Partially ordered set

Transitivebinary relations
Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Total, Semiconnex Anti- reflexive Equivalence relation Y ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Preorder(Quasiorder) ✗ ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Partial order ✗ Y ✗ ✗ ✗ ✗ Y ✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict weak order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict total order ✗ Y Y ✗ ✗ ✗ ✗ Y Y Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Definitions, for all$a,b$and$S\neq \varnothing:$ ${\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}$ ${\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}$ ${\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}$ ${\begin{aligned}\min S\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}$ $aRa$ ${\text{not }}aRa$ ${\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}$
Yindicates that the column's property is always true for the row's term (at the very left), while✗indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated byYin the "Symmetric" column and✗in the "Antisymmetric" column, respectively. All definitions tacitly require thehomogeneous relation$R$betransitive:for all$a,b,c,$if$aRb$and$bRc$then$aRc.$ A term's definition may require additional properties that are not listed in this table.

Fig. 1TheHasse diagramof theset of all subsetsof a three-element set$\{x,y,z\},$ordered byinclusion.Sets connected by an upward path, like$\emptyset$and$\{x,y\}$,are comparable, while e.g.$\{x\}$and$\{y\}$are not.

Inmathematics,especiallyorder theory,apartial orderon asetis an arrangement such that, for certain pairs of elements, one precedes the other. The wordpartialis 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 generalizetotal orders,in which every pair is comparable.

Formally, a partial order is ahomogeneous binary relationthat isreflexive,antisymmetric,andtransitive.Apartially ordered set(posetfor short) is an ordered pair$P=(X,\leq )$consisting of a set$X$(called theground setof$P$) and a partial order$\leq$on$X$.When the meaning is clear from context and there is no ambiguity about the partial order, the set$X$itself is sometimes called a poset.

Partial order relations

The termpartial orderusually refers to the reflexive partial order relations, referred to in this article asnon-strictpartial 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 aone-to-one correspondence,so for every strict partial order there is a unique corresponding non-strict partial order, and vice versa.

Partial orders

Areflexive,weak,ornon-strict partial order,commonly referred to simply as apartial order,is ahomogeneous relation≤ on aset$P$that isreflexive,antisymmetric,andtransitive.That is, for all$a,b,c\in P,$it must satisfy:

1. Reflexivity:$a\leq a$,i.e. every element is related to itself.
2. Antisymmetry:if$a\leq b$and$b\leq a$then$a=b$,i.e. no two distinct elements precede each other.
3. Transitivity:if$a\leq b$and$b\leq c$then$a\leq c$.

A non-strict partial order is also known as an antisymmetricpreorder.

Strict partial orders

Anirreflexive,strong,orstrict partial orderis a homogeneous relation < on a set$P$that istransitive,irreflexive,andasymmetric;that is, it satisfies the following conditions for all$a,b,c\in P:$

1. Transitivity:if$a<b$and$b<c$then$a<c$.
2. Irreflexivity:$\neg \left(a<a\right)$,i.e. no element is related to itself (also called anti-reflexive).
3. Asymmetry:if$a<b$then not$b<a$.

A transitive relation is asymmetric if and only if it is irreflexive.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 asymmetricstrict preorder.

Correspondence of strict and non-strict partial order relations

Fig. 2Commutative diagramabout 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 itslogical matrixfor the poset whoseHasse diagramis depicted in the center. For example$3\not \leq 4$so row 3, column 4 of the bottom left matrix is empty.

Strict and non-strict partial orders on a set$P$are closely related. A non-strict partial order$\leq$may be converted to a strict partial order by removing all relationships of the form$a\leq a;$that is, the strict partial order is the set$<\;:=\ \leq \ \setminus \ \Delta _{P}$where$\Delta _{P}:=\{(p,p):p\in P\}$is theidentity relationon$P\times P$and$\;\setminus \;$denotesset subtraction.Conversely, a strict partial order < on$P$may be converted to a non-strict partial order by adjoining all relationships of that form; that is,$\leq \;:=\;\Delta _{P}\;\cup \;<\;$is a non-strict partial order. Thus, if$\leq$is a non-strict partial order, then the corresponding strict partial order < is theirreflexive kernelgiven by $$a<b{\text{ if }}a\leq b{\text{ and }}a\neq b.$$ Conversely, if < is a strict partial order, then the corresponding non-strict partial order$\leq$is thereflexive closuregiven by: $$a\leq b{\text{ if }}a<b{\text{ or }}a=b.$$

Dual orders

Main article:Duality (order theory)

Thedual(oropposite)$R^{\text{op}}$of a partial order relation$R$is defined by letting$R^{\text{op}}$be theconverse relationof$R$,i.e.$xR^{\text{op}}y$if and only if$yRx$.The dual of a non-strict partial order is a non-strict partial order,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$P$and a partial order relation, typically the non-strict partial order$\leq$,we may uniquely extend our notation to define four partial order relations$\leq,$$<,$$\geq,$and$>$,where$\leq$is a non-strict partial order relation on$P$,$<$is the associated strict partial order relation on$P$(theirreflexive kernelof$\leq$),$\geq$is the dual of$\leq$,and$>$is the dual of$<$.Strictly speaking, the termpartially ordered setrefers to a set with all of these relations defined appropriately. But practically, one need only consider a single relation,$(P,\leq )$or$(P,<)$,or, in rare instances, the non-strict and strict relations together,$(P,\leq,<)$.

The termordered setis sometimes used as a shorthand forpartially ordered set,as long as it is clear from the context that no other kind of order is meant. In particular,totally ordered setscan also be referred to as "ordered sets", especially in areas where these structures are more common than posets. Some authors use different symbols than$\leq$such as$\sqsubseteq$or$\preceq$to distinguish partial orders from total orders.

When referring to partial orders,$\leq$should not be taken as thecomplementof$>$.The relation$>$is the converse of the irreflexive kernel of$\leq$,which is always a subset of the complement of$\leq$,but$>$is equal to the complement of$\leq$if, and only if,$\leq$is a total order.

Alternative definitions

Another way of defining a partial order, found incomputer science,is via a notion ofcomparison.Specifically, given$\leq,<,\geq,{\text{ and }}>$as defined previously, it can be observed that two elementsxandymay stand in any of fourmutually exclusiverelationships to each other: eitherx<y,orx=y,orx>y,orxandyareincomparable.This can be represented by a function${\text{compare}}:P\times P\to \{<,>,=,\vert \}$that returns one of four codes when given two elements.This definition is equivalent to apartial order on asetoid,where equality is taken to be a definedequivalence relationrather than set equality.

Wallis defines a more general notion of apartial order relationas anyhomogeneous relationthat istransitiveandantisymmetric.This includes both reflexive and irreflexive partial orders as subtypes.

A finite poset can be visualized through itsHasse diagram.Specifically, taking a strict partial order relation$(P,<)$,adirected acyclic graph(DAG) may be constructed by taking each element of$P$to be a node and each element of$<$to be an edge. Thetransitive reductionof this DAGis 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

Fig. 3Graph 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:

• Thereal numbers,or in general any totally ordered set, ordered by the standardless-than-or-equalrelation ≤, is a partial order.
• On the real numbers$\mathbb {R}$,the usualless thanrelation < is a strict partial order. The same is also true of the usualgreater thanrelation > on$\mathbb {R}$.
• By definition, everystrict weak orderis a strict partial order.
• The set ofsubsetsof a given set (itspower set) ordered byinclusion(see Fig. 1). Similarly, the set ofsequencesordered bysubsequence,and the set ofstringsordered bysubstring.
• The set ofnatural numbersequipped with the relation ofdivisibility.(see Fig. 3 and Fig. 6)
• The vertex set of adirected acyclic graphordered byreachability.
• The set ofsubspacesof avector spaceordered by inclusion.
• For a partially ordered setP,thesequence spacecontaining allsequencesof elements fromP,where sequenceaprecedes sequencebif every item inaprecedes the corresponding item inb.Formally,$\left(a_{n}\right)_{n\in \mathbb {N} }\leq \left(b_{n}\right)_{n\in \mathbb {N} }$if and only if$a_{n}\leq b_{n}$for all$n\in \mathbb {N}$;that is, acomponentwise order.
• For a setXand a partially ordered setP,thefunction spacecontaining all functions fromXtoP,wheref≤gif and only iff(x) ≤g(x)for all$x\in X.$
• Afence,a partially ordered set defined by an alternating sequence of order relationsa<b>c<d...
• The set of events inspecial relativityand, in most cases,general relativity,where for two eventsXandY,X≤Yif and only ifYis in the futurelight coneofX.An eventYcan be causally affected byXonly ifX≤Y.

One familiar example of a partially ordered set is a collection of people ordered bygenealogicaldescendancy. 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

Fig. 4aLexicographic order on$\mathbb {N} \times \mathbb {N}$

Fig. 4bProduct order on$\mathbb {N} \times \mathbb {N}$

Fig. 4cReflexive closure of strict direct product order on$\mathbb {N} \times \mathbb {N}.$Elementscoveredby(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 theCartesian productof two partially ordered sets are (see Fig. 4):

• thelexicographical order:(a,b) ≤ (c,d)ifa<cor (a=candb≤d);
• theproduct order:(a,b) ≤ (c,d) ifa≤candb≤d;
• thereflexive closureof thedirect productof the corresponding strict orders:(a,b) ≤ (c,d)if (a<candb<d) or (a=candb=d).

All three can similarly be defined for the Cartesian product of more than two sets.

Applied toordered vector spacesover the samefield,the result is in each case also an ordered vector space.

See alsoorders on the Cartesian product of totally ordered sets.

Sums of partially ordered sets

Another way to combine two (disjoint) posets is theordinal sum(orlinear sum),Z=X⊕Y,defined on the union of the underlying setsXandYby the ordera≤_Zbif and only if:

• a,b∈Xwitha≤_Xb,or
• a,b∈Ywitha≤_Yb,or
• a∈Xandb∈Y.

If two posets arewell-ordered,then so is their ordinal sum.

Series-parallel partial ordersare formed from the ordinal sum operation (in this context called series composition) and another operation called parallel composition. Parallel composition is thedisjoint unionof 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$({\mathcal {P}}(\{x,y,z\}),\subseteq )$consisting of theset of all subsetsof a three-element set$\{x,y,z\},$ordered by set inclusion (see Fig. 1).

• aisrelated tobwhena≤b.This does not imply thatbis also related toa,because the relation need not besymmetric.For example,$\{x\}$is related to$\{x,y\},$but not the reverse.
• aandbarecomparableifa≤borb≤a.Otherwise they areincomparable.For example,$\{x\}$and$\{x,y,z\}$are comparable, while$\{x\}$and$\{y\}$are not.
• Atotal orderorlinear orderis a partial order under which every pair of elements is comparable, i.e.trichotomyholds. For example, the natural numbers with their standard order.
• Achainis a subset of a poset that is a totally ordered set. For example,$\{\{\,\},\{x\},\{x,y,z\}\}$is a chain.
• Anantichainis a subset of a poset in which no two distinct elements are comparable. For example, the set ofsingletons$\{\{x\},\{y\},\{z\}\}.$
• An elementais said to bestrictly less thanan elementb,ifa≤band$a\neq b.$For example,$\{x\}$is strictly less than$\{x,y\}.$
• An elementais said to becoveredby another elementb,writtena⋖b(ora<:b), ifais strictly less thanband no third elementcfits between them; formally: if botha≤band$a\neq b$are true, anda≤c≤bis false for eachcwith$a\neq c\neq b.$Using the strict order <, the relationa⋖bcan be equivalently rephrased as "a<bbut nota<c<bfor anyc".For example,$\{x\}$is covered by$\{x,z\},$but is not covered by$\{x,y,z\}.$

Extrema

Fig. 5The figure above with the greatest and least elements removed. In this reduced poset, the top row of elements are allmaximalelements, and the bottom row are allminimalelements, but there is nogreatestand noleastelement.

There are several notions of "greatest" and "least" element in a poset$P,$notably:

• Greatest elementand least element: An element$g\in P$is agreatest elementif$a\leq g$for every element$a\in P.$An element$m\in P$is aleast elementif$m\leq a$for every element$a\in P.$A poset can only have one greatest or least element. In our running example, the set$\{x,y,z\}$is the greatest element, and$\{\,\}$is the least.
• Maximal elementsand 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. In our running example,$\{x,y,z\}$and$\{\,\}$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 subsetAofP,an elementxinPis an upper bound ofAifa≤x,for each elementainA.In particular,xneed not be inAto be an upper bound ofA.Similarly, an elementxinPis a lower bound ofAifa≥x,for each elementainA.A greatest element ofPis an upper bound ofPitself, and a least element is a lower bound ofP.In our example, the set$\{x,y\}$is anupper boundfor the collection of elements$\{\{x\},\{y\}\}.$

Fig. 6Nonnegative integers,ordered by divisibility

As another example, consider the positiveintegers,ordered by divisibility: 1 is a least element, as itdividesall 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 anygdivides for instance 2g,which is distinct from it, sogis 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 anyprime numberis 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 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. 6).

Mappings between partially ordered sets

Fig. 7aOrder-preserving, but not order-reflecting (sincef(u) ≼f(v),but not u$\leq$v) map.

Fig. 7bOrder 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)

Given two partially ordered sets(S,≤)and(T,≼),a function$f:S\to T$is calledorder-preserving,ormonotone,orisotone,if for all$x,y\in S,$$x\leq y$impliesf(x) ≼f(y). If(U,≲)is also a partially ordered set, and both$f:S\to T$and$g:T\to U$are order-preserving, theircomposition$g\circ f:S\to U$is order-preserving, too. A function$f:S\to T$is calledorder-reflectingif for all$x,y\in S,$f(x) ≼f(y)implies$x\leq y.$ Iffis both order-preserving and order-reflecting, then it is called anorder-embeddingof(S,≤)into(T,≼). In the latter case,fis necessarilyinjective,since$f(x)=f(y)$implies$x\leq y{\text{ and }}y\leq x$and in turn$x=y$according to the antisymmetry of$\leq.$If an order-embedding between two posetsSandTexists, one says thatScan beembeddedintoT.If an order-embedding$f:S\to T$isbijective,it is called anorder isomorphism,and the partial orders(S,≤)and(T,≼)are said to beisomorphic.Isomorphic orders have structurally similarHasse diagrams(see Fig. 7a). It can be shown that if order-preserving maps$f:S\to T$and$g:T\to U$exist such that$g\circ f$and$f\circ g$yields theidentity functiononSandT,respectively, thenSandTare order-isomorphic.

For example, a mapping$f:\mathbb {N} \to \mathbb {P} (\mathbb {N} )$from the set of natural numbers (ordered by divisibility) to thepower setof natural numbers (ordered by set inclusion) can be defined by taking each number to the set of itsprime divisors.It is order-preserving: ifxdividesy,then each prime divisor ofxis also a prime divisor ofy.However, it is neither injective (since it maps both 12 and 6 to$\{2,3\}$) nor order-reflecting (since 12 does not divide 6). Taking instead each number to the set of itsprime powerdivisors defines a map$g:\mathbb {N} \to \mathbb {P} (\mathbb {N} )$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$\{4\}$), but it can be made one byrestricting its codomainto$g(\mathbb {N} ).$Fig. 7b shows a subset of$\mathbb {N}$and its isomorphic image underg.The construction of such an order-isomorphism into a power set can be generalized to a wide class of partial orders, calleddistributive lattices;seeBirkhoff's representation theorem.

Number of partial orders

SequenceA001035inOEISgives the number of partial orders on a set ofnlabeled elements:

Number ofn-element binary relations of different types

Elem­ents	Any	Transitive	Reflexive	Symmetric	Preorder	Partial order	Total preorder	Total order	Equivalence relation
0	1	1	1	1	1	1	1	1	1
1	2	2	1	2	1	1	1	1	1
2	16	13	4	8	4	3	3	2	2
3	512	171	64	64	29	19	13	6	5
4	65,536	3,994	4,096	1,024	355	219	75	24	15
n	2n2		2n(n−1)	2n(n+1)/2			∑n k=0k!S(n,k)	n!	∑n k=0S(n,k)
OEIS	A002416	A006905	A053763	A006125	A000798	A001035	A000670	A000142	A000110

Note thatS(n,k)refers toStirling numbers of the second kind.

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

If the count is made onlyup toisomorphism, the sequence 1, 1, 2, 5, 16, 63, 318,... (sequenceA000112in theOEIS) is obtained.

Subposets

A poset$P^{*}=(X^{*},\leq ^{*})$is called asubposetof another poset$P=(X,\leq )$provided that$X^{*}$is asubsetof$X$and$\leq ^{*}$is a subset of$\leq$.The latter condition is equivalent to the requirement that for any$x$and$y$in$X^{*}$(and thus also in$X$), if$x\leq ^{*}y$then$x\leq y$.

If$P^{*}$is a subposet of$P$and furthermore, for all$x$and$y$in$X^{*}$,whenever$x\leq y$we also have$x\leq ^{*}y$,then we call$P^{*}$the subposet of$P$inducedby$X^{*}$,and write$P^{*}=P[X^{*}]$.

Linear extension

A partial order$\leq ^{*}$on a set$X$is called anextensionof another partial order$\leq$on$X$provided that for all elements$x,y\in X,$whenever$x\leq y,$it is also the case that$x\leq ^{*}y.$Alinear extensionis 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).

Incomputer science,algorithms for finding linear extensions of partial orders (represented as thereachabilityorders ofdirected acyclic graphs) are calledtopological sorting.

In category theory

Main article:Posetal category

Every poset (and everypreordered set) may be considered as acategorywhere, for objects$x$and$y,$there is at most onemorphismfrom$x$to$y.$More explicitly, lethom(x,y) = {(x,y)}ifx≤y(and otherwise theempty set) and$(y,z)\circ (x,y)=(x,z).$Such categories are sometimes calledposetal.

Posets areequivalentto one another if and only if they areisomorphic.In a poset, the smallest element, if it exists, is aninitial object,and the largest element, if it exists, is aterminal object.Also, every preordered set is equivalent to a poset. Finally, every subcategory of a poset isisomorphism-closed.

Partial orders in topological spaces

Main article:Partially ordered space

If$P$is a partially ordered set that has also been given the structure of atopological space,then it is customary to assume that$\{(a,b):a\leq b\}$is aclosedsubset of the topologicalproduct space$P\times P.$Under this assumption partial order relations are well behaved atlimitsin the sense that if$\lim _{i\to \infty }a_{i}=a,$and$\lim _{i\to \infty }b_{i}=b,$and for all$i,$$a_{i}\leq b_{i},$then$a\leq b.$

Intervals

See also:Interval (mathematics)

Aconvex setin a posetPis a subsetIofPwith the property that, for anyxandyinIand anyzinP,ifx≤z≤y,thenzis also inI.This definition generalizes the definition ofintervalsofreal numbers.When there is possible confusion withconvex setsofgeometry,one usesorder-convexinstead of "convex".

Aconvex sublatticeof alatticeLis a sublattice ofLthat is also a convex set ofL.Every nonempty convex sublattice can be uniquely represented as the intersection of afilterand anidealofL.

Anintervalin a posetPis a subset that can be defined with interval notation:

• Fora≤b,theclosed interval[a,b]is the set of elementsxsatisfyinga≤x≤b(that is,a≤xandx≤b). It contains at least the elementsaandb.
• Using the corresponding strict relation "<", theopen interval(a,b)is the set of elementsxsatisfyinga<x<b(i.e.a<xandx<b). An open interval may be empty even ifa<b.For example, the open interval(0, 1)on the integers is empty since there is no integerxsuch that0 <x< 1.
• Thehalf-open intervals[a,b)and(a,b]are defined similarly.

Whenevera≤bdoes 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 {1, 2, 4, 5, 8} is convex, but not an interval.

An intervalIis bounded if there exist elements$a,b\in P$such thatI⊆[a,b].Every interval that can be represented in interval notation is obviously bounded, but the converse is not true. For example, letP=(0, 1)∪(1, 2)∪(2, 3)as a subposet of the real numbers. The subset(1, 2)is a bounded interval, but it has noinfimumorsupremuminP,so it cannot be written in interval notation using elements ofP.

A poset is calledlocally finiteif every bounded interval is finite. For example, the integers are locally finite under their natural ordering. The lexicographical order on the cartesian product$\mathbb {N} \times \mathbb {N}$is not locally finite, since(1, 2) ≤ (1, 3) ≤ (1, 4) ≤ (1, 5) ≤... ≤ (2, 1). Using the interval notation, the property "ais covered byb"can be rephrased equivalently as$[a,b]=\{a,b\}.$

This concept of an interval in a partial order should not be confused with the particular class of partial orders known as theinterval orders.

See also

• 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
• Comparability graph– Graph linking pairs of comparable elements in a partial order
• Complete partial order– Mathematical phrase
• Directed set– Mathematical ordering with upper bounds
• Graded poset– partially ordered set equipped with a rank functionPages displaying wikidata descriptions as a fallback
• Incidence algebra– Associative algebra used in combinatorics, a branch of mathematics
• Lattice– Set whose pairs have minima and maxima
• Locally finite poset– MathematicsPages displaying wikidata descriptions as a fallbackPages displaying short descriptions with no spaces
• Möbius function on posets– Associative algebra used in combinatorics, a branch of mathematics
• Nested set collection
• Order polytope
• Ordered field– Algebraic object with an ordered structure
• Ordered group– Group with a compatible partial orderPages displaying short descriptions of redirect targets
• Ordered vector space– Vector space with a partial order
• Poset topology,a kind of topological space that can be defined from any poset
• Scott continuity– continuity of a function between two partial orders.
• Semilattice– Partial order with joins
• Semiorder– Numerical ordering with a margin of error
• Szpilrajn extension theorem– every partial order is contained in some total order.
• Stochastic dominance– Partial order between random variables
• Strict weak ordering– strict partial order "<" in which the relation"neithera<bnorb<a"is transitive.
• Total order– Order whose elements are all comparable
• Zorn's lemma– Mathematical proposition equivalent to the axiom of choice

References

• Davey, B. A.; Priestley, H. A. (2002).Introduction to Lattices and Order(2nd ed.). New York: Cambridge University Press.ISBN978-0-521-78451-1.
• Deshpande, Jayant V. (1968)."On Continuity of a Partial Order".Proceedings of the American Mathematical Society.19(2): 383–386.doi:10.1090/S0002-9939-1968-0236071-7.
• Schmidt, Gunther(2010).Relational Mathematics.Encyclopedia of Mathematics and its Applications. Vol. 132. Cambridge University Press.ISBN978-0-521-76268-7.
• Bernd Schröder (11 May 2016).Ordered Sets: An Introduction with Connections from Combinatorics to Topology.Birkhäuser.ISBN978-3-319-29788-0.
• Stanley, Richard P.(1997).Enumerative Combinatorics 1.Cambridge Studies in Advanced Mathematics. Vol. 49. Cambridge University Press.ISBN0-521-66351-2.
• Eilenberg, S.(2016).Foundations of Algebraic Topology.Princeton University Press.
• Kalmbach, G. (1976). "Extension of Homology Theory to Partially Ordered Sets".J. Reine Angew. Math.280:134–156.

External links

Media related toHasse diagramsat Wikimedia Commons; each of which shows an example for a partial order

• OEISsequence A001035 (Number of posets withnlabeled elements)
• OEISsequence A000112 (Number of partially ordered sets ( "posets" ) with n unlabeled elements.)