Y 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 byY in the "Symmetric" column and✗ in the "Antisymmetric" column, respectively.
All definitions tacitly require thehomogeneous relation betransitive: for all if and then 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 ordered byinclusion. Sets connected by an upward path, like and, are comparable, while e.g. and 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 pair consisting of a set (called theground set of) and a partial order on. When the meaning is clear from context and there is no ambiguity about the partial order, the set itself is sometimes called a poset.
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.
Anirreflexive,strong,[1] orstrict partial order is a homogeneous relation < on a set that isirreflexive,asymmetric andtransitive; that is, it satisfies the following conditions for all
Irreflexivity:, i.e. no element is related to itself (also called anti-reflexive).
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 an asymmetricstrict preorder.
Correspondence of strict and non-strict partial order relations
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 example so row 3, column 4 of the bottom left matrix is empty.
Strict and non-strict partial orders on a set are closely related. A non-strict partial order may be converted to a strict partial order by removing all relationships of the form that is, the strict partial order is the set where is theidentity relation on and denotesset subtraction. Conversely, a strict partial order < on may be converted to a non-strict partial order by adjoining all relationships of that form; that is, is a non-strict partial order. Thus, if is a non-strict partial order, then the corresponding strict partial order < is theirreflexive kernel given byConversely, if < is a strict partial order, then the corresponding non-strict partial order is thereflexive closure given by:
Thedual (oropposite) of a partial order relation is defined by letting be theconverse relation of, i.e. if and only if. 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.
Given a set and a partial order relation, typically the non-strict partial order, we may uniquely extend our notation to define four partial order relations and, where is a non-strict partial order relation on, is the associated strict partial order relation on (theirreflexive kernel of), is the dual of, and is the dual of. 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, or, or, in rare instances, the non-strict and strict relations together,.[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 such as[6] or[7] to distinguish partial orders from total orders.
When referring to partial orders, should not be taken as thecomplement of. The relation is the converse of the irreflexive kernel of, which is always a subset of the complement of, but is equal to the complement ofif, and only if, is a total order.[a]
Another way of defining a partial order, found incomputer science, is via a notion ofcomparison. Specifically, given 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 function 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, adirected acyclic graph (DAG) may be constructed by taking each element of to be a node and each element of 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.
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:
Thereal numbers, or in general any totally ordered set, ordered by the standardless-than-or-equal relation ≤, is a partial order.
On the real numbers, the usualless than relation < is a strict partial order. The same is also true of the usualgreater than relation > on.
For a partially ordered setP, thesequence space containing allsequences of elements fromP, where sequencea precedes sequenceb if every item ina precedes the corresponding item inb. Formally, if and only if for all; that is, acomponentwise order.
For a setX and a partially ordered setP, thefunction space containing all functions fromX toP, wheref ≤g if and only iff(x) ≤g(x) for all
Afence, a partially ordered set defined by an alternating sequence of order relationsa <b >c <d ...
The set of events inspecial relativity and, in most cases,[c]general relativity, where for two eventsX andY,X ≤Y if and only ifY is in the futurelight cone ofX. An eventY can be causally affected byX only ifX ≤Y.
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
Fig. 4c Reflexive closure of strict direct product order on 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):
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.
The examples use the poset consisting of theset of all subsets of a three-element set ordered by set inclusion (see Fig. 1).
a isrelated tob whena ≤b. This does not imply thatb is also related toa, because the relation need not besymmetric. For example, is related to but not the reverse.
a andb arecomparable ifa ≤b orb ≤a. Otherwise they areincomparable. For example, and are comparable, while and are not.
Atotal order orlinear order is a partial order under which every pair of elements is comparable, i.e.trichotomy holds. For example, the natural numbers with their standard order.
Achain is a subset of a poset that is a totally ordered set. For example, is a chain.
Anantichain is a subset of a poset in which no two distinct elements are comparable. For example, the set ofsingletons
An elementa is said to bestrictly less than an elementb, ifa ≤b and For example, is strictly less than
An elementa is said to becovered by another elementb, writtena ⋖b (ora <:b), ifa is strictly less thanb and no third elementc fits between them; formally: if botha ≤b and are true, anda ≤c ≤b is false for eachc with Using the strict order <, the relationa ⋖b can be equivalently rephrased as "a <b but nota <c <b for anyc". For example, is covered by but is not covered by
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 poset notably:
Greatest element and least element: An element is agreatest element if for every element An element is aleast element if for every element A poset can only have one greatest or least element. In our running example, the set is the greatest element, and is the least.
Maximal elements and minimal elements: An element is a maximal element if there is no element such that Similarly, an element is a minimal element if there is no element such that 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, 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 subsetA ofP, an elementx inP is an upper bound ofA ifa ≤ x, for each elementa inA. In particular,x need not be inA to be an upper bound ofA. Similarly, an elementx inP is a lower bound ofA ifa ≥ x, for each elementa inA. A greatest element ofP is an upper bound ofP itself, and a least element is a lower bound ofP. In our example, the set is anupper bound for the collection of elements
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 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).
Fig. 7a Order-preserving, but not order-reflecting (sincef(u) ≼f(v), but not u v) map.
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 function is calledorder-preserving, ormonotone, orisotone, if for all impliesf(x) ≼f(y).If(U, ≲) is also a partially ordered set, and both and are order-preserving, theircomposition is order-preserving, too.A function is calledorder-reflecting if for allf(x) ≼f(y) impliesIff is both order-preserving and order-reflecting, then it is called anorder-embedding of(S, ≤) into(T, ≼).In the latter case,f is necessarilyinjective, since implies and in turn according to the antisymmetry of If an order-embedding between two posetsS andT exists, one says thatS can beembedded intoT. If an order-embedding 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 and exist such that and yields theidentity function onS andT, respectively, thenS andT are order-isomorphic.[15]
For example, a mapping 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) nor order-reflecting (since 12 does not divide 6). Taking instead each number to the set of itsprime power divisors defines a map 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), but it can be made one byrestricting its codomain to Fig. 7b shows a subset of 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.
A poset is called asubposet of another poset provided that is asubset of and is a subset of. The latter condition is equivalent to the requirement that for any and in (and thus also in), if then.
If is a subposet of and furthermore, for all and in, whenever we also have, then we call the subposet ofinduced by, and write.
A partial order on a set is called anextension of another partial order on provided that for all elements whenever it is also the case that 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]
Every poset (and everypreordered set) may be considered as acategory where, for objects and there is at most onemorphism from to More explicitly, lethom(x,y) = {(x,y)} ifx ≤y (and otherwise theempty set) and 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.
If is a partially ordered set that has also been given the structure of atopological space, then it is customary to assume that is aclosed subset of the topologicalproduct space Under this assumption partial order relations are well behaved atlimits in the sense that if and and for all then[17]
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 elements 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 product 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
This concept of an interval in a partial order should not be confused with the particular class of partial orders known as theinterval orders.
Order polytope – convex polytope associated to a finite poset, whose points are monotonic functions from the poset to [0,1], whose vertices are upper sets of the poset, and whose dimension is the cardinality of the posetPages displaying wikidata descriptions as a fallback
Ordered field – Algebraic object with an ordered structure
Ordered group – Group with a compatible partial orderPages displaying short descriptions of redirect targets
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^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".
^Avigad, Jeremy; Lewis, Robert Y.; van Doorn, Floris (29 March 2021). "13.2. More on Orderings".Logic and Proof (Release 3.18.4 ed.). 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.
^Rounds, William C. (7 March 2002)."Lectures slides"(PDF).EECS 203: DISCRETE MATHEMATICS. Retrieved23 July 2021.
^"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.
^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.
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