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{{Short description|Mathematical set with an ordering}}
{{Stack|{{Binary relations}}}}
[[Image:Hasse diagram of powerset of 3.svg|right|thumb|upright=1.15|'''Fig. 1''' The [[Hasse diagram]] of the [[Power set|set of all subsets]] of a three-element set <math>\{x, y, z\},</math> ordered by [[set inclusion|inclusion]]. Sets connected by an upward path, like <math>\emptyset</math> and <math>\{x,y\}</math>, are comparable, while e.g. <math>\{x\}</math> and <math>\{y\}</math> are not.]]
In [[mathematics]], especially [[order theory]], a '''partial order''' on a [[Set (mathematics)|set]] is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize [[total order]]s, in which every pair is comparable.
Formally, a partial order is a [[homogeneous binary relation]] that is [[Reflexive relation|reflexive]], [[
== Partial order relations ==
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A '''reflexive''', '''weak''',<ref name=Wallis/> or '''{{visible anchor|non-strict partial order|Non-strict partial order}}''',<ref>{{cite book|chapter=Partially Ordered Sets|title=Mathematical Tools for Data Mining: Set Theory, Partial Orders, Combinatorics|publisher=Springer|year=2008|isbn=9781848002012|chapter-url=https://books.google.com/books?id=6i-F3ZNcub4C&pg=PA127|author1=Simovici, Dan A. |author2=Djeraba, Chabane |name-list-style=amp }}</ref> commonly referred to simply as a '''partial order''', is a [[homogeneous relation]] ≤ on a [[Set (mathematics)|set]] <math>P</math> that is [[Reflexive relation|reflexive]], [[Antisymmetric relation|antisymmetric]], and [[Transitive relation|transitive]]. That is, for all <math>a, b, c \in P,</math> it must satisfy:
# [[Reflexive relation|Reflexivity]]: <math>a \leq a</math>, i.e. every element is related to itself.
# [[Antisymmetric relation|Antisymmetry]]: if <math>a \leq b</math> and <math>b \leq a</math> then <math>a = b</math>, i.e. no two distinct elements precede each other.
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=== Strict partial orders ===
An '''irreflexive''', '''strong''',<ref name=Wallis/> or '''{{visible anchor|strict partial order|Strict partial order|Irreflexive partial order}}''' is a homogeneous relation < on a set <math>P</math> that is [[Irreflexive relation|irreflexive]], [[Asymmetric relation|asymmetric]]
# [[Irreflexive relation|Irreflexivity]]: <math>\neg\left( a < a \right)</math>, i.e. no element is related to itself (also called anti-reflexive).
# [[
# [[
A strict partial order is also known as an asymmetric [[strict preorder]].
=== Correspondence of strict and non-strict partial order relations ===
[[File:PartialOrders redundencies svg.svg|thumb|upright=1.25|'''Fig. 2''' [[Commutative diagram]] about the connections between strict/non-strict relations and their duals, via the operations of reflexive closure (''cls''), irreflexive kernel (''ker''), and converse relation (''cnv''). Each relation is depicted by its [[logical matrix]] for the poset whose [[Hasse diagram]] is depicted in the center. For example <math>3 \not\leq 4</math> so row 3, column 4 of the bottom left matrix is empty.]]
Strict and non-strict partial orders on a set <math>P</math> are closely related. A non-strict partial order <math>\leq</math> may be converted to a strict partial order by removing all relationships of the form <math>a \leq a;</math> that is, the strict partial order is the set <math>< \; := \ \leq\ \setminus \ \Delta_P</math> where <math>\Delta_P := \{ (p, p) : p \in P \}</math> is the [[identity relation]] on <math>P \times P</math> and <math>\;\setminus\;</math> denotes [[set subtraction]]. Conversely, a strict partial order < on <math>P</math> may be converted to a non-strict partial order by adjoining all relationships of that form; that is, <math>\leq\; := \;\Delta_P\; \cup \;<\;</math> is a non-strict partial order. Thus, if <math>\leq</math> is a non-strict partial order, then the corresponding strict partial order < is the [[irreflexive kernel]] given by
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{{Main|Duality (order theory)}}
The ''dual'' (or ''opposite'') <math>R^{\text{op}}</math> of a partial order relation <math>R</math> is defined by letting <math>R^{\text{op}}</math> be the [[converse relation]] of <math>R</math>, i.e. <math>x R^{\text{op}} y</math> if and only if <math>y R x</math>. The dual of a non-strict partial order is a non-strict partial order,{{sfnp|Davey|Priestley|2002|pp=[https://books.google.com/books?id=vVVTxeuiyvQC&pg=PA14
== Notation ==
Given a set <math>P</math> and a partial order relation, typically the non-strict partial order <math>\leq</math>, we may uniquely extend our notation to define four partial order relations <math>\leq,</math> <math><,</math> <math>\geq,
The term ''ordered set'' is sometimes used as a shorthand for ''partially ordered set'', as long as it is clear from the context that no other kind of order is meant. In particular, [[Total order|totally ordered sets]] can also be referred to as "ordered sets", especially in areas where these structures are more common than posets. Some authors use different symbols than <math>\leq</math> such as <math>\sqsubseteq</math><ref>{{cite web |last1=Rounds |first1=William C. |title=Lectures slides |url=http://www.eecs.umich.edu/courses/eecs203-1/203-Mar7.pdf |website=EECS 203: DISCRETE MATHEMATICS |access-date=23 July 2021 |date=7 March 2002}}</ref> or <math>\preceq</math><ref>{{cite book |last1=Kwong |first1=Harris |title=A Spiral Workbook for Discrete Mathematics |date=25 April 2018 |url=https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/07%3A_Relations/7.04%3A_Partial_and_Total_Ordering |access-date=23 July 2021 |language=en |chapter=7.4: Partial and Total Ordering}}</ref> to distinguish partial orders from total orders.
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== Alternative definitions ==
Another way of defining a partial order, found in [[computer science]], is via a notion of [[Comparability|comparison]]. Specifically, given <math>\leq, <, \geq, \text{ and } ></math> as defined previously, it can be observed that two elements ''x'' and ''y'' may stand in any of four [[mutually exclusive]] relationships to each other: either {{nowrap|''x''
Wallis defines a more general notion of a ''partial order relation'' as any [[homogeneous relation]] that is [[Transitive relation|transitive]] and [[Antisymmetric relation|antisymmetric]]. This includes both reflexive and irreflexive partial orders as subtypes.<ref name=Wallis>{{cite book |last1=Wallis |first1=W. D. |title=A Beginner's Guide to Discrete Mathematics |date=14 March 2013 |publisher=Springer Science & Business Media |isbn=978-1-4757-3826-1 |page=100 |url=https://books.google.com/books?id=ONgRBwAAQBAJ&dq=%22partial%20order%20relation%22&pg=PA100 |language=en}}</ref>
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== Examples ==
[[File:Division relation 4.
Standard examples of posets arising in mathematics include:
* The [[real number]]s, or in general any totally ordered set, ordered by the standard ''less-than-or-equal'' relation ≤, is a partial order.
* On the real numbers <math>\mathbb{R}</math>, the usual [[less than]] relation < is a strict partial order. The same is also true of the usual [[greater than]] relation > on <math>\R</math>.
* By definition, every [[strict weak order]] is a strict partial order.
* The set of [[subset]]s of a given set (its [[power set]]) ordered by [[subset|inclusion]] (see Fig. 1). Similarly, the set of [[sequence]]s ordered by [[subsequence]], and the set of [[string (computer science)|string]]s ordered by [[substring]].
* The set of [[natural number]]s equipped with the relation of [[divisor|divisibility]]. (see Fig. 3 and Fig. 6)
* The vertex set of a [[directed acyclic graph]] ordered by [[reachability]].
* The set of [[Linear subspace|subspaces]] of a [[vector space]] ordered by inclusion.
* For a partially ordered set ''P'', the [[sequence space]] containing all [[sequence]]s of elements from ''P'', where sequence ''a'' precedes sequence ''b'' if every item in ''a'' precedes the corresponding item in ''b''. Formally, <math>\left(a_n\right)_{n \in \N} \leq \left(b_n\right)_{n \in \N}</math> if and only if <math>a_n \leq b_n</math> for all <math>n \in \N</math>; that is, a [[componentwise order]].
* For a set ''X'' and a partially ordered set ''P'', the [[function space]] containing all functions from ''X'' to ''P'', where {{nowrap|''f'' ≤ ''g''}} if and only if {{nowrap|''f''(''x'') ≤ ''g''(''x'')}} for all <math>x \in X.</math>
* A [[Fence (mathematics)|fence]], a partially ordered set defined by an alternating sequence of order relations {{nowrap|''a'' < ''b'' > ''c'' < ''d'' ...}}
* The set of events in [[special relativity]] and, in most cases,{{efn|See
One familiar example of a partially ordered set is a collection of people ordered by [[genealogy|genealogical]] descendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs of people are incomparable, with neither being a descendant of the other.
=== Orders on the Cartesian product of partially ordered sets ===
{{multiple image
| align = right
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| caption2 = '''Fig. 4b''' Product order on <math>\N \times \N</math>
| image3 = Strict product order on pairs of natural numbers.svg|
| caption3 = '''Fig. 4c''' Reflexive closure of strict direct product order on <math>\N \times \N.</math> Elements [[#Formal definition|covered]] by {{nowrap|(3, 3)}} and covering {{nowrap|(3, 3)}} are highlighted in green and red, respectively.
}}
In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on the [[Cartesian product]] of two partially ordered sets are (see Fig. 4):
* the [[lexicographical order]]: {{nowrap|(''a'', ''b'') ≤ (''c'', ''d'')}} if {{nowrap|''a'' < ''c''}} or ({{nowrap|1=''a'' = ''c''}} and {{nowrap|''b'' ≤ ''d''}});
* the [[product order]]: (''a'', ''b'') ≤ (''c'', ''d'') if ''a'' ≤ ''c'' and ''b'' ≤ ''d'';
* the [[reflexive closure]] of the [[Direct product#Direct product of binary relations|direct product]] of the corresponding strict orders: {{nowrap|(''a'', ''b'') ≤ (''c'', ''d'')}} if ({{nowrap|''a'' < ''c''}} and {{nowrap|''b'' < ''d''}}) or ({{nowrap|1=''a'' = ''c''}} and {{nowrap|1=''b'' = ''d''}}).
All three can similarly be defined for the Cartesian product of more than two sets.
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=== Sums of partially ordered sets ===
{{anchor|sum}}
Another way to combine two (disjoint) posets is the '''ordinal sum'''<ref>
{{citation | last1 = Neggers | first1 = J.
| last2 = Kim | first2 = Hee Sik
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| publisher = World Scientific
| title = Basic Posets
| year = 1998
* ''a'', ''b'' ∈ ''X'' with ''a'' ≤<sub>''X''</sub> ''b'', or
* ''a'', ''b'' ∈ ''Y'' with ''a'' ≤<sub>''Y''</sub> ''b'', or
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== Derived notions ==
The examples use the poset <math>(\mathcal{P}(\{x, y, z\}),\subseteq)</math> consisting of the [[Power set|set of all subsets]] of a three-element set <math>\{x, y, z\},</math> ordered by set inclusion (see Fig. 1).
* ''a'' is ''related to'' ''b'' when ''a'' ≤ ''b''. This does not imply that ''b'' is also related to ''a'', because the relation need not be [[Symmetric relation|symmetric]]. For example, <math>\{x\}</math> is related to <math>\{x, y\},</math> but not the reverse.
* ''a'' and ''b'' are ''[[Comparability|comparable]]'' if {{nowrap|''a'' ≤ ''b''}} or {{nowrap|''b'' ≤ ''a''}}. Otherwise they are ''incomparable''. For example, <math>\{x\}</math> and <math>\{x, y, z\}</math> are comparable, while <math>\{x\}</math> and <math>\{y\}</math> are not.
* A ''[[Totally ordered set|total order]]'' or ''linear order'' is a partial order under which every pair of elements is comparable, i.e. [[Trichotomy (mathematics)|trichotomy]] holds. For example, the natural numbers with their standard order.
* A ''[[Chain (order theory)|chain]]'' is a subset of a poset that is a totally ordered set. For example, <math>\{ \{\,\}, \{x\}, \{x, y, z\} \}</math> is a chain.
* An ''[[antichain]]'' is a subset of a poset in which no two distinct elements are comparable. For example, the set of [[Singleton (mathematics)|singleton]]s <math>\{\{x\}, \{y\}, \{z\}\}.</math>
* An element ''a'' is said to be ''strictly less than'' an element ''b'', if ''a'' ≤ ''b'' and <math>a \neq b.</math> For example, <math>\{x\}</math> is strictly less than <math>\{x, y\}.</math>
* An element ''a'' is said to be ''[[Covering relation|covered]]'' by another element ''b'', written ''a'' ⋖ ''b'' (or ''a'' <: ''b''), if ''a'' is strictly less than ''b'' and no third element ''c'' fits between them; formally: if both ''a'' ≤ ''b'' and <math>a \neq b</math> are true, and ''a'' ≤ ''c'' ≤ ''b'' is false for each ''c'' with <math>a \neq c \neq b.</math> Using the strict order <, the relation ''a'' ⋖ ''b'' can be equivalently rephrased as "{{nowrap|''a'' < ''b''}} but not {{nowrap|''a'' < ''c'' < ''b''}} for any ''c''". For example, <math>\{x\}</math> is covered by <math>\{x, z\},</math> but is not covered by <math>\{x, y, z\}.</math>
=== Extrema ===
[[File:Hasse diagram of powerset of 3 no greatest or least.svg|thumb|upright=1.5|'''Fig. 5''' The figure above with the greatest and least elements removed. In this reduced poset, the top row of elements are all {{em|maximal}} elements, and the bottom row are all {{em|minimal}} elements, but there is no {{em|greatest}} and no {{em|least}} element.]]
There are several notions of "greatest" and "least" element in a poset <math>P,</math> notably:
* [[Greatest element]] and least element: An element <math>g \in P</math> is a {{em|greatest element}} if <math>a \leq g</math> for every element <math>a \in P.</math> An element <math>m \in P</math> is a {{em|least element}} if <math>m \leq a</math> for every element <math>a \in P.</math> A poset can only have one greatest or least element. In our running example, the set <math>\{x, y, z\}</math> is the greatest element, and <math>\{\,\}</math> is the least.
* [[Maximal element]]s and minimal elements: An element <math>g \in P</math> is a maximal element if there is no element <math>a \in P</math> such that <math>a > g.</math> Similarly, an element <math>m \in P</math> is a minimal element if there is no element <math>a \in P</math> such that <math>a < m.</math> If a poset has a greatest element, it must be the unique maximal element, but otherwise there can be more than one maximal element, and similarly for least elements and minimal elements. In our running example, <math>\{x, y, z\}</math> and <math>\{\,\}</math> are the maximal and minimal elements. Removing these, there are 3 maximal elements and 3 minimal elements (see Fig. 5).
* [[Upper and lower bounds]]: For a subset ''A'' of ''P'', an element ''x'' in ''P'' is an upper bound of ''A'' if ''a'' ≤ ''x'', for each element ''a'' in ''A''. In particular, ''x'' need not be in ''A'' to be an upper bound of ''A''. Similarly, an element ''x'' in ''P'' is a lower bound of ''A'' if ''a'' ≥ ''x'', for each element ''a'' in ''A''. A greatest element of ''P'' is an upper bound of ''P'' itself, and a least element is a lower bound of ''P''. In our example, the set <math>\{x, y\}</math> is an {{em|upper bound}} for the collection of elements <math>\{\{x\}, \{y\}\}.</math>
[[File:Infinite lattice of divisors.svg|thumb|upright|'''Fig. 6''' [[Nonnegative integer]]s, ordered by divisibility]]
As another example, consider the positive [[integer]]s, ordered by divisibility: 1 is a least element, as it [[divisor|divides]] all other elements; on the other hand this poset does not have a greatest element. This partially ordered set does not even have any maximal elements, since any ''g'' divides for instance 2''g'', which is distinct from it, so ''g'' is not maximal. If the number 1 is excluded, while keeping divisibility as ordering on the elements greater than 1, then the resulting poset does not have a least element, but any [[prime number]] is a minimal element for it. In this poset, 60 is an upper bound (though not a least upper bound) of the subset <math>\{2, 3, 5, 10\},</math> which does not have any lower bound (since 1 is not in the poset); on the other hand 2 is a lower bound of the subset of powers of 2, which does not have any upper bound. If the number 0 is
== Mappings between partially ordered sets ==
{{multiple image
| align = right
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| total_width = 580
| image1 = Monotonic but nonhomomorphic map between lattices.gif
| caption1 = '''Fig. 7a''' Order-preserving, but not order-reflecting (since {{nowrap|''f''(''u'') ≼ ''f''(''v'')}}, but not u {{small|<math>\leq</math>}} v) map.
| image2 = Birkhoff120.svg
| caption2 = '''Fig. 7b''' Order isomorphism between the divisors of 120 (partially ordered by divisibility) and the divisor-closed subsets of {{nowrap|{{mset|2, 3, 4, 5, 8}}}} (partially ordered by set inclusion)
}}
Given two partially ordered sets {{math|1=(''S'', ≤)}} and {{math|(''T'', ≼)}}, a function <math>f : S \to T</math> is called '''[[order-preserving]]''', or '''[[Monotonic function#In order theory|monotone]]''', or '''isotone''', if for all <math>x, y \in S,</math> <math>x \leq y</math> implies {{math|1=''f''(''x'') ≼ ''f''(''y'')}}.
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A function <math>f : S \to T</math> is called '''order-reflecting''' if for all <math>x, y \in S,</math> {{math|1=''f''(''x'') ≼ ''f''(''y'')}} implies <math>x \leq y.</math>
If {{mvar|f}} is both order-preserving and order-reflecting, then it is called an '''[[order-embedding]]''' of {{math|1=(''S'', ≤)}} into {{math|1=(''T'', ≼)}}.
In the latter case, {{mvar|f}} is necessarily [[injective]], since <math>f(x) = f(y)</math> implies <math>x \leq y \text{ and } y \leq x</math> and in turn <math>x = y</math> according to the antisymmetry of <math>\leq.</math> If an order-embedding between two posets ''S'' and ''T'' exists, one says that ''S'' can be '''embedded''' into ''T''. If an order-embedding <math>f : S \to T</math> is [[bijective]], it is called an '''[[order isomorphism]]''', and the partial orders {{math|1=(''S'', ≤)}} and {{math|1=(''T'', ≼)}} are said to be '''isomorphic'''. Isomorphic orders have structurally similar [[Hasse diagram]]s (see Fig. 7a). It can be shown that if order-preserving maps <math>f : S \to T</math> and <math>g : T \to U</math> exist such that <math>g \circ f</math> and <math>f \circ g</math> yields the [[identity function]] on ''S'' and ''T'', respectively, then ''S'' and ''T'' are order-isomorphic.{{sfnp|Davey|Priestley|2002|pp=23–24}}
For example, a mapping <math>f : \N \to \mathbb{P}(\N)</math> from the set of natural numbers (ordered by divisibility) to the [[power set]] of natural numbers (ordered by set inclusion) can be defined by taking each number to the set of its [[prime divisor]]s. It is order-preserving: if {{mvar|x}} divides {{mvar|y}}, then each prime divisor of {{mvar|x}} is also a prime divisor of {{mvar|y}}. However, it is neither injective (since it maps both 12 and 6 to <math>\{2, 3\}</math>) nor order-reflecting (since 12 does not divide 6). Taking instead each number to the set of its [[prime power]] divisors defines a map <math>g : \N \to \mathbb{P}(\N)</math> that is order-preserving, order-reflecting, and hence an order-embedding. It is not an order-isomorphism (since it, for instance, does not map any number to the set <math>\{4\}</math>), but it can be made one by [[Injective function#Injections may be made invertible|restricting its codomain]] to <math>g(\N).</math> Fig. 7b shows a subset of <math>\N</math> and its isomorphic image under {{mvar|g}}. The construction of such an order-isomorphism into a power set can be generalized to a wide class of partial orders, called [[distributive lattice]]s
== Number of partial orders ==
Sequence [{{fullurl:OEIS:A001035}} A001035] in [[On-Line Encyclopedia of Integer Sequences|OEIS]] gives the number of partial orders on a set of ''n'' labeled elements:
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If the count is made only [[up to]] isomorphism, the sequence 1, 1, 2, 5, 16, 63, 318, ... {{OEIS|A000112}} is obtained.
== Subposets ==
A poset <math>P^*=(X^*, \leq^*)</math> is called a '''subposet''' of another poset <math>P=(X, \leq)</math> provided that <math>X^*</math> is a [[subset]] of <math>X</math> and <math>\leq^*</math> is a subset of <math>\leq</math>. The latter condition is equivalent to the requirement that for any <math>x</math> and <math>y</math> in <math>X^*</math> (and thus also in <math>X</math>), if <math>x \leq^* y</math> then <math>x \leq y</math>.
If <math>P^*</math> is a subposet of <math>P</math> and furthermore, for all <math>x</math> and <math>y</math> in <math>X^*</math>, whenever <math>x \leq y</math> we also have <math>x \leq^* y</math>, then we call <math>P^*</math> the subposet of <math>P</math> '''induced''' by <math>X^*</math>, and write <math>P^* = P[X^*]</math>.
== Linear extension ==
A partial order <math>\leq^*</math> on a set <math>X</math> is called an '''extension''' of another partial order <math>\leq</math> on <math>X</math> provided that for all elements <math>x, y \in X,</math> whenever <math>x \leq y,</math> it is also the case that <math>x \leq^* y.</math> A [[linear extension]] is an extension that is also a linear (that is, total) order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order. Every partial order can be extended to a total order ([[order-extension principle]]).<ref>{{cite book |last=Jech |first=Thomas |author-link=Thomas Jech |title=The Axiom of Choice |year=2008 |orig-year=1973 |publisher=[[Dover Publications]] |isbn=978-0-486-46624-8}}</ref>▼
▲A partial order <math>\leq^*</math> on a set <math>X</math> is an '''extension''' of another partial order <math>\leq</math> on <math>X</math> provided that for all elements <math>x, y \in X,</math> whenever <math>x \leq y,</math> it is also the case that <math>x \leq^* y.</math> A [[linear extension]] is an extension that is also a linear (that is, total) order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order. Every partial order can be extended to a total order ([[order-extension principle]]).<ref>{{cite book |last=Jech |first=Thomas |author-link=Thomas Jech |title=The Axiom of Choice |year=2008 |orig-year=1973 |publisher=[[Dover Publications]] |isbn=978-0-486-46624-8}}</ref>
In [[computer science]], algorithms for finding linear extensions of partial orders (represented as the [[reachability]] orders of [[directed acyclic graph]]s) are called [[topological sorting]].
== In category theory ==
{{main|Posetal category}}
Every poset (and every [[Preorder|preordered set]]) may be considered as a [[Category (mathematics)|category]] where, for objects <math>x</math> and <math>y,</math> there is at most one [[morphism]] from <math>x</math> to <math>y.</math> More explicitly, let {{nowrap|1=hom(''x'', ''y'') = {{mset|(''x'', ''y'')}}}} if {{nowrap|''x'' ≤ ''y''}} (and otherwise the [[empty set]]) and <math>(y, z) \circ (x, y) = (x, z).</math> Such categories are sometimes called ''[[Posetal category|posetal]]''.
Posets are [[Equivalence of categories|equivalent]] to one another if and only if they are [[Isomorphism of categories|isomorphic]]. In a poset, the smallest element, if it exists, is an [[initial object]], and the largest element, if it exists, is a [[terminal object]]. Also, every preordered set is equivalent to a poset. Finally, every subcategory of a poset is [[isomorphism-closed]].
▲Every poset (and every [[Preorder|preordered set]]) may be considered as a [[Category (mathematics)|category]] where, for objects <math>x</math> and <math>y,</math> there is at most one [[morphism]] from <math>x</math> to <math>y.</math> More explicitly, let hom(''x'', ''y'') = {(''x'', ''y'')} if ''x'' ≤ ''y'' (and otherwise the [[empty set]]) and <math>(y, z) \circ (x, y) = (x, z).</math> Such categories are sometimes called ''[[Posetal category|posetal]]''. In differential topology, homology theory (HT) is used for classifying equivalent smooth manifolds M, related to the geometrical shapes of M.
== Partial orders in topological spaces ==
{{Main|Partially ordered space}}
If <math>P</math> is a partially ordered set that has also been given the structure of a [[topological space]], then it is customary to assume that <math>\{ (a,b) : a \le b \}</math> is a [[Closed (mathematics)|closed]] subset of the topological [[product space]] <math>P \times P.</math> Under this assumption partial order relations are well behaved at [[Limit of a sequence|limits]] in the sense that if <math>\lim_{i \to \infty} a_i = a,</math> and <math>\lim_{i \to \infty} b_i = b,</math> and for all <math>i,</math> <math>a_i \leq b_i,</math> then <math>a \leq b.</math><ref name="ward-1954">{{Cite journal|first=L. E. Jr|last=Ward|title=Partially Ordered Topological Spaces|journal=Proceedings of the American Mathematical Society|volume=5 |year=1954|pages= 144–161|issue= 1|doi=10.1090/S0002-9939-1954-0063016-5|hdl=10338.dmlcz/101379|doi-access=free}}</ref>
== Intervals ==
{{See also|Interval (mathematics)}}
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An '''interval''' in a poset ''P'' is a subset that can be defined with interval notation:
* For ''a'' ≤ ''b'', the ''closed interval'' {{closed-closed|''a'', ''b''}} is the set of elements ''x'' satisfying {{nowrap|''a'' ≤ ''x'' ≤ ''b''}} (that is, {{nowrap|''a'' ≤ ''x''}} and {{nowrap|''x'' ≤ ''b''}}). It contains at least the elements ''a'' and ''b''.
* Using the corresponding strict relation "<", the ''open interval'' {{open-open|''a'', ''b''}} is the set of elements ''x'' satisfying {{nowrap|''a'' < ''x'' < ''b''}} (i.e. {{nowrap|''a'' < ''x''}} and {{nowrap|''x'' < ''b''}}). An open interval may be empty even if {{nowrap|''a'' < ''b''}}. For example, the open interval {{open-open|0, 1}} on the integers is empty since there is no integer {{mvar|x}} such that {{math|0 < {{var|x}} < 1}}.
* The ''half-open intervals'' {{closed-open|''a'', ''b''}} and {{open-closed|''a'', ''b''}} are defined similarly.
Whenever {{nowrap|''a'' ≤ ''b''}} does not hold, all these intervals are empty. Every interval is a convex set, but the converse does not hold; for example, in the poset of divisors of 120, ordered by divisibility (see Fig. 7b), the set {{mset|1, 2, 4, 5, 8}} is convex, but not an interval.
An interval {{mvar|I}} is bounded if there exist elements <math>a, b \in P</math> such that {{math|{{var|I}} ⊆ {{closed-closed|''a'', ''b''}}}}. Every interval that can be represented in interval notation is obviously bounded, but the converse is not true. For example, let {{math|1=''P'' = {{open-open|0, 1}} ∪ {{open-open|1, 2}} ∪ {{open-open|2, 3}}}} as a subposet of the real numbers. The subset {{open-open|1, 2}} is a bounded interval, but it has no [[infimum]] or [[supremum]] in
A poset is called [[Locally finite poset|locally finite]] if every bounded interval is finite. For example, the integers are locally finite under their natural ordering. The lexicographical order on the cartesian product <math>\N \times \N</math> is not locally finite, since {{math|(1, 2) ≤ (1, 3) ≤ (1, 4) ≤ (1, 5) ≤ ... ≤ (2, 1)}}.
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== See also ==
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* [[Antimatroid]], a formalization of orderings on a set that allows more general families of orderings than posets
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* {{annotated link|Semilattice}}
* {{annotated link|Semiorder}}
* [[Szpilrajn extension theorem]]
* {{annotated link|Stochastic dominance}}
* [[Strict weak ordering]] – strict partial order "<" in which the relation {{nowrap|"neither ''a'' < ''b''}} {{nowrap|nor ''b'' < ''a''"}} is transitive.
* {{annotated link|Total order}}
* {{annotated link|Zorn's lemma}}
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== Notes ==
{{notelist}}
== Citations ==
{{reflist}}
== References ==
{{refbegin}}
* {{cite book |last1=Davey |first1=B. A. |last2=Priestley |first2=H. A. |title=Introduction to Lattices and Order |edition=2nd |___location=New York |publisher=Cambridge University Press |year=2002 |isbn=978-0-521-78451-1 |title-link= Introduction to Lattices and Order}}
* {{Cite journal|first=Jayant V. |last=Deshpande|title=On Continuity of a Partial Order|journal=Proceedings of the American Mathematical Society|volume=19|year=1968|pages=383–386|issue=2 |doi=10.1090/S0002-9939-1968-0236071-7 |doi-access= free}}
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* {{cite book|first=S.|last=Eilenberg|author-link=N. Steenrod|title=Foundations of Algebraic Topology|publisher=Princeton University Press|year=2016}}
* {{cite journal|first=G.|last=Kalmbach|title=Extension of Homology Theory to Partially Ordered Sets|journal=J. Reine Angew. Math.|volume=280|year=1976|pages=134–156}}
{{refend}}
==External links==▼
▲== External links ==
{{Commons category inline|Hasse diagrams}}; each of which shows an example for a partial order
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* {{OEIS el|1=A001035|2= Number of posets with ''n'' labeled elements|formalname=Number of partially ordered sets ("posets") with n labeled elements (or labeled acyclic transitive digraphs)}}
* {{OEIS el|1=A000112|2=Number of partially ordered sets ("posets") with n unlabeled elements.}}
{{refend}}
{{Order theory}}
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[[Category:Order theory]]
[[Category:Binary relations]]
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