preorders|total preorder]]:
-* [[Preference]], according to common models.
-==Uses==
-Preorders play a pivotal role in several situations:
-* Every preorder can be given a topology, the [[Alexandrov topology]]; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrov topology on that set.
-* Preorders may be used to define [[interior algebra]]s.
-* Preorders provide the [[Kripke semantics]] for certain types of [[modal logic]].
-==Constructions==
-Every binary relation R on a set S can be extended to a preorder on S by taking the [[transitive closure]] and [[Binary relation#Operations on binary relations|reflexive closure]], R<sup>+=</sup>.  The transitive closure indicates path connection in R: ''x'' R<sup>+</sup> ''y'' if and only if there is an R-[[Path (graph theory)|path]] from ''x'' to y.
-Given a preorder <math>\lesssim</math> on S one may define an [[equivalence relation]] ~ on S such that ''a'' ~ ''b'' if and only if ''a'' <math>\lesssim</math> ''b'' and ''b'' <math>\lesssim</math> ''a''.  (The resulting relation is reflexive since a preorder is reflexive, transitive by applying transitivity of the preorder twice, and symmetric by definition.)
-Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, S / ~, the set of all [[equivalence class]]es of ~.  Note that if the preorder is R<sup>+=</sup>, S / ~ is the set of R-[[Cycle (graph theory)|cycle]] equivalence classes: ''x'' ∈ [''y''] if and only if ''x'' = ''y'' or ''x'' is in an R-cycle with y.  In any case, on S / ~ we can define [''x''] ≤ [''y''] if and only if ''x'' <math>\lesssim</math> ''y''. By the construction of ~, this definition is independent of the chosen representatives and the corresponding relation is indeed well-defined. It is readily verified that this yields a partially ordered set.
-Conversely, from a partial order on a partition of a set S one can construct a preorder on S. There is a 1-to-1 correspondence between preorders and pairs (partition, partial order).
-For a preorder "<math>\lesssim</math>", a relation "<" can be defined as ''a'' < ''b'' if and only if (''a'' <math>\lesssim</math> ''b'' and not ''b'' <math>\lesssim</math> ''a''), or equivalently, using the equivalence relation introduced above, (''a'' <math>\lesssim</math> ''b'' and not ''a'' ~ ''b''). It is a [[strict partial order]]; every strict partial order can be the result of such a construction. If the preorder is anti-symmetric, hence a partial order "≤", the equivalence is equality, so the relation "<" can also be defined as ''a'' < ''b'' if and only if (''a'' ≤ ''b'' and ''a'' ≠ ''b'').
-(Alternatively, for a preorder "<math>\lesssim</math>", a relation "<" can be defined as ''a'' < ''b'' if and only if (''a'' <math>\lesssim</math> ''b'' and ''a'' ≠ ''b''). The result is the reflexive reduction of the preorder. However, if the preorder is not anti-symmetric the result is not transitive, and if it is, as we have seen, it is the same as before.)
-Conversely we have ''a'' <math>\lesssim</math> ''b'' if and only if ''a'' < ''b'' or ''a'' ~ ''b''. This is the reason for using the notation "<math>\lesssim</math>"; "≤" can be confusing for a preorder that is not anti-symmetric, it may suggest that ''a'' ≤ ''b'' implies that ''a'' < ''b'' or ''a'' = ''b''.
-Note that with this construction multiple preorders "<math>\lesssim</math>" can give the same relation "<", so without more information, such as the equivalence relation, "<math>\lesssim</math>" cannot be reconstructed from "<". Possible preorders include the following:
-*Define ''a'' ≤ ''b'' as ''a'' < ''b'' or ''a'' = ''b'' (i.e., take the reflexive closure of the relation). This gives the partial order associated with the strict partial order "<" through reflexive closure; in this case the equivalence is equality, so we don't need the notations <math>\lesssim</math> and ~.
-*Define ''a'' <math>\lesssim</math> ''b'' as "not ''b'' < ''a''" (i.e., take the inverse complement of the relation), which corresponds to defining ''a'' ~ ''b'' as "neither ''a'' < ''b'' nor ''b'' < ''a''"; these relations <math>\lesssim</math> and ~ are in general not transitive; however, if they are, ~ is an equivalence; in that case "<" is a [[strict weak order]]. The resulting preorder is [[total relation|total]], that is, a [[total preorder]].
-==Number of preorders==
-{{Number of relations}}
-As explained above, there is a 1-to-1 correspondence between preorders and pairs (partition, partial order). Thus the number of preorders is the sum of the number of partial orders on every partition. For example:
-*for n=3:
-**1 partition of 3, giving 1 preorder
-**3 partitions of 2+1, giving 3 × 3 = 9 preorders
-**1 partition of 1+1+1, giving 19 preorders
-:i.e. together 29 preorders.
-*for n=4:
-**1 partition of 4, giving 1 preorder
-**7 partitions with two classes (4  of 3+1 and 3 of 2+2), giving 7 × 3 = 21 preorders
-**6 partitions of 2+1+1, giving 6 × 19 = 114 preorders
-**1 partition of 1+1+1+1, giving 219 preorders
-:i.e. together 355 preorders.
-==Interval==
-For ''a'' <math>\lesssim</math> ''b'', the [[interval (mathematics)|interval]] [''a'',''b''] is the set of points ''x'' satisfying ''a'' <math>\lesssim</math> ''x'' and ''x'' <math>\lesssim</math> ''b'', also written ''a'' <math>\lesssim</math> ''x'' <math>\lesssim</math> ''b''. It contains at least the points ''a'' and ''b''. One may choose to extend the definition to all pairs (''a'',''b''). The extra intervals are all empty.
-Using the corresponding strict relation "<", one can also define the interval (''a'',''b'') as the set of points ''x'' satisfying ''a'' < ''x'' and ''x'' < ''b'', also written ''a'' < ''x'' < ''b''. An open interval may be empty even if ''a'' < ''b''.
-Also [''a'',''b'') and (''a'',''b''] can be defined similarly.
-==See also==
-*[[partially ordered set|partial order]] - preorder that is [[antisymmetric relation|antisymmetric]]
-*[[equivalence relation]] - preorder that is [[Symmetric relation|symmetric]]
-*[[Strict weak ordering#Total preorders|total preorder]] - preorder that is [[Total relation|total]]
-*[[total order]] - preorder that is antisymmetric and total
-*[[directed set]]
-*[[category of preordered sets]]
-*[[prewellordering]]
-*[[Well-quasi-ordering]]
-==References==
-<references />
-{{refbegin}}
-* {{Citation
-  | last = Schröder | first = Bernd S. W.
-   | title = Ordered Sets: An Introduction
-  | place = Boston
-  | publisher = Birkhäuser
-  | year = 2002
-  | isbn = 0-8176-4128-9}}
-{{refend}}
-[[Category:Order theory]]
-[[Category:Mathematical relations]]

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