Content deleted Content added
→Extrema: added → included; spaces |
spacing |
||
Line 1:
{{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.
Line 32:
=== 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
Line 65:
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 ''f'' ≤ ''g'' if and only if ''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 [[General relativity#Time travel]].}} [[general relativity]], where for two events ''X'' and ''Y'', {{nowrap|''X'' ≤ ''Y''}} if and only if ''Y'' is in the future [[light cone]] of ''X''. An event ''Y'' can only be causally affected by ''X'' if {{nowrap|''X'' ≤ ''Y''}}.
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
Line 92 ⟶ 91:
| caption3 = '''Fig. 4c''' Reflexive closure of strict direct product order on <math>\N \times \N.</math> Elements [[#Formal definition|covered]] by (3, 3) and covering (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]]: (''a'', ''b'') ≤ (''c'', ''d'') if ''a'' < ''c'' or (''a'' = ''c'' and ''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: (''a'', ''b'') ≤ (''c'', ''d'') if (''a'' < ''c'' and ''b'' < ''d'') or (''a'' = ''c'' and ''b'' = ''d'').
All three can similarly be defined for the Cartesian product of more than two sets.
Line 124 ⟶ 123:
== 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 ''a'' ≤ ''b'' or ''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.
Line 151 ⟶ 149:
| 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'')}}.
Line 159 ⟶ 157:
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; 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:
Line 184 ⟶ 182:
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]], 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)}}
Line 201 ⟶ 199:
* The ''half-open intervals'' {{closed-open|''a'', ''b''}} and {{open-closed|''a'', ''b''}} are defined similarly.
Whenever ''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)}}.
Line 211 ⟶ 209:
== See also ==
{{div col|colwidth=27em}}
* [[Antimatroid]], a formalization of orderings on a set that allows more general families of orderings than posets
Line 240 ⟶ 237:
{{div col end}}
== Notes ==
{{notelist}}
== Citations ==
{{reflist}}
== References ==
* {{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}}
Line 256 ⟶ 251:
* {{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}}
== External links ==
{{Commons|Hasse diagram}}
* {{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.}}
| |||