In

_{''Z''} ''b'' if and only if:
* ''a'', ''b'' ∈ ''X'' with ''a'' ≤_{''X''} ''b'', or
* ''a'', ''b'' ∈ ''Y'' with ''a'' ≤_{''Y''} ''b'', or
* ''a'' ∈ ''X'' and ''b'' ∈ ''Y''.
If two posets are

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, especially order theory
Order theory is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...

, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation
Binary may refer to:
Science and technology
Mathematics
* Binary number
In mathematics and digital electronics
Digital electronics is a field of electronics
The field of electronics is a branch of physics and electrical engineeri ...

indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order."
The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total order
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s, in which every pair is comparable.
Informal definition

A partial order defines a notion ofcomparison
Comparison or comparing is the act of evaluating two or more things by determining the relevant, comparable characteristics of each thing, and then determining which characteristics of each are similar to the other, which are different, and t ...

. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive
In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails, ...

relationships to each other: either ''x'' < ''y'', or ''x'' = ''y'', or ''x'' > ''y'', or ''x'' and ''y'' are ''incomparable''.
A set with a partial order is called a partially ordered set (also called a poset). The term ''ordered set'' is sometimes also used, 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.
A poset can be visualized through its Hasse diagram
In order theory
Order theory is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they ...

, which depicts the ordering relation.
Partial order relation

A partial order relation is ahomogeneous relation In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

that is transitive
Transitivity or transitive may refer to:
Grammar
* Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects
* Transitive verb, a verb which takes an object
* Transitive case, a grammatical case to mark arg ...

and antisymmetric. There are two common sub-definitions for a partial order relation, for reflexive and irreflexive partial order relations, also called "non-strict" and "strict" respectively. The two definitions can be put into a one-to-one correspondence
In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is paired with exactly on ...

, so for every strict partial order there is a unique corresponding non-strict partial order, and vice-versa. The term partial order typically refers to a non-strict partial order relation.
Non-strict partial order

A reflexive, weak, or is ahomogeneous relation In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

≤ over a set $P$ that is reflexive, antisymmetric, and transitive
Transitivity or transitive may refer to:
Grammar
* Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects
* Transitive verb, a verb which takes an object
* Transitive case, a grammatical case to mark arg ...

. That is, for all $a,\; b,\; c\; \backslash in\; P,$ it must satisfy:
# reflexivity: $a\; \backslash leq\; a$, i.e. every element is related to itself.
# antisymmetry
In linguistics
Linguistics is the scientific study of language
A language is a structured system of communication
Communication (from Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic br ...

: if $a\; \backslash leq\; b\; \backslash text\; b\; \backslash leq\; a\; \backslash text\; a\; =\; b$, i.e. no two distinct elements precede each other.
# transitivity: if $a\; \backslash leq\; b\; \backslash text\; b\; \backslash leq\; c\; \backslash text\; a\; \backslash leq\; c$.
A non-strict partial order is also known as an antisymmetric preorder
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

.
Strict partial order

An irreflexive, strong, or is a homogeneous relation < over a set $P$ that isirreflexive
In mathematics, a homogeneous binary relation ''R'' over a set (mathematics), set ''X'' is reflexive if it relates every element of ''X'' to itself.
An example of a reflexive relation is the relation "equality (mathematics), is equal to" on the se ...

, transitive
Transitivity or transitive may refer to:
Grammar
* Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects
* Transitive verb, a verb which takes an object
* Transitive case, a grammatical case to mark arg ...

and asymmetric; that is, it satisfies the following conditions for all $a,\; b,\; c\; \backslash in\; P:$
#Irreflexivity
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

: not $a\; <\; a$, i.e. no element is related to itself
# Transitivity: if $a\; <\; b\; \backslash text\; b\; <\; c\; \backslash text\; a\; <\; c,$
#Asymmetry
Asymmetry is the absence of, or a violation of, symmetry
Symmetry (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is ...

: if $a\; <\; b$ then not $b\; <\; a$.
Irreflexivity and transitivity together imply asymmetry. Also, asymmetry implies irreflexivity. In other words, a transitive relation is asymmetric if and only if it is irreflexive. Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric". So the definition is the same if it omits either irreflexivity or asymmetry (but not both).
A strict partial order is also known as a strict preorder.
Correspondence of strict and non-strict partial order relations

Strict and non-strict partial orders on a set $P$ are closely related. A non-strict partial order $\backslash leq$ may be converted to a strict partial order by removing all relationships of the form $(a,a);$ that is, the strict partial order is the set $<\; \backslash ;\; :=\; \backslash \; \backslash leq\backslash \; \backslash setminus\; \backslash \; \backslash Delta\_P$ where $\backslash Delta\_P\; :=\; \backslash $ is theidentity relation
Identity may refer to:
Social sciences
* Identity (social science)
Identity is the qualities, beliefs, personality, looks and/or expressions that make a person (self-identity as emphasized in psychology)
or group (collective identity as p ...

on $P\; \backslash times\; P$ and $\backslash ;\backslash setminus\backslash ;$ denotes set subtraction
In set theory, the complement of a Set (mathematics), set , often denoted by (or ), are the Element (mathematics), elements not in .
When all sets under consideration are considered to be subsets of a given set , the absolute complement of is t ...

. 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, $\backslash leq\backslash ;\; :=\; \backslash ;\backslash Delta\_P\backslash ;\; \backslash cup\; \backslash ;<\backslash ;$ is a non-strict partial order. Thus, if $\backslash leq$ is a non-strict partial order, then the corresponding strict partial order < is the irreflexive kernel
In mathematics, a homogeneous binary relation ''R'' over a set (mathematics), set ''X'' is reflexive if it relates every element of ''X'' to itself.
An example of a reflexive relation is the relation "equality (mathematics), is equal to" on the se ...

given by
$$a\; <\; b\; \backslash text\; a\; \backslash leq\; b\; \backslash text\; a\; \backslash neq\; b.$$
Conversely, if < is a strict partial order, then the corresponding non-strict partial order $\backslash leq$ is the reflexive closureIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

given by:
$$a\; \backslash leq\; b\; \backslash text\; a\; <\; b\; \backslash text\; a\; =\; b.$$
Dual orders

The ''dual'' (or ''opposite'') $R^$ of a partial order relation $R$ is defined by letting $R^$ be theconverse relation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

of $R$, i.e. $x\; R^\; y$ if and only if $y\; R\; x$. 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

We can consider a poset as a 3-tuple $(P,\backslash leq,<)$, or even a 5-tuple $(P,\backslash leq,<,\backslash geq,>)$, where $\backslash leq$ and $\backslash geq$ are non-strict partial order relations over $P$, $<$ and $>$ are strict partial order relations over $P$, $<$ is theirreflexive kernel
In mathematics, a homogeneous binary relation ''R'' over a set (mathematics), set ''X'' is reflexive if it relates every element of ''X'' to itself.
An example of a reflexive relation is the relation "equality (mathematics), is equal to" on the se ...

of $\backslash leq$, $\backslash geq$ is the dual of $\backslash leq$, and $>$ is the dual of $<$.
Any one of the four partial order relations $\backslash leq,\; <,\; \backslash geq,\; \backslash text\; >$ on a given set uniquely determines the other three. Hence, as a matter of notation, we may write $(P,\backslash leq)$ or $(P,<)$, and assume that the other relations are defined appropriately. Defining via a non-strict partial order $\backslash leq$ is most common. Some authors use different symbols than $\backslash leq$ such as $\backslash sqsubseteq$ or $\backslash preceq$ to distinguish partial orders from total orders.
When referring to partial orders, $\backslash leq$ should not be taken as the complement
A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be:
* Complement (linguistics), a word or phrase having a particular syntactic role
** Subject complement, a word or phrase addi ...

of $>$. The relation $>$ is the converse of the irreflexive kernel of $\backslash leq$, which is always a subset of the complement of $\backslash leq$, but $>$ is equal to the complement of $\backslash leq$ if, and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...

, $\backslash leq$ is a total order.
Examples

Standard examples of posets arising in mathematics include: * Thereal number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s, or in general any totally ordered set, ordered by the standard ''less-than-or-equal'' relation ≤, is a non-strict partial order.
* On the real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s $P\; :=\; \backslash R,$ the usual less than
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different not ...

relation < is a strict partial order and the same is also true of the usual greater than
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different not ...

relation > on $\backslash R.$
* By definition, every strict weak order
The 13 possible strict weak orderings on a set of three elements . The only total orders are shown in black. Two orderings are connected by an edge if they differ by a single dichotomy.
In mathematics
Mathematics (from Ancient Greek, Gre ...

is a strict partial order.
* The set of subset
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s of a given set (its power set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

) ordered by inclusion
Inclusion or Include may refer to:
Sociology
* Social inclusion, affirmative action to change the circumstances and habits that leads to social exclusion
** Inclusion (disability rights), including people with and without disabilities, people of ...

(see Fig.1). Similarly, the set of sequence
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s ordered by subsequence
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, and the set of string
String or strings may refer to:
*String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects
Arts, entertainment, and media Films
* Strings (1991 film), ''Strings'' (1991 fil ...

s ordered by .
* The set of natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s equipped with the relation of divisibility
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

.
* The vertex set of a directed acyclic graph
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

ordered by reachability
In graph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph the ...

.
* The set of subspaces of a vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

ordered by inclusion.
* For a partially ordered set ''P'', the sequence space
In functional analysis
200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis.
Functional analy ...

containing all sequence
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s of elements from ''P'', where sequence ''a'' precedes sequence ''b'' if every item in ''a'' precedes the corresponding item in ''b''. Formally, $\backslash left(a\_n\backslash right)\_\; \backslash leq\; \backslash left(b\_n\backslash right)\_$ if and only if $a\_n\; \backslash leq\; b\_n$ for all $n\; \backslash in\; \backslash N$; that is, a componentwise order.
* For a set ''X'' and a partially ordered set ''P'', the function space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

containing all functions from ''X'' to ''P'', where ''f'' ≤ ''g'' if and only if ''f''(''x'') ≤ ''g''(''x'') for all $x\; \backslash in\; X.$
* A fence
A fence is a structure that encloses an area, typically outdoors, and is usually constructed from posts that are connected by boards, wire, rails or netting. A fence differs from a wall
A wall is a structure and a surface that defines an ...

, a partially ordered set defined by an alternating sequence of order relations ''a'' < ''b'' > ''c'' < ''d'' ...
* The set of events in special relativity
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...

and, in most cases, general relativity
General relativity, also known as the general theory of relativity, is the geometric
Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...

, where for two events ''X'' and ''Y'', ''X'' ≤ ''Y'' if and only if ''Y'' is in the future light cone
In special and general relativity, a light cone is the path that a flash of light, emanating from a single event
Event may refer to:
Gatherings of people
* Ceremony
A ceremony (, ) is a unified ritual
A ritual is a sequence of activities ...

of ''X''. An event ''Y'' can only be causally affected by ''X'' if ''X'' ≤ ''Y''.
One familiar example of a partially ordered set is a collection of people ordered by genealogical
Genealogy (from el, γενεαλογία ' "the making of a pedigree") is the study of families, family history, and the tracing of their lineages. Genealogists use oral interviews, historical records, genetic analysis, and other records to ob ...

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

In order of increasing strength, i.e., decreasing sets of pairs, three of the possible partial orders on theCartesian product
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of two partially ordered sets are (see Fig.3-5):
*the lexicographical order
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

: (''a'',''b'') ≤ (''c'',''d'') if ''a'' < ''c'' or (''a'' = ''c'' and ''b'' ≤ ''d'');
*the product order
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

: (''a'',''b'') ≤ (''c'',''d'') if ''a'' ≤ ''c'' and ''b'' ≤ ''d'';
*the reflexive closureIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

of the direct productIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

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.
Applied to ordered vector space
In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations.
Definition
Given a vector space ''X'' over the real numbers R and a preo ...

s over the same field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

, the result is in each case also an ordered vector space.
See also orders on the Cartesian product of totally ordered sets.
Sums of partially ordered sets

Another way to combine two (disjoint) posets is the ordinal sum (or linear sum), ''Z'' = ''X'' ⊕ ''Y'', defined on the union of the underlying sets ''X'' and ''Y'' by the order ''a'' ≤well-ordered
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, then so is their ordinal sum.
Series-parallel partial order
In order-theoretic mathematics, a series-parallel partial order is a partially ordered set
250px, The set of all subsets of a three-element set , ordered by inclusion. Distinct sets on the same horizontal level are incomparable with each other. ...

s are formed from the ordinal sum operation (in this context called series composition) and another operation called parallel composition. Parallel composition is the disjoint union
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of 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 $(\backslash mathcal(\backslash ),\backslash subseteq)$ consisting of the set of all subsets of a three-element set $\backslash ,$ 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 besymmetric
Symmetry (from Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more pre ...

. For example, $\backslash $ is related to $\backslash ,$ but not the reverse.
* ''a'' and ''b'' are '' comparable'' if ''a'' ≤ ''b'' or ''b'' ≤ ''a''. Otherwise they are ''incomparable''. For example, $\backslash $ and $\backslash $ are comparable, while $\backslash $ and $\backslash $ are not.
* A ''total order
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

'' or ''linear 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.
* A ''chain
A chain is a wikt:series#Noun, serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression (physics), compression but line (g ...

'' is a subset of a poset that is a totally ordered set. For example, $\backslash $ is a chain.
* An ''antichain In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

'' is a subset of a poset in which no two distinct elements are comparable. For example, the set of singletons $\backslash .$
* An element ''a'' is said to be ''strictly less than'' an element ''b'', if ''a'' ≤ ''b'' and $a\; \backslash neq\; b.$ For example, $\backslash $ is strictly less than $\backslash .$
* An element ''a'' is said to be ''covered
Cover or covers may refer to:
Packaging
* Another name for a lid
* Cover (philately), generic term for envelope or package
* Album cover, the front of the packaging
* Book cover or magazine cover
* CD and DVD cover, CD and DVD packaging
* Smar ...

'' by another element ''b'', written ''a'' ⋖ ''b'' (or ''a'' <: ''b''), if ''a'' is strictly less than ''b'' and no third element ''c'' fits between them; formally: if both ''a'' ≤ ''b'' and $a\; \backslash neq\; b$ are true, and ''a'' ≤ ''c'' ≤ ''b'' is false for each ''c'' with $a\; \backslash neq\; c\; \backslash neq\; b.$ Using the strict order <, the relation ''a'' ⋖ ''b'' can be equivalently rephrased as "''a'' < ''b'' but not ''a'' < ''c'' < ''b'' for any ''c''". For example, $\backslash $ is covered by$\backslash ,$ but is not covered by $\backslash .$
Extrema

There are several notions of "greatest" and "least" element in a poset $P,$ notably: *Greatest element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined duality (order theory), dually, ...

and least element: An element $g\; \backslash in\; P$ is a if for every element $a\; \backslash in\; P,\; a\; \backslash leq\; g.$ An element $m\; \backslash in\; P$ is a if for every element $a\; \backslash in\; P,\; m\; \backslash leq\; a.$ A poset can only have one greatest or least element. In our running example, the set $\backslash $ is the greatest element, and $\backslash $ is the least.
* Maximal element
In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is define ...

s and minimal elements: An element $g\; \backslash in\; P$ is a maximal element if there is no element $a\; \backslash in\; P$ such that $a\; >\; g.$ Similarly, an element $m\; \backslash in\; P$ is a minimal element if there is no element $a\; \backslash in\; P$ 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, $\backslash $ and $\backslash $ are the maximal and minimal elements. Removing these, there are 3 maximal elements and 3 minimal elements (see Fig.6).
* Upper and lower bounds
In mathematics, particularly in order theory
Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than ...

: For a subset ''A'' of ''P'', an element ''x'' in ''P'' is an upper bound of ''A'' if ''a'' ≤ ''x'', for each element ''a'' in ''A''. In particular, ''x'' need not be in ''A'' to be an upper bound of ''A''. Similarly, an element ''x'' in ''P'' is a lower bound of ''A'' if ''a'' ≥ ''x'', for each element ''a'' in ''A''. A greatest element of ''P'' is an upper bound of ''P'' itself, and a least element is a lower bound of ''P''. In our example, the set $\backslash $ is an for the collection of elements $\backslash .$
As another example, consider the positive integer
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s, ordered by divisibility: 1 is a least element, as it divides
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

all other elements; on the other hand this poset does not have a greatest element (although if one would include 0 in the poset, which is a multiple of any integer, that would be a greatest element; see Fig.7). This partially ordered set does not even have any maximal elements, since any ''g'' divides for instance 2''g'', which is distinct from it, so ''g'' is not maximal. If the number 1 is excluded, while keeping divisibility as ordering on the elements greater than 1, then the resulting poset does not have a least element, but any prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

is a minimal element for it. In this poset, 60 is an upper bound (though not a least upper bound) of the subset $\backslash ,$ 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.
Mappings between partially ordered sets

Given two partially ordered sets (''S'', ≤) and (''T'', ≼), a function $f\; :\; S\; \backslash to\; T$ is calledorder-preserving
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

, or monotone, or isotone, if for all $x,\; y\; \backslash in\; S,$ $x\; \backslash leq\; y$ implies ''f''(''x'') ≼ ''f''(''y'').
If (''U'', ≲) is also a partially ordered set, and both $f\; :\; S\; \backslash to\; T$ and $g\; :\; T\; \backslash to\; U$ are order-preserving, their composition
Composition or Compositions may refer to:
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Dance is a performing art art form, form consisting of sequences of movement, either improvised or pur ...

$g\; \backslash circ\; f\; :\; S\; \backslash to\; U$ is order-preserving, too.
A function $f\; :\; S\; \backslash to\; T$ is called order-reflecting if for all $x,\; y\; \backslash in\; S,$ ''f''(''x'') ≼ ''f''(''y'') implies $x\; \backslash leq\; y.$
If $f$ is both order-preserving and order-reflecting, then it is called an order-embedding of (''S'', ≤) into (''T'', ≼).
In the latter case, $f$ is necessarily injective
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, since $f(x)\; =\; f(y)$ implies $x\; \backslash leq\; y\; \backslash text\; y\; \backslash leq\; x$ and in turn $x\; =\; y$ according to the antisymmetry of $\backslash leq.$ If an order-embedding between two posets ''S'' and ''T'' exists, one says that ''S'' can be embedded into ''T''. If an order-embedding $f\; :\; S\; \backslash to\; T$ is bijective
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, it is called an order isomorphism In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

, and the partial orders (''S'', ≤) and (''T'', ≼) are said to be isomorphic. Isomorphic orders have structurally similar Hasse diagram
In order theory
Order theory is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they ...

s (see Fig.8). It can be shown that if order-preserving maps $f\; :\; S\; \backslash to\; T$ and $g\; :\; T\; \backslash to\; U$ exist such that $g\; \backslash circ\; f$ and $f\; \backslash circ\; g$ yields the identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function (mathematics), function that always returns the same value that was ...

on ''S'' and ''T'', respectively, then ''S'' and ''T'' are order-isomorphic.
For example, a mapping $f\; :\; \backslash N\; \backslash to\; \backslash mathbb(\backslash N)$ from the set of natural numbers (ordered by divisibility) to the power set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of natural numbers (ordered by set inclusion) can be defined by taking each number to the set of its prime divisor
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

s. It is order-preserving: if $x$ divides $y,$ then each prime divisor of $x$ is also a prime divisor of $y.$ However, it is neither injective (since it maps both 12 and 6 to $\backslash $) nor order-reflecting (since 12 does not divide 6). Taking instead each number to the set of its prime power
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

divisors defines a map $g\; :\; \backslash N\; \backslash to\; \backslash mathbb(\backslash N)$ 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 $\backslash $), but it can be made one by restricting its codomain to $g(\backslash N).$ Fig.9 shows a subset of $\backslash N$ and its isomorphic image under $g.$ The construction of such an order-isomorphism into a power set can be generalized to a wide class of partial orders, called distributive lattice
In mathematics, a distributive lattice is a lattice (order), lattice in which the operations of join and meet distributivity, distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice ope ...

s, see "Birkhoff's representation theorem
:''This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).''
In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite set, finite distributi ...

".
Number of partial orders

Sequence inOEIS
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...

gives the number of partial orders on a set of ''n'' labeled elements:
The number of strict partial orders is the same as that of partial orders.
If the count is made only up to Two mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

isomorphism, the sequence 1, 1, 2, 5, 16, 63, 318, ... is obtained.
Linear extension

A partial order $\backslash leq^*$ on a set $X$ is an extension of another partial order $\backslash leq$ on $X$ provided that for all elements $x,\; y\; \backslash in\; X,$ whenever $x\; \backslash leq\; y,$ it is also the case that $x\; \backslash leq^*\; y.$ Alinear extension
In order theory
Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". ...

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).
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of , , and . Computer science ...

, algorithms for finding linear extensions of partial orders (represented as the reachability
In graph theory
In mathematics, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph the ...

orders of directed acyclic graph
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s) are called topological sorting
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of , , a ...

.
Directed acyclic graphs

Strict partial orders correspond directly todirected acyclic graph
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s (DAGs). If a graph is constructed by taking each element of $P$ to be a node and each element of $\backslash leq$ to be an edge, then every strict partial order is a DAG, and the transitive closure
In mathematics, the transitive closure of a binary relation ''R'' on a set (mathematics), set ''X'' is the smallest relation on ''X'' that contains ''R'' and is Transitive relation, transitive. For finite sets, "smallest" can be taken in its usua ...

of a DAG is both a strict partial order and also a DAG itself. In contrast a non-strict partial order would have self loops at every node and therefore not be a DAG.
In category theory

Every poset (and everypreordered set
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive relation, reflexive and Transitive relation, transitive. Preorders are more general than equivalence relations and (non-strict) partial ...

) may be considered as a category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

where, for objects $x$ and $y,$ there is at most one morphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

from $x$ to $y.$ More explicitly, let hom(''x'', ''y'') = if ''x'' ≤ ''y'' (and otherwise the empty set) and $(y,\; z)\; \backslash circ\; (x,\; y)\; =\; (x,\; z).$ Such categories are sometimes called '' posetal''.
Posets are equivalent
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*''Equivalent ...

to one another if and only if they are isomorphic
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

. In a poset, the smallest element, if it exists, is an initial object
In category theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...

, and the largest element, if it exists, is a terminal object
In category theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...

. Also, every preordered set is equivalent to a poset. Finally, every subcategory of a poset is isomorphism-closed.
Partial orders in topological spaces

If $P$ is a partially ordered set that has also been given the structure of atopological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

, then it is customary to assume that $\backslash $ is a closed subset of the topological product space
Product may refer to:
Business
* Product (business)
In marketing, a product is an object or system made available for consumer use; it is anything that can be offered to a Market (economics), market to satisfy the desire or need of a customer ...

$P\; \backslash times\; P.$ Under this assumption partial order relations are well behaved at limits
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* Limit (manga), ''Limit'' (manga), a manga by Keiko Suenobu
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* Limits (song), "Limits" (song), ...

in the sense that if $a\_i\; \backslash to\; a,$ and $b\_i\; \backslash to\; b,$ and for all $i$ $a\_i\; \backslash leq\; b\_i,$ then $a\; \backslash leq\; b.$
Intervals

An ''interval'' in a poset ''P'' is a subset of ''P'' with the property that, for any ''x'' and ''y'' in and any ''z'' in ''P'', if ''x'' ≤ ''z'' ≤ ''y'', then ''z'' is also in . (This definition generalizes the '' interval'' definition for real numbers.) For ''a'' ≤ ''b'', theclosed interval
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

is the set of elements ''x'' satisfying ''a'' ≤ ''x'' ≤ ''b'' (that is, ''a'' ≤ ''x'' and ''x'' ≤ ''b''). It contains at least the elements ''a'' and ''b''.
Using the corresponding strict relation "<", the open interval
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

is the set of elements ''x'' satisfying ''a'' < ''x'' < ''b'' (i.e. ''a'' < ''x'' and ''x'' < ''b''). An open interval may be empty even if ''a'' < ''b''. For example, the open interval on the integers is empty since there are no integers such that .
The ''half-open intervals'' and are defined similarly.
Sometimes the definitions are extended to allow ''a'' > ''b'', in which case the interval is empty.
An interval is bounded if there exist elements $a,\; b\; \backslash in\; P$ such that . Every interval that can be represented in interval notation is obviously bounded, but the converse is not true. For example, let as a subposet of the real numbers
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

. The subset is a bounded interval, but it has no infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to all elements of S, if such an element exists. Consequently, the term ''greatest low ...

or supremum
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

in ''P'', so it cannot be written in interval notation using elements of ''P''.
A poset is called locally finite if every bounded interval is finite. For example, the integers
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power ...

are locally finite under their natural ordering. The lexicographical order on the cartesian product $\backslash N\; \backslash times\; \backslash N$ is not locally finite, since .
Using the interval notation, the property "''a'' is covered by ''b''" can be rephrased equivalently as $$, b
The comma is a punctuation
Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...

= \.
This concept of an interval in a partial order should not be confused with the particular class of partial orders known as the interval order
In mathematics, especially order theory,
the interval order for a collection of intervals on the real line
is the partial order corresponding to their left-to-right precedence relation—one interval, ''I''1, being considered less than another, '' ...

s.
See also

*Antimatroid
In mathematics, an antimatroid is a formal system that describes processes in which a set (mathematics), set is built up by including elements one at a time, and in which an element, once available for inclusion, remains available until it is inclu ...

, a formalization of orderings on a set that allows more general families of orderings than posets
* Causal set
The causal sets program is an approach to quantum gravity. Its founding principles are that spacetime is fundamentally discrete (a collection of discrete spacetime points, called the elements of the causal set) and that spacetime events are related ...

, a poset-based approach to quantum gravity
*
*
*
*
*
*
*
*
* Nested Set Collection
*
*
*
*
* Poset topologyIn mathematics, the poset topology associated to a poset (''S'', ≤) is the Alexandrov topology (open sets are upper sets) on the poset of finite Total_order#Chains, chains of (''S'', ≤), ordered by inclusion.
Let ''V'' be a set of vertices. An ...

, a kind of topological space that can be defined from any poset
* Scott continuity In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

– continuity of a function between two partial orders.
*
*
*
* Strict weak ordering
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

– strict partial order "<" in which the relation is transitive.
*
* Tree
In botany, a tree is a perennial plant with an elongated Plant stem, stem, or trunk (botany), trunk, supporting branches and leaves in most species. In some usages, the definition of a tree may be narrower, including only wood plants with se ...

– Data structure of set inclusion
*
Notes

Citations

References

* * * * *External links

* * {{Authority control Order theory Binary relations de:Ordnungsrelation#Halbordnung