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| {{redirect|Ordered set|totally ordered sets|total order}}
| | == Nike Cortez cricketer == |
| [[Image:Hasse diagram of powerset of 3.svg|right|thumb|250px|The [[Hasse diagram]] of the [[power set|set of all subsets]] of a three-element set {x, y, z}, ordered by inclusion.]]
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| In [[mathematics]], especially [[order theory]], a '''partially ordered set''' (or '''poset''') formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a [[Set (mathematics)|set]]. A poset consists of a set together with a [[Relation (mathematics)|binary relation]] that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. Such a relation is called a ''partial order'' to reflect the fact that not every pair of elements need be related: for some pairs, it may be that neither element precedes the other in the poset.
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| Thus, partial orders generalize the more familiar [[total order]]s, in which every pair is related. A finite poset can be visualized through its [[Hasse diagram]], which depicts the ordering relation.<ref>{{cite book |last1=Merrifield |first1=Richard E. |last2=Simmons |first2=Howard E. |authorlink2=Howard Ensign Simmons, Jr. |title=Topological Methods in Chemistry |year=1989 |publisher=John Wiley & Sons |location=New York |isbn=0-471-83817-9 |url=http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471838179.html |accessdate=27 July 2012 |pages=28 |quote=A partially ordered set is conveniently represented by a ''Hasse diagram''...}}</ref>
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| A familiar real-life 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 bear no such relationship.
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| == Formal definition == | | == Buy Timberland Australia please == |
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| A (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|url=http://books.google.com/books?id=6i-F3ZNcub4C&pg=PA127|author=Simovici, Dan A. & Djeraba, Chabane}}</ref> is a [[binary relation]] "≤" over a [[Set (mathematics)|set]] ''P'' which is [[reflexive relation|reflexive]], [[antisymmetric relation|antisymmetric]], and [[transitive relation|transitive]], i.e., which satisfies for all ''a'', ''b'', and ''c'' in ''P'':
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| *''a ≤ a'' (reflexivity);
| | <li>[http://verdamilio.net/tonio/spip.php?article1536/ http://verdamilio.net/tonio/spip.php?article1536/]</li> |
| *if ''a ≤ b'' and ''b ≤ a'' then ''a'' = ''b'' (antisymmetry);
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| *if ''a ≤ b'' and ''b ≤ c'' then ''a ≤ c'' (transitivity).
| | <li>[http://www.57162.com/forum.php?mod=viewthread&tid=260336 http://www.57162.com/forum.php?mod=viewthread&tid=260336]</li> |
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| In other words, a partial order is an antisymmetric [[preorder]].
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| A set with a partial order is called a '''partially ordered set''' (also called a '''poset'''). The term ''ordered set'' is sometimes also used for posets, as long as it is clear from the context that no other kinds of orders are 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.
| | <li>[http://www.jamiatou.com/spip.php?article21 http://www.jamiatou.com/spip.php?article21]</li> |
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| For ''a, b'', elements of a partially ordered set ''P'', if ''a ≤ b'' or ''b ≤ a'', then ''a'' and ''b'' are '''[[Comparability|comparable]]'''. Otherwise they are '''incomparable'''. In the figure on top-right, e.g. {x} and {x,y,z} are comparable, while {x} and {y} are not. A partial order under which every pair of elements is comparable is called a '''[[totally ordered set|total order]]''' or '''linear order'''; a totally ordered set is also called a '''chain''' (e.g., the natural numbers with their standard order). A subset of a poset in which no two distinct elements are comparable is called an '''[[antichain]]''' (e.g. the set of [[singleton (mathematics)|singleton]]s {{x}, {y}, {z}} in the top-right figure). An element ''a'' is said to be '''[[Covering relation|covered]]''' by another element ''b'', written ''a''<:''b'', if ''a'' is strictly less than ''b'' and no third element ''c'' fits between them; formally: if both ''a''≤''b'' and ''a''≠''b'' are true, and ''a''≤''c''≤''b'' is false for each ''c'' with ''a''≠''c''≠''b''. A more concise definition will be given [[#Strict and non-strict partial orders|below]] using the strict order corresponding to "≤". For example, {x} is covered by {x,z} in the top-right figure, but not by {x,y,z}.
| | <li>[http://gift.xueersi.org/home.php?mod=space&uid=10293&do=blog&quickforward=1&id=5405207 http://gift.xueersi.org/home.php?mod=space&uid=10293&do=blog&quickforward=1&id=5405207]</li> |
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| == Examples ==
| | </ul> |
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| Standard examples of posets arising in mathematics include:
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| * The [[real number]]s ordered by the standard ''less-than-or-equal'' relation ≤ (a totally ordered set as well).
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| * The set of [[subset]]s of a given set (its [[power set]]) ordered by [[subset|inclusion]] (see the figure on top-right). Similarly, the set of [[sequence]]s ordered by [[subsequence]], and the set of [[string (computer science)|string]]s ordered by [[substring]].
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| * The set of [[natural number]]s equipped with the relation of [[divisor#Divisibility of numbers|divisibility]].
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| * The vertex set of a [[directed acyclic graph]] ordered by [[reachability]].
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| * The set of [[Linear subspace|subspaces]] of a [[vector space]] ordered by inclusion.
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| * 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, <big>(''a''<sub>''n''</sub>)<sub>''n''∈ℕ</sub> ≤ (''b''<sub>''n''</sub>)<sub>n∈ℕ</sub></big> if and only if <big>''a''<sub>n</sub> ≤ ''b''<sub>n</sub></big> for all ''n'' in ℕ.
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| * For a set ''X'' and a partially ordered set ''P'', the [[function space]] containing all functions from ''X'' to ''P'', where ''f'' ≤ ''g'' if and only if ''f''(''x'') ≤ ''g''(''x'') for all ''x'' in ''X''.
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| * A [[Fence (mathematics)|fence]], a partially ordered set defined by an alternating sequence of order relations ''a'' < ''b'' > ''c'' < ''d'' ...
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| == Extrema ==
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| {| style="float:right"
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| | [[File:Infinite lattice of divisors.svg|thumb|x150px|Nonnegative integers, ordered by divisibility]]
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| |}
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| {| style="float:right"
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| | [[File:Hasse diagram of powerset of 3 no greatest or least.svg|thumb|x150px|The figure above with the greatest and least elements removed. In this reduced poset, the top row of elements are all ''maximal'' elements, and the bottom row are all ''minimal'' elements, but there is no ''greatest'' and no ''least'' element. The set {x, y} is an ''upper bound'' for the collection of elements {{x}, {y}}.]]
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| |}
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| There are several notions of "greatest" and "least" element in a poset ''P'', notably:
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| * [[Greatest element]] and least element: An element ''g'' in ''P'' is a greatest element if for every element ''a'' in ''P'', ''a'' ≤ ''g''. An element ''m'' in ''P'' is a least element if for every element ''a'' in ''P'', ''a'' ≥ ''m''. A poset can only have one greatest or least element.
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| * [[Maximal element]]s and minimal elements: An element ''g'' in P is a maximal element if there is no element ''a'' in ''P'' such that ''a'' > ''g''. Similarly, an element ''m'' in ''P'' is a minimal element if there is no element ''a'' in P such that ''a'' < ''m''. 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.
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| * [[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''.
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| For example, consider the [[positive integer]]s, ordered by divisibility: 1 is a least element, as it divides all other elements; on the other hand this poset does not have a greatest element (although if one would include 0 in the poset, which is a multiple of any integer, that would be a greatest element; see figure). 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 {2,3,5,10}, which subset 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 subset does not have any upper bound.
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| ==Orders on the Cartesian product of partially ordered sets==
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| {| style="float:right"
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| |[[File:Strict product order on pairs of natural numbers.svg|thumb|x150px|Reflexive closure of strict direct product order on ℕ×ℕ. Elements covered by (3,3) and covering (3,3) are highlighted in green and red, respectively.]]
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| {| style="float:right"
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| |[[File:N-Quadrat, gedreht.svg|thumb|x150px|Product order on ℕ×ℕ]]
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| |}
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| {| style="float:right"
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| |[[File:Lexicographic order on pairs of natural numbers.svg|thumb|x150px|Lexicographic order on ℕ×ℕ]]
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| |}
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| 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 figures):
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| *the [[lexicographical order]]: (''a'',''b'') ≤ (''c'',''d'') if ''a'' < ''c'' or (''a'' = ''c'' and ''b'' ≤ ''d'');
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| *the [[product order]]: (''a'',''b'') ≤ (''c'',''d'') if ''a'' ≤ ''c'' and ''b'' ≤ ''d'';
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| *the [[reflexive closure]] of the [[Direct product#Direct product of binary relations|direct product]] of the corresponding strict orders: (''a'',''b'') ≤ (''c'',''d'') if (''a'' < ''c'' and ''b'' < ''d'') or (''a'' = ''c'' and ''b'' = ''d'').
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| All three can similarly be defined for the Cartesian product of more than two sets.
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| Applied to [[ordered vector space]]s over the same [[Field (mathematics)|field]], the result is in each case also an ordered vector space.
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| See also [[Total order#Orders on the Cartesian product of totally ordered sets|orders on the Cartesian product of totally ordered sets]].
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| == Strict and non-strict partial orders ==
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| In some contexts, the partial order defined above is called a '''non-strict''' (or '''reflexive''', or '''weak''') '''partial order'''. In these contexts a '''strict''' (or '''irreflexive''') '''partial order''' "<" is a binary relation that is [[irreflexive relation|irreflexive]], [[transitive relation|transitive]] and [[asymmetric relation|asymmetric]], i.e. which satisfies for all ''a'', ''b'', and ''c'' in ''P'':
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| *not ''a < a'' (irreflexivity),
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| *if ''a < b'' and ''b < c'' then ''a < c'' (transitivity), and
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| *if ''a < b'' then not ''b < a'' (asymmetry; implied by irreflexivity and transitivity<ref>{{cite book|last1=Flaška|first1=V.|last2=Ježek|first2=J.|last3=Kepka|first3=T.|last4=Kortelainen|first4=J.|title=Transitive Closures of Binary Relations I|year=2007|publisher=School of Mathematics - Physics Charles University|location=Prague|page=1|url=http://www.karlin.mff.cuni.cz/~jezek/120/transitive1.pdf?}} Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric".</ref>).
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| There is a 1-to-1 correspondence between all non-strict and strict partial orders.
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| If "≤" is a non-strict partial order, then the corresponding strict partial order "<" is the irreflexive reduction given by:
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| :''a'' < ''b'' if ''a'' ≤ ''b'' and ''a'' ≠ ''b''
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| Conversely, if "<" is a strict partial order, then the corresponding non-strict partial order "≤" is the [[Binary relation#Operations on binary relations|reflexive closure]] given by:
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| : ''a'' ≤ ''b'' if ''a'' < ''b'' or ''a'' = ''b''.
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| This is the reason for using the notation "≤".
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| Using the strict order "<", the relation "''a'' is covered by ''b''" can be equivalently rephrased as "''a''<''b'', but not ''a''<''c''<''b'' for any ''c''".
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| Strict partial orders are useful because they correspond more directly to [[directed acyclic graph]]s (dags): every strict partial order is a dag, and the [[transitive closure]] of a dag is both a strict partial order and also a dag itself.
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| ==Inverse and order dual==
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| The [[inverse relation|inverse]] or converse ≥ of a partial order relation ≤ satisfies ''x''≥''y'' if and only if ''y''≤''x''. The inverse of a partial order relation is reflexive, transitive, and antisymmetric, and hence itself a partial order relation. The [[Duality (order theory)|order dual]] of a partially ordered set is the same set with the partial order relation replaced by its inverse. The irreflexive relation > is to ≥ as < is to ≤.
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| Any one of these four relations ≤, <, ≥, and > on a given set uniquely determine the other three.
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| In general two elements ''x'' and ''y'' of a partial order may stand in any of four mutually exclusive relationships to each other: either ''x'' < ''y'', or ''x'' = ''y'', or ''x'' > ''y'', or ''x'' and ''y'' are ''incomparable'' (none of the other three). A [[total order|totally ordered]] set is one that rules out this fourth possibility: all pairs of elements are comparable and we then say that [[Trichotomy (mathematics)|trichotomy]] holds. The [[natural number]]s, the [[integer]]s, the [[Rational number|rationals]], and the [[Real number|real]]s are all totally ordered by their algebraic (signed) magnitude whereas the [[complex number]]s are not. This is not to say that the complex numbers cannot be totally ordered; we could for example order them lexicographically via ''x''+'''i'''''y'' < ''u''+'''i'''''v'' if and only if ''x'' < ''u'' or (''x'' = ''u'' and ''y'' < ''v''), but this is not ordering by magnitude in any reasonable sense as it makes 1 greater than 100'''i'''. Ordering them by absolute magnitude yields a preorder in which all pairs are comparable, but this is not a partial order since 1 and '''i''' have the same absolute magnitude but are not equal, violating antisymmetry.
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| ==Mappings between partially ordered sets==
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| {| style="float:right"
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| |[[File:Birkhoff120.svg|thumb|x150px|Order isomorphism between the divisors of 120 (partially ordered by divisibility) and the divisor-closed subsets of {2,3,4,5,8} (partially ordered by set inclusion)]]
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| |}
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| {| style="float:right"
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| |[[File:Monotonic but nonhomomorphic map between lattices.gif|thumb|x150px|Order-preserving, but not order-reflecting (since ''f''(''u'')≤''f''(''v''), but not ''u''≤''v'') map.]]
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| |}
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| Given two partially ordered sets (''S'',≤) and (''T'',≤), a function ''f'': ''S'' → ''T'' is called '''[[order-preserving]]''', or '''[[Monotonic function#Monotonicity in order theory|monotone]]''', or '''isotone''', if for all ''x'' and ''y'' in ''S'', ''x''≤''y'' implies ''f''(''x'') ≤ ''f''(''y'').
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| If (''U'',≤) is also a partially ordered set, and both ''f'': ''S'' → ''T'' and ''g'': ''T'' → ''U'' are order-preserving, their [[function composition|composition]] (''g''∘''f''): ''S'' → ''U'' is order-preserving, too.
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| A function ''f'': ''S'' → ''T'' is called '''order-reflecting''' if for all ''x'' and ''y'' in ''S'', ''f''(''x'') ≤ ''f''(''y'') implies ''x''≤''y''.
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| If ''f'' is both order-preserving and order-reflecting, then it is called an '''[[order-embedding]]''' of (''S'',≤) into (''T'',≤).
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| In the latter case, ''f'' is necessarily [[injective]], since ''f''(''x'') = ''f''(''y'') implies ''x'' ≤ ''y'' and ''y'' ≤ ''x''. If an order-embedding between two posets ''S'' and ''T'' exists, one says that ''S'' can be '''embedded''' into ''T''. If an order-embedding ''f'': ''S'' → ''T'' is [[bijective]], it is called an '''[[order isomorphism]]''', and the partial orders (''S'',≤) and (''T'',≤) are said to be '''isomorphic'''. Isomorphic orders have structurally similar [[Hasse diagram]]s (cf. right picture). It can be shown that if order-preserving maps ''f'': ''S'' → ''T'' and ''g'': ''T'' → ''S'' exist such that ''g''∘''f'' and ''f''∘''g'' yields the [[identity function]] on ''S'' and ''T'', respectively, then ''S'' and ''T'' are order-isomorphic.
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| <ref name="dp02">{{Cite book
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| | last1 = Davey | first1 = B. A.
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| | last2 = Priestley | first2 = H. A.
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| | contribution = Maps between ordered sets
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| | edition = 2nd
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| | isbn = 0-521-78451-4
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| | location = New York
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| | mr = 1902334
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| | pages = 23–24
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| | publisher = Cambridge University Press
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| | title = Introduction to Lattices and Order
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| | url = http://books.google.com/books?id=vVVTxeuiyvQC&pg=PA23
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| | year = 2002
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| | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}.</ref>
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| For example, a mapping ''f'': ℕ → ℙ(ℕ) 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 ''x'' divides ''y'', then each prime divisor of ''x'' is also a prime divisor of ''y''. However, it is neither injective (since it maps both 12 and 6 to {2,3}) nor order-reflecting (since besides 12 doesn't divide 6). Taking instead each number to the set of its [[prime power]] divisors defines a map ''g'': ℕ → ℙ(ℕ) that is order-preserving, order-reflecting, and hence an order-embedding. It is not an order-isomorphism (since it e.g. doesn't map any number to the set {4}), but it can be made one by [[Injective function#Injections may be made invertible|restricting its codomain]] to ''g''(ℕ). The right picture shows a subset of ℕ and its isomorphic image under ''g''. The construction of such an order-isomophism into a power set can be generalized to a wide class of partial orders, called [[distributive lattice]]s, see "[[Birkhoff's representation theorem]]".
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| ==Number of partial orders==
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| [[Image:poset6.jpg|right|thumb|250px| Partially ordered set of [[power set|set of all subsets]] of a six-element set {a, b, c, d, e, f}, ordered by the subset relation.]]
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| 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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| {{Number of relations}}
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| The number of strict partial orders is the same as that of partial orders.
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| If we count only [[up to]] isomorphism, we get 1, 1, 2, 5, 16, 63, 318, … {{OEIS|A000112}}.
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| == Linear extension ==
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| A partial order ≤<sup>*</sup> on a set ''X'' is an '''extension''' of another partial order ≤ on ''X'' provided that for all elements ''x'' and ''y'' of ''X'', whenever <math>x \leq y</math>, it is also the case that ''x'' ≤<sup>*</sup> ''y''. A [[linear extension]] is an extension that is also a linear (i.e., total) order. Every partial order can be extended to a total order ([[order-extension principle]]).<ref>{{cite book |last=Jech |first=Thomas |authorlink=Thomas Jech |title=The Axiom of Choice |year=2008 |origyear=1973 |publisher=[[Dover Publications]] |isbn=0-486-46624-8}}</ref>
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| 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]].
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| == In category theory ==
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| Every poset (and every [[preorder]]) may be considered as a [[category (mathematics)|category]] in which every hom-set has at most one element. More explicitly, let hom(''x'', ''y'') = {(''x'', ''y'')} if ''x'' ≤ ''y'' (and otherwise the empty set) and (''y'', ''z'')∘(''x'', ''y'') = (''x'', ''z''). Posets are [[Equivalence of categories|equivalent]] to one another if and only if they are [[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]].
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| A [[functor]] from a poset category (a [[diagram (category theory)|diagram]] indexed by a poset category) is a [[commutative diagram]].
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| ==Partial orders in topological spaces==
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| If ''P'' 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'' ≤ ''b''} }} 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>a_i\to a</math>, <math>b_i\to b</math> and <span class="texhtml">''a''<sub>''i''</sub> ≤ ''b''<sub>''i''</sub></span> for all ''i'', then <span class="texhtml">''a'' ≤ ''b''</span>.<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|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref>
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| ==Interval==
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| For ''a'' ≤ ''b'', the [[interval (mathematics)|closed interval]] {{closed-closed|''a'',''b''}} is the set of elements ''x'' satisfying ''a'' ≤ ''x'' ≤ ''b'' (i.e. ''a'' ≤ ''x'' and ''x'' ≤ ''b''). It contains at least the elements ''a'' and ''b''.
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| Using the corresponding strict relation "<", the [[open interval]] {{open-open|''a'',''b''}} is the set of elements ''x'' satisfying ''a'' < ''x'' < ''b'' (i.e. ''a'' < ''x'' and ''x'' < ''b''). An open interval may be empty even if ''a'' < ''b''. For example, the open interval {{open-open|1,2}} on the integers is empty since there are no integers ''i'' such that 1 < ''i'' < 2.
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| Sometimes the definitions are extended to allow ''a'' > ''b'', in which case the interval is empty.
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| The ''half-open intervals'' {{closed-open|''a'',''b''}} and {{open-closed|''a'',''b''}} are defined similarly.
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| A poset is [[Locally finite poset|locally finite]] if every interval is finite. For example, the [[integers]] are locally finite under their natural ordering. The lexicographical order on the cartesian product ℕ×ℕ is not locally finite, since e.g. (1,2)≤(1,3)≤(1,4)≤(1,5)≤...≤(2,1).
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| Using the interval notation, the property "''a'' is covered by ''b''" can be rephrased equivalently as [''a'',''b''] = {''a'',''b''}.
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| 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]]s.
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| == See also ==
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| {{colbegin|3}}
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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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| *[[causal set]]
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| *[[comparability graph]]
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| *[[directed set]]
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| *[[graded poset]]
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| *[[semilattice]]
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| *[[lattice (order)|lattice]]
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| *[[ordered group]]
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| *[[poset topology]], a kind of topological space that can be defined from any poset
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| *[[semiorder]]
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| *[[series-parallel partial order]]
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| *[[strict weak ordering]] - strict partial order "<" in which the relation "neither ''a'' < ''b'' nor ''b'' < ''a''" is transitive.
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| *[[complete partial order]]
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| *[[Zorn's lemma]]
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| {{colend}}
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| ==Notes==
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| <references/>
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| ==References==
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| * {{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|postscript=.}}
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| * {{cite book|first=Bernd S. W. |last=Schröder|title= Ordered Sets: An Introduction |publisher=Birkhäuser, Boston|year=2003}}
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| * {{cite book|first=Richard P.|last=Stanley|authorlink=Richard P. Stanley|title=Enumerative Combinatorics 1|series=Cambridge Studies in Advanced Mathematics|volume=49|publisher=Cambridge University Press|isbn=0-521-66351-2}}
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| ==External links==
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| {{Commons|Hasse diagram}}
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| * {{OEIS link|A001035}}: Number of posets with ''n'' labeled elements in the [[On-Line Encyclopedia of Integer Sequences|OEIS]]
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| * {{OEIS link|A000112}}: Number of posets with ''n'' unlabeled elements in the OEIS
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| [[Category:Order theory]]
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| [[Category:Mathematical relations]]
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