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Partially ordered set

From Wikipedia, the free encyclopedia
Mathematical set with an ordering
Transitive binary relations
SymmetricAntisymmetricConnectedWell-foundedHas joinsHas meetsReflexiveIrreflexiveAsymmetric
Total,
Semiconnex		 Anti-
reflexive
Equivalence relationGreen tickY✗✗✗✗✗Green tickY✗✗
Preorder(Quasiorder)✗✗✗✗✗✗Green tickY✗✗
Partial order✗Green tickY✗✗✗✗Green tickY✗✗
Total preorder✗✗Green tickY✗✗✗Green tickY✗✗
Total order✗Green tickYGreen tickY✗✗✗Green tickY✗✗
Prewellordering✗✗Green tickYGreen tickY✗✗Green tickY✗✗
Well-quasi-ordering✗✗✗Green tickY✗✗Green tickY✗✗
Well-ordering✗Green tickYGreen tickYGreen tickY✗✗Green tickY✗✗
Lattice✗Green tickY✗✗Green tickYGreen tickYGreen tickY✗✗
Join-semilattice✗Green tickY✗✗Green tickY✗Green tickY✗✗
Meet-semilattice✗Green tickY✗✗✗Green tickYGreen tickY✗✗
Strict partial order✗Green tickY✗✗✗✗✗Green tickYGreen tickY
Strict weak order✗Green tickY✗✗✗✗✗Green tickYGreen tickY
Strict total order✗Green tickYGreen tickY✗✗✗✗Green tickYGreen tickY
SymmetricAntisymmetricConnectedWell-foundedHas joinsHas meetsReflexiveIrreflexiveAsymmetric
Definitions,
for alla,b{\displaystyle a,b} andS≠∅:{\displaystyle S\neq \varnothing :}		aRb⇒bRa{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}aRb and bRa⇒a=b{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}a≠b⇒aRb or bRa{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}minSexists{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}a∨bexists{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}a∧bexists{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}aRa{\displaystyle aRa}not aRa{\displaystyle {\text{not }}aRa}aRb⇒not bRa{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}
Green tickY indicates 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 byGreen tickY in the "Symmetric" column and✗ in the "Antisymmetric" column, respectively.

All definitions tacitly require thehomogeneous relationR{\displaystyle R} betransitive: for alla,b,c,{\displaystyle a,b,c,} ifaRb{\displaystyle aRb} andbRc{\displaystyle bRc} thenaRc.{\displaystyle aRc.}
A term's definition may require additional properties that are not listed in this table.

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

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

Formally, a partial order is ahomogeneous binary relation that isreflexive,antisymmetric, andtransitive. Apartially ordered set (poset for short) is anordered pairP=(X,≤){\displaystyle P=(X,\leq )} consisting of a setX{\displaystyle X} (called theground set ofP{\displaystyle P}) and a partial order≤{\displaystyle \leq } onX{\displaystyle X}. When the meaning is clear from context and there is no ambiguity about the partial order, the setX{\displaystyle X} itself is sometimes called a poset.

Partial order relations

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

Partial orders

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Areflexive,weak,[1] ornon-strict partial order,[2] commonly referred to simply as apartial order, is ahomogeneous relation ≤ on asetP{\displaystyle P} that isreflexive,antisymmetric, andtransitive. That is, for alla,b,c∈P,{\displaystyle a,b,c\in P,} it must satisfy:

  1. Reflexivity:a≤a{\displaystyle a\leq a}, i.e. every element is related to itself.
  2. Antisymmetry: ifa≤b{\displaystyle a\leq b} andb≤a{\displaystyle b\leq a} thena=b{\displaystyle a=b}, i.e. no two distinct elements precede each other.
  3. Transitivity: ifa≤b{\displaystyle a\leq b} andb≤c{\displaystyle b\leq c} thena≤c{\displaystyle a\leq c}.

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

Strict partial orders

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Anirreflexive,strong,[1] orstrict partial order is a homogeneous relation < on a setP{\displaystyle P} that isirreflexive,asymmetric andtransitive; that is, it satisfies the following conditions for alla,b,c∈P:{\displaystyle a,b,c\in P:}

  1. Irreflexivity:¬(a<a){\displaystyle \neg \left(a<a\right)}, i.e. no element is related to itself (also called anti-reflexive).
  2. Asymmetry: ifa<b{\displaystyle a<b} then notb<a{\displaystyle b<a}.
  3. Transitivity: ifa<b{\displaystyle a<b} andb<c{\displaystyle b<c} thena<c{\displaystyle a<c}.

A transitive relation is asymmetric if and only if it is irreflexive.[3] So the definition is the same if it omits either irreflexivity or asymmetry (but not both).

A strict partial order is also known as astrict preorder.

Correspondence of strict and non-strict partial order relations

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Fig. 2Commutative 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 itslogical matrix for the poset whoseHasse diagram is depicted in the center. For example3≰4{\displaystyle 3\not \leq 4} so row 3, column 4 of the bottom left matrix is empty.

Strict and non-strict partial orders on a setP{\displaystyle P} are closely related. A non-strict partial order≤{\displaystyle \leq } may be converted to a strict partial order by removing all relationships of the forma≤a;{\displaystyle a\leq a;} that is, the strict partial order is the set<:= ≤ ∖ ΔP{\displaystyle <\;:=\ \leq \ \setminus \ \Delta _{P}} whereΔP:={(p,p):p∈P}{\displaystyle \Delta _{P}:=\{(p,p):p\in P\}} is theidentity relation onP×P{\displaystyle P\times P} and∖{\displaystyle \;\setminus \;} denotesset subtraction. Conversely, a strict partial order < onP{\displaystyle P} may be converted to a non-strict partial order by adjoining all relationships of that form; that is,≤:=ΔP∪<{\displaystyle \leq \;:=\;\Delta _{P}\;\cup \;<\;} is a non-strict partial order. Thus, if≤{\displaystyle \leq } is a non-strict partial order, then the corresponding strict partial order < is theirreflexive kernel given bya<b if a≤b and a≠b.{\displaystyle 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≤{\displaystyle \leq } is thereflexive closure given by:a≤b if a<b or a=b.{\displaystyle a\leq b{\text{ if }}a<b{\text{ or }}a=b.}

Dual orders

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Main article:Duality (order theory)

Thedual (oropposite)Rop{\displaystyle R^{\text{op}}} of a partial order relationR{\displaystyle R} is defined by lettingRop{\displaystyle R^{\text{op}}} be theconverse relation ofR{\displaystyle R}, i.e.xRopy{\displaystyle xR^{\text{op}}y} if and only ifyRx{\displaystyle yRx}. The dual of a non-strict partial order is a non-strict partial order,[4] 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

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Given a setP{\displaystyle P} and a partial order relation, typically the non-strict partial order≤{\displaystyle \leq }, we may uniquely extend our notation to define four partial order relations≤,{\displaystyle \leq ,}<,{\displaystyle <,}≥,{\displaystyle \geq ,} and>{\displaystyle >}, where≤{\displaystyle \leq } is a non-strict partial order relation onP{\displaystyle P},<{\displaystyle <} is the associated strict partial order relation onP{\displaystyle P} (theirreflexive kernel of≤{\displaystyle \leq }),≥{\displaystyle \geq } is the dual of≤{\displaystyle \leq }, and>{\displaystyle >} is the dual of<{\displaystyle <}. Strictly speaking, the termpartially ordered set refers to a set with all of these relations defined appropriately. But practically, one need only consider a single relation,(P,≤){\displaystyle (P,\leq )} or(P,<){\displaystyle (P,<)}, or, in rare instances, the non-strict and strict relations together,(P,≤,<){\displaystyle (P,\leq ,<)}.[5]

The termordered set is 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 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≤{\displaystyle \leq } such as⊑{\displaystyle \sqsubseteq }[6] or⪯{\displaystyle \preceq }[7] to distinguish partial orders from total orders.

When referring to partial orders,≤{\displaystyle \leq } should not be taken as thecomplement of>{\displaystyle >}. The relation>{\displaystyle >} is the converse of the irreflexive kernel of≤{\displaystyle \leq }, which is always a subset of the complement of≤{\displaystyle \leq }, but>{\displaystyle >} is equal to the complement of≤{\displaystyle \leq }if, and only if,≤{\displaystyle \leq } is a total order.[a]

Alternative definitions

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Another way of defining a partial order, found incomputer science, is via a notion ofcomparison. Specifically, given≤,<,≥, and >{\displaystyle \leq ,<,\geq ,{\text{ and }}>} as defined previously, it can be observed that two elementsx andy may stand in any of fourmutually exclusive relationships to each other: eitherx <y, orx =y, orx >y, orx andy areincomparable. This can be represented by a functioncompare:P×P→{<,>,=,|}{\displaystyle {\text{compare}}:P\times P\to \{<,>,=,\vert \}} that returns one of four codes when given two elements.[8][9] This definition is equivalent to apartial order on asetoid, where equality is taken to be a definedequivalence relation rather than set equality.[10]

Wallis defines a more general notion of apartial order relation as anyhomogeneous relation that istransitive andantisymmetric. This includes both reflexive and irreflexive partial orders as subtypes.[1]

A finite poset can be visualized through itsHasse diagram.[11] Specifically, taking a strict partial order relation(P,<){\displaystyle (P,<)}, adirected acyclic graph (DAG) may be constructed by taking each element ofP{\displaystyle P} to be a node and each element of<{\displaystyle <} to be an edge. Thetransitive reduction of this DAG[b] 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

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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:

One familiar example of a partially ordered set is a collection of people ordered bygenealogical 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

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Fig. 4a Lexicographic order onN×N{\displaystyle \mathbb {N} \times \mathbb {N} }
Fig. 4c Reflexive closure of strict direct product order onN×N.{\displaystyle \mathbb {N} \times \mathbb {N} .} Elementscovered 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 theCartesian product of two partially ordered sets are (see Fig. 4):

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

Applied toordered vector spaces over 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

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Another way to combine two (disjoint) posets is theordinal sum[12] (orlinear sum),[13]Z =X ⊕Y, defined on the union of the underlying setsX andY by the ordera ≤Zb if and only if:

  • a,b ∈X witha ≤Xb, or
  • a,b ∈Y witha ≤Yb, or
  • a ∈X andb ∈Y.

If two posets arewell-ordered, then so is their ordinal sum.[14]

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

Derived notions

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The examples use the poset(P({x,y,z}),⊆){\displaystyle ({\mathcal {P}}(\{x,y,z\}),\subseteq )} consisting of theset of all subsets of a three-element set{x,y,z},{\displaystyle \{x,y,z\},} ordered by set inclusion (see Fig. 1).

Extrema

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Fig. 5 The figure above with the greatest and least elements removed. In this reduced poset, the top row of elements are allmaximal elements, and the bottom row are allminimal elements, but there is nogreatest and noleast element.

There are several notions of "greatest" and "least" element in a posetP,{\displaystyle P,} notably:

Fig. 6 A portion of the lattice ofnonnegative integers ordered by divisibility

As another example, consider the positiveintegers, ordered by divisibility: 1 is a least element, as itdivides 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 anyg divides for instance 2g, which is distinct from it, sog 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 anyprime 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},{\displaystyle \{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

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Fig. 7a Order-preserving, but not order-reflecting map (sincef(u) ≼f(v), but not u≤{\displaystyle \leq } v)
Fig. 7b 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)

Given two partially ordered sets(S, ≤) and(T, ≼), a functionf:S→T{\displaystyle f:S\to T} is calledorder-preserving, ormonotone, orisotone, if for allx,y∈S,{\displaystyle x,y\in S,}x≤y{\displaystyle x\leq y} impliesf(x) ≼f(y).If(U, ≲) is also a partially ordered set, and bothf:S→T{\displaystyle f:S\to T} andg:T→U{\displaystyle g:T\to U} are order-preserving, theircompositiong∘f:S→U{\displaystyle g\circ f:S\to U} is order-preserving, too.A functionf:S→T{\displaystyle f:S\to T} is calledorder-reflecting if for allx,y∈S,{\displaystyle x,y\in S,}f(x) ≼f(y) impliesx≤y.{\displaystyle x\leq y.}Iff is both order-preserving and order-reflecting, then it is called anorder-embedding of(S, ≤) into(T, ≼).In the latter case,f is necessarilyinjective, sincef(x)=f(y){\displaystyle f(x)=f(y)} impliesx≤y and y≤x{\displaystyle x\leq y{\text{ and }}y\leq x} and in turnx=y{\displaystyle x=y} according to the antisymmetry of≤.{\displaystyle \leq .} If an order-embedding between two posetsS andT exists, one says thatS can beembedded intoT. If an order-embeddingf:S→T{\displaystyle 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 mapsf:S→T{\displaystyle f:S\to T} andg:T→U{\displaystyle g:T\to U} exist such thatg∘f{\displaystyle g\circ f} andf∘g{\displaystyle f\circ g} yields theidentity function onS andT, respectively, thenS andT are order-isomorphic.[15]

For example, a mappingf:N→P(N){\displaystyle f:\mathbb {N} \to \mathbb {P} (\mathbb {N} )} from the set of natural numbers (ordered by divisibility) to thepower set of natural numbers (ordered by set inclusion) can be defined by taking each number to the set of itsprime divisors. It is order-preserving: ifx dividesy, then each prime divisor ofx is also a prime divisor ofy. However, it is neither injective (since it maps both 12 and 6 to{2,3}{\displaystyle \{2,3\}}) nor order-reflecting (since 12 does not divide 6). Taking instead each number to the set of itsprime power divisors defines a mapg:N→P(N){\displaystyle 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}{\displaystyle \{4\}}), but it can be made one byrestricting its codomain tog(N).{\displaystyle g(\mathbb {N} ).} Fig. 7b shows a subset ofN{\displaystyle \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

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SequenceA001035 inOEIS gives the number of partial orders on a set ofn labeled elements:

Number ofn-element binary relations of different types
Elem­entsAnyTransitiveReflexiveSymmetricPreorderPartial orderTotal preorderTotal orderEquivalence relation
0111111111
1221211111
216134843322
3512171646429191365
465,5363,9944,0961,024355219752415
n2n22n(n−1)2n(n+1)/2∑n
k=0k!S(n,k)		n!∑n
k=0S(n,k)
OEISA002416A006905A053763A006125A000798A001035A000670A000142A000110

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 to isomorphism, the sequence 1, 1, 2, 5, 16, 63, 318, ... (sequenceA000112 in theOEIS) is obtained.

Subposets

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A posetP∗=(X∗,≤∗){\displaystyle P^{*}=(X^{*},\leq ^{*})} is called asubposet of another posetP=(X,≤){\displaystyle P=(X,\leq )} provided thatX∗{\displaystyle X^{*}} is asubset ofX{\displaystyle X} and≤∗{\displaystyle \leq ^{*}} is a subset of≤{\displaystyle \leq }. The latter condition is equivalent to the requirement that for anyx{\displaystyle x} andy{\displaystyle y} inX∗{\displaystyle X^{*}} (and thus also inX{\displaystyle X}), ifx≤∗y{\displaystyle x\leq ^{*}y} thenx≤y{\displaystyle x\leq y}.

IfP∗{\displaystyle P^{*}} is a subposet ofP{\displaystyle P} and furthermore, for allx{\displaystyle x} andy{\displaystyle y} inX∗{\displaystyle X^{*}}, wheneverx≤y{\displaystyle x\leq y} we also havex≤∗y{\displaystyle x\leq ^{*}y}, then we callP∗{\displaystyle P^{*}} the subposet ofP{\displaystyle P}induced byX∗{\displaystyle X^{*}}, and writeP∗=P[X∗]{\displaystyle P^{*}=P[X^{*}]}.

Linear extension

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A partial order≤∗{\displaystyle \leq ^{*}} on a setX{\displaystyle X} is called anextension of another partial order≤{\displaystyle \leq } onX{\displaystyle X} provided that for all elementsx,y∈X,{\displaystyle x,y\in X,} wheneverx≤y,{\displaystyle x\leq y,} it is also the case thatx≤∗y.{\displaystyle x\leq ^{*}y.} Alinear 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).[16]

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

In category theory

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Main article:Posetal category

Every poset (and everypreordered set) may be considered as acategory where, for objectsx{\displaystyle x} andy,{\displaystyle y,} there is at most onemorphism fromx{\displaystyle x} toy.{\displaystyle y.} More explicitly, lethom(x,y) = {(x,y)} ifx ≤y (and otherwise theempty set) and(y,z)∘(x,y)=(x,z).{\displaystyle (y,z)\circ (x,y)=(x,z).} Such categories are sometimes calledposetal.

Posets areequivalent to 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

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Main article:Partially ordered space

IfP{\displaystyle 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≤b}{\displaystyle \{(a,b):a\leq b\}} is aclosed subset of the topologicalproduct spaceP×P.{\displaystyle P\times P.} Under this assumption partial order relations are well behaved atlimits in the sense that iflimi→∞ai=a,{\displaystyle \lim _{i\to \infty }a_{i}=a,} andlimi→∞bi=b,{\displaystyle \lim _{i\to \infty }b_{i}=b,} and for alli,{\displaystyle i,}ai≤bi,{\displaystyle a_{i}\leq b_{i},} thena≤b.{\displaystyle a\leq b.}[17]

Intervals

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See also:Interval (mathematics)

Aconvex set in a posetP is a subsetI ofP with the property that, for anyx andy inI and anyz inP, ifx ≤z ≤y, thenz is also inI. This definition generalizes the definition ofintervals ofreal numbers. When there is possible confusion withconvex sets ofgeometry, one usesorder-convex instead of "convex".

Aconvex sublattice of alatticeL is a sublattice ofL that is also a convex set ofL. Every nonempty convex sublattice can be uniquely represented as the intersection of afilter and anideal ofL.

Aninterval in a posetP is a subset that can be defined with interval notation:

  • Fora ≤b, theclosed interval[a,b] is the set of elementsx satisfyinga ≤x ≤b (that is,a ≤x andx ≤b). It contains at least the elementsa andb.
  • Using the corresponding strict relation "<", theopen interval(a,b) is the set of elementsx satisfyinga <x <b (i.e.a <x andx <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 integerx such that0 <x < 1.
  • Thehalf-open intervals[a,b) and(a,b] are defined similarly.

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

An intervalI is bounded if there exist elementsa,b∈P{\displaystyle 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 noinfimum orsupremum in P, so it cannot be written in interval notation using elements of P.

A poset is calledlocally finite if every bounded interval is finite. For example, the integers are locally finite under their natural ordering. The lexicographical order on the cartesian productN×N{\displaystyle \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 "a is covered byb" can be rephrased equivalently as[a,b]={a,b}.{\displaystyle [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

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Notes

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  1. ^A proof can be foundhere.
  2. ^which always exists and is unique, sinceP{\displaystyle P} is assumed to be finite
  3. ^SeeGeneral relativity § Time travel.

Citations

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  1. ^abcWallis, W. D. (14 March 2013).A Beginner's Guide to Discrete Mathematics. Springer Science & Business Media. p. 100.ISBN 978-1-4757-3826-1.
  2. ^Simovici, Dan A. & Djeraba, Chabane (2008)."Partially Ordered Sets".Mathematical Tools for Data Mining: Set Theory, Partial Orders, Combinatorics. Springer.ISBN 9781848002012.
  3. ^Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007)."Transitive Closures of Binary Relations I".Acta Universitatis Carolinae. Mathematica et Physica.48 (1). Prague: School of Mathematics – Physics Charles University:55–69. Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric".
  4. ^Davey & Priestley (2002), pp. 14–15.
  5. ^Avigad, Jeremy; Lewis, Robert Y.; van Doorn, Floris (29 March 2021). "13.2. More on Orderings".Logic and Proof (Release 3.18.4 ed.). Archived fromthe original on 3 April 2023. Retrieved24 July 2021.So we can think of every partial order as really being a pair, consisting of a weak partial order and an associated strict one.
  6. ^Rounds, William C. (7 March 2002)."Lectures slides"(PDF).EECS 203: DISCRETE MATHEMATICS. Retrieved23 July 2021.
  7. ^Kwong, Harris (25 April 2018). "7.4: Partial and Total Ordering".A Spiral Workbook for Discrete Mathematics. Retrieved23 July 2021.
  8. ^"Finite posets".Sage 9.2.beta2 Reference Manual: Combinatorics. Retrieved5 January 2022.compare_elements(x,y): Comparex andy in the poset. Ifx <y, return −1. Ifx =y, return 0. Ifx >y, return 1. Ifx andy are not comparable, return None.
  9. ^Chen, Peter; Ding, Guoli; Seiden, Steve.On Poset Merging(PDF) (Technical report). p. 2. Retrieved5 January 2022.A comparison between two elements s, t in S returns one of three distinct values, namely s≤t, s>t or s|t.
  10. ^Prevosto, Virgile; Jaume, Mathieu (11 September 2003).Making proofs in a hierarchy of mathematical structures. CALCULEMUS-2003 – 11th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning. Roma, Italy: Aracne. pp. 89–100.
  11. ^Merrifield, Richard E.;Simmons, Howard E. (1989).Topological Methods in Chemistry. New York: John Wiley & Sons. pp. 28.ISBN 0-471-83817-9. Retrieved27 July 2012.A partially ordered set is conveniently represented by aHasse diagram...
  12. ^Neggers, J.; Kim, Hee Sik (1998), "4.2 Product Order and Lexicographic Order",Basic Posets, World Scientific, pp. 62–63,ISBN 9789810235895
  13. ^Davey & Priestley (2002), pp. 17–18.
  14. ^P. R. Halmos (1974).Naive Set Theory. Springer. p. 82.ISBN 978-1-4757-1645-0.
  15. ^Davey & Priestley (2002), pp. 23–24.
  16. ^Jech, Thomas (2008) [1973].The Axiom of Choice.Dover Publications.ISBN 978-0-486-46624-8.
  17. ^Ward, L. E. Jr (1954)."Partially Ordered Topological Spaces".Proceedings of the American Mathematical Society.5 (1):144–161.doi:10.1090/S0002-9939-1954-0063016-5.hdl:10338.dmlcz/101379.

References

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External links

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

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