View source for Partially ordered set

{{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|Hasse diagram]] of the [[Power set|set of all subsets]] of a three-element set <math>\{x, y, z\},</math> ordered by 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|mathematics]], especially [[Order theory|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|total order]]s, in which every pair is comparable.

Formally, a partial order is a homogeneous binary relation that is [[Reflexive relation|reflexive]], [[Antisymmetric relation|antisymmetric]], and [[Transitive relation|transitive]]. A '''partially ordered set''' ('''poset''' for short) is an ordered pair <math>P=(X,\leq)</math> of a set <math>X</math> (called the ''ground set'' of <math>P</math>) and a partial order <math>\leq</math> on <math>X</math>. When the meaning is clear from context and there is no ambiguity about the partial order, the set <math>X</math> itself is sometimes called a poset.

== Partial order relations ==
{{anchor|Partial order}}The term ''partial order'' usually refers to the reflexive partial order relations, referred to in this article as ''non-strict'' partial orders. However some authors use the term for the other common type of partial order relations, the irreflexive partial order relations, also called strict partial orders. Strict and non-strict partial orders can be put into a one-to-one correspondence, so for every strict partial order there is a unique corresponding non-strict partial order, and vice versa.

=== Partial orders ===

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|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.
# [[Transitive relation|Transitivity]]: if <math>a \leq b </math> and <math>b \leq c</math> then <math>a \leq c</math>.

A non-strict partial order is also known as an antisymmetric [[Preorder|preorder]].

=== 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, [[Asymmetric relation|asymmetric]], and [[Transitive relation|transitive]]; that is, it satisfies the following conditions for all <math>a, b, c \in P:</math>
# Irreflexivity: not <math>a < a</math>, i.e. no element is related to itself (also called anti-reflexive).
# [[Asymmetric relation|Asymmetry]]: if <math>a < b</math> then not <math>b < a</math>.
# [[Transitive relation|Transitivity]]: if <math>a < b</math> and <math>b < c</math> then <math>a < c</math>.

Irreflexivity and transitivity together imply asymmetry. Also, asymmetry implies irreflexivity. In other words, a transitive relation is asymmetric if and only if it is irreflexive.<ref name="Flaška 2007">{{cite journal |last1=Flaška |first1=V. |last2=Ježek |first2=J. |last3=Kepka |first3=T. |last4=Kortelainen |first4=J. |title=Transitive Closures of Binary Relations I |journal=Acta Universitatis Carolinae. Mathematica et Physica |year=2007 |volume=48 |issue=1 |pages=55–69 |publisher=School of Mathematics – Physics Charles University |location=Prague |url=http://dml.cz/dmlcz/142762 }} Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric".</ref> So the definition is the same if it omits either irreflexivity or asymmetry (but not both).

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|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|logical matrix]] for the poset whose [[Hasse diagram|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
<math display=block>a < b \text{ if } a \leq b \text{ and } a \neq b.</math>
Conversely, if < is a strict partial order, then the corresponding non-strict partial order <math>\leq</math> is the [[Reflexive closure|reflexive closure]] given by:
<math display=block>a \leq b \text{ if } a < b \text{ or } a = b.</math>

=== Dual orders ===
{{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|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 14–15]}} and the dual of a strict partial order is a strict partial order. The dual of a dual of a relation is the original relation.

== 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,</math> and <math>></math>, where <math>\leq</math> is a non-strict partial order relation on <math>P</math>, <math> < </math> is the associated strict partial order relation on <math>P</math> (the irreflexive kernel of <math>\leq</math>), <math>\geq</math> is the dual of <math>\leq</math>, and <math> > </math> is the dual of <math> < </math>. Strictly speaking, the term ''partially ordered set'' refers to a set with all of these relations defined appropriately. But practically, one need only consider a single relation, <math>(P,\leq)</math> or <math>(P,<)</math>, or, in rare instances, the strict and non-strict relations together, <math>(P,\leq,<)</math>.<ref>{{cite book |last1=Avigad |first1=Jeremy |last2=Lewis |first2=Robert Y. |last3=van Doorn |first3=Floris |title=Logic and Proof |date=29 March 2021 |edition=Release 3.18.4 |url=https://leanprover.github.io/logic_and_proof/relations.html#more-on-orderings |access-date=24 July 2021 |chapter=13.2. More on Orderings|quote=So we can think of every partial order as really being a pair, consisting of a weak partial order and an associated strict one.}}</ref> 

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.

When referring to partial orders, <math>\leq</math> should not be taken as the complement of <math> > </math>. The relation <math> > </math> is the converse of the irreflexive kernel of <math>\leq</math>, which is always a subset of the complement of <math>\leq</math>, but <math> > </math> is equal to the complement of <math>\leq</math> if, and only if, <math>\leq</math> is a total order.{{efn|A proof can be found [[:File:PartialOrders redundencies.pdf|here]].}}

== Alternative definitions ==

Another way of defining a partial order, found in [[Computer science|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'' < ''y''}}, or {{nowrap|1=''x'' = ''y''}}, or {{nowrap|''x'' > ''y''}}, or ''x'' and ''y'' are ''incomparable''. This can be represented by a function <math>\text{compare}: P \times P \to \{<,>,=,\vert \}</math> that returns one of four codes when given two elements.<ref>{{cite web |title=Finite posets |url=http://match.stanford.edu/reference/combinat/sage/combinat/posets/posets.html#sage.combinat.posets.posets.FinitePoset.compare_elements |website=Sage 9.2.beta2 Reference Manual: Combinatorics |access-date=5 January 2022|quote=compare_elements(''x'', ''y''): Compare ''x'' and ''y'' in the poset. If {{nowrap|''x'' < ''y''}}, return −1. If {{nowrap|1=''x'' = ''y''}}, return 0. If {{nowrap|''x'' > ''y''}}, return 1. If ''x'' and ''y'' are not comparable, return None.}}</ref><ref>{{cite tech report |last1=Chen |first1=Peter |last2=Ding |first2=Guoli |last3=Seiden |first3=Steve |title=On Poset Merging |page=2 |url=https://www.math.lsu.edu/~ding/poset.pdf |access-date=5 January 2022 |quote=A comparison between two elements s, t in S returns one of three distinct values, namely s≤t, s>t or s<nowiki>|</nowiki>t.}}</ref> This definition is equivalent to a ''partial order on a [[Setoid|setoid]]'', where equality is taken to be a defined [[Equivalence relation|equivalence relation]] rather than the primitive notion of set equality.<ref>{{cite conference |conference=CALCULEMUS-2003 – 11th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning|location=Roma, Italy |date=11 September 2003 |url=https://hal.science/hal-02549766/document#page=98 |publisher=Aracne |language=en|title=Making proofs in a hierarchy of mathematical structures|first1=Virgile|last1=Prevosto|first2=Mathieu|last2=Jaume|pages=89–100}}</ref>

Wallis defines a more general notion of a ''partial order relation'' as any [[Homogeneous relation|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>

A finite poset can be visualized through its [[Hasse diagram]].<ref>{{cite book |last1=Merrifield |first1=Richard E. |last2=Simmons |first2=Howard E. |title=Topological Methods in Chemistry |year=1989 |publisher=John Wiley & Sons |location=New York |isbn=0-471-83817-9 |url=https://archive.org/details/topologicalmetho00merr/page/28 |access-date=27 July 2012 |pages=[https://archive.org/details/topologicalmetho00merr/page/28 28] |quote=A partially ordered set is conveniently represented by a ''Hasse diagram''... |url-access=registration }}</ref> Specifically, taking a strict partial order relation <math>(P,<)</math>, a [[Directed acyclic graph|directed acyclic graph]] (DAG) may be constructed by taking each element of <math>P</math> to be a node and each element of <math> < </math> to be an edge. The [[Transitive reduction|transitive reduction]] of this DAG{{efn|which always exists and is unique, since <math>P</math> is assumed to be finite}} is then the Hasse diagram. Similarly this process can be reversed to construct strict partial orders from certain DAGs. In contrast, the graph associated to a non-strict partial order has self-loops at every node and therefore is not a DAG; when a non-strict order is said to be depicted by a Hasse diagram, actually the corresponding strict order is shown.

== Examples ==

[[File:Division relation 4.png|thumb|alt=Division Relationship Up to 4|'''Fig. 3''' Graph of the divisibility of numbers from 1 to 4. This set is partially, but not totally, ordered because there is a relationship from 1 to every other number, but there is no relationship from 2 to 3 or 3 to 4]]

Standard examples of posets arising in mathematics include:
* The [[Real number|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|subset]]s of a given set (its [[Power set|power set]]) ordered by [[Subset|inclusion]] (see Fig.&nbsp;1). Similarly, the set of [[Sequence|sequence]]s ordered by [[Subsequence|subsequence]], and the set of [[String (computer science)|string]]s ordered by [[Substring|substring]].
* The set of [[Natural number|natural number]]s equipped with the relation of [[Divisor|divisibility]]. (see Fig.&nbsp;3 and Fig.&nbsp;6)
* The vertex set of a [[Directed acyclic graph|directed acyclic graph]] ordered by [[Reachability|reachability]].
* The set of [[Linear subspace|subspaces]] of a [[Vector space|vector space]] ordered by inclusion.
* For a partially ordered set ''P'', the [[Sequence space|sequence space]] containing all [[Sequence|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|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 [[Physics:Special relativity|special relativity]] and, in most cases,{{efn|See ''{{slink|General relativity#Time travel}}''.}} [[Physics:General relativity|general relativity]], where for two events ''X'' and ''Y'', {{nowrap|''X'' ≤ ''Y''}} if and only if ''Y'' is in the future [[Astronomy:Light cone|light cone]] of ''X''. An event ''Y'' can be causally affected by ''X'' only if {{nowrap|''X'' ≤ ''Y''}}.

One familiar example of a partially ordered set is a collection of people ordered by [[Social: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
| dirction = horizontal
| total_width = 550
| image1 = Lexicographic order on pairs of natural numbers.svg
| caption1 = '''Fig. 4a''' Lexicographic order on <math>\N \times \N</math>
| image2 = N-Quadrat, gedreht.svg|
| 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 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.&nbsp;4):
* the [[Lexicographical order|lexicographical order]]: &nbsp; {{nowrap|(''a'', ''b'') ≤ (''c'', ''d'')}} if {{nowrap|''a'' < ''c''}} or ({{nowrap|1=''a'' = ''c''}} and {{nowrap|''b'' ≤ ''d''}});
* the [[Product order|product order]]:  &nbsp; (''a'', ''b'') ≤ (''c'', ''d'') if ''a'' ≤ ''c'' and ''b'' ≤ ''d'';
* the [[Reflexive closure|reflexive closure]] of the [[Direct product#Direct product of binary relations|direct product]] of the corresponding strict orders:  &nbsp; {{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.

Applied to [[Ordered vector space|ordered vector space]]s over the same [[Field (mathematics)|field]], the result is in each case also an ordered vector space.

See also [[Total order#Orders on the Cartesian product of totally ordered sets|orders on the Cartesian product of totally ordered sets]].

=== 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
 | contribution = 4.2 Product Order and Lexicographic Order
 | isbn = 9789810235895
 | pages = 62–63
 | publisher = World Scientific
 | title = Basic Posets
 | year = 1998
}}</ref> (or '''linear sum'''),{{sfnp|Davey|Priestley|2002|pp=[https://books.google.com/books?id=vVVTxeuiyvQC&pg=PA17 17–18]}} {{nowrap|1=''Z'' = ''X'' ⊕ ''Y''}}, defined on the union of the underlying sets ''X'' and ''Y'' by the order {{nowrap|''a'' ≤<sub>''Z''</sub> ''b''}} if and only if:
* ''a'', ''b'' ∈ ''X'' with ''a'' ≤<sub>''X''</sub> ''b'', or
* ''a'', ''b'' ∈ ''Y'' with ''a'' ≤<sub>''Y''</sub> ''b'', or
* ''a'' ∈ ''X'' and ''b'' ∈ ''Y''.

If two posets are well-ordered, then so is their ordinal sum.<ref>{{cite book|author=P. R. Halmos|title=Naive Set Theory|url=https://archive.org/details/naivesettheory0000halm_r4g0|url-access=registration|year=1974|publisher=Springer |isbn=978-1-4757-1645-0|page=[https://archive.org/details/naivesettheory0000halm_r4g0/page/82 82]}}</ref>

[[Series-parallel partial order]]s are formed from the ordinal sum operation (in this context called series composition) and another operation called parallel composition. Parallel composition is the [[Disjoint union|disjoint union]] of two partially ordered sets, with no order relation between elements of one set and elements of the other set.

== 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.&nbsp;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 ''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'' 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|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 elements 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.&nbsp;5).
* [[Upper and lower bounds]]: For a subset ''A'' of ''P'', an element ''x'' in ''P'' is an upper bound of ''A'' if ''a''&nbsp;≤&nbsp;''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''&nbsp;≥&nbsp;''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 integers, ordered by divisibility]]
As another example, consider the positive [[Integer|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|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 included, this will be the greatest element, since this is a multiple of every integer (see Fig.&nbsp;6).

== Mappings between partially ordered sets ==
{{multiple image
| align = right
| direction = horizontal
| 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'')}}.
If {{math|1=(''U'', ≲)}} is also a partially ordered set, and both <math>f : S \to T</math> and <math>g : T \to U</math> are order-preserving, their [[Function composition|composition]] <math>g \circ f : S \to U</math> is order-preserving, too.
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|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.&nbsp;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|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|power set]] of natural numbers (ordered by set inclusion) can be defined by taking each number to the set of its prime divisors. 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|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.&nbsp;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|distributive lattice]]s; see ''[[Birkhoff's representation theorem]]''.

== 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:

{{Number of relations}}

The number of strict partial orders is the same as that of partial orders.

If the count is made only [[Up to|up to]] isomorphism, the sequence 1, 1, 2, 5, 16, 63, 318, ... {{OEIS|A000112}} is obtained.

== Linear extension ==

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|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|order-extension principle]]).<ref>{{cite book |last=Jech |first=Thomas |title=The Axiom of Choice |year=2008 |orig-year=1973 |publisher=[[Social:Dover Publications|Dover Publications]] |isbn=978-0-486-46624-8}}</ref>

In [[Computer science|computer science]], algorithms for finding linear extensions of partial orders (represented as the [[Reachability|reachability]] orders of [[Directed acyclic graph|directed acyclic graph]]s) are called [[Topological sorting|topological sorting]].

== In category theory ==

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|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|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.  

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. In differential topology, homology theory (HT) is used for classifying equivalent smooth manifolds M, related to the geometrical shapes of M.  In homology theory is given an axiomatic HT approach, especially to singular homology.{{clarify|date=May 2023}} The HT members are algebraic invariants under diffeomorphisms. The axiomatic HT category is taken in G. Kalmbach from the book Eilenberg–Steenrod (see the references) in order to show that the set theoretical topological concept for the HT definition can be extended to partial ordered sets P. Important are chains and filters in P (replacing shapes of M) for defining HT classifications, available for many P applications not related to set theory.

== 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|topological space]], then it is customary to assume that <math>\{ (a,b) : a \le b \}</math> is a 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 ==

A '''convex set''' in a poset ''P'' is a subset {{mvar|I}} of ''P'' with the property that, for any ''x'' and ''y'' in {{mvar|I}} and any ''z'' in ''P'', if ''x'' ≤ ''z'' ≤ ''y'', then ''z'' is also in {{mvar|I}}. This definition generalizes the definition of [[Interval (mathematics)|interval]]s of [[Real number|real number]]s. When there is possible confusion with [[Convex set|convex set]]s of [[Geometry|geometry]], one uses '''order-convex''' instead of "convex".

A '''convex sublattice''' of a lattice ''L'' is a sublattice of ''L'' that is also a convex set of ''L''. Every nonempty convex sublattice can be uniquely represented as the intersection of a [[Filter (mathematics)|filter]] and an [[Ideal (order theory)|ideal]] of ''L''.

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.&nbsp;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&nbsp;''P'', so it cannot be written in interval notation using elements of&nbsp;''P''.

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)}}.
Using the interval notation, the property "''a'' is covered by ''b''" can be rephrased equivalently as <math>[a, b] = \{a, b\}.</math>

This concept of an interval in a partial order should not be confused with the particular class of partial orders known as the [[Interval order|interval order]]s.

== See also ==
{{div col|colwidth=27em}}
* [[Antimatroid]], a formalization of orderings on a set that allows more general families of orderings than posets
* Causal set, a poset-based approach to quantum gravity
* {{annotated link|Comparability graph}}
* {{annotated link|Complete partial order}}
* {{annotated link|Directed set}}
* {{annotated link|Graded poset}}
* {{annotated link|Incidence algebra}}
* {{annotated link|Lattice (order)|Lattice}}
* {{annotated link|Locally finite poset}}
* {{annotated link|Incidence algebra|Möbius function on posets}}
* [[Nested set collection#Formal definition|Nested set collection]]
* {{annotated link|Order polytope}}
* {{annotated link|Ordered field}}
* {{annotated link|Ordered vector space}}
* [[Poset topology]], a kind of topological space that can be defined from any poset
* [[Scott continuity]] – continuity of a function between two partial orders.
* {{annotated link|Semilattice}}
* {{annotated link|Semiorder}}
* [[Szpilrajn extension theorem]] – every partial order is contained in some total order.
* {{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}}
* [[Tree (data structure)#Using set inclusion|Tree]] – Data structure of set inclusion
* {{annotated link|Zorn's lemma}}
{{div col end}}

== 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}}
* {{cite book|first=Gunther|last=Schmidt|year=2010|title=Relational Mathematics |series=Encyclopedia of Mathematics and its Applications|volume= 132|publisher=Cambridge University Press |isbn=978-0-521-76268-7}}
* {{cite book|author=Bernd Schröder|title=Ordered Sets: An Introduction with Connections from Combinatorics to Topology|url=https://books.google.com/books?id=66oqDAAAQBAJ&q=%22Partially+ordered+set%22+OR+poset |date=11 May 2016 |publisher=Birkhäuser|isbn=978-3-319-29788-0}}
* {{cite book|first=Richard P.|last=Stanley|title=Enumerative Combinatorics 1 |series=Cambridge Studies in Advanced Mathematics|volume=49|publisher=Cambridge University Press|isbn=0-521-66351-2 |year=1997}}
* {{cite book|first=S.|last=Eilenberg|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 ==
{{Commons|Hasse diagram}}
{{refbegin}}
* {{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}}


[[Category:Order theory]]
[[Category:Binary relations]]

de:Ordnungsrelation#Halbordnung

{{Sourceattribution|Partially ordered set}}