right|thumb|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. Some other pairs, such as and , are also incomparable.]]
In [[mathematics]], especially [[order theory]], 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 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 orders, in which every pair is comparable.
Formally, a partial order is any binary relation that is
reflexive (each element is comparable to itself),
antisymmetric (no two different elements precede each other), and
transitive (the start of a chain of precedence relations must precede the end of the chain).
One familiar example of a partially ordered set is a collection of people ordered by
genealogical descendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs of people are incomparable, with neither being a descendant of the other.
A poset can be visualized through its
Hasse diagram, which depicts the ordering relation.
Formal definition
A (non-strict) partial order is a
homogeneous binary relation ≤ over a
set ''P'' satisfying particular axioms which are discussed below. When ''a'' ≤ ''b'', we say that ''a'' is related to ''b''. (This does not imply that ''b'' is also related to ''a'', because the relation need not be
symmetric.)
The axioms for a non-strict partial order state that the relation ≤ is
reflexive,
antisymmetric, and
transitive. That is, for all ''a'', ''b'', and ''c'' in ''P'', it must satisfy:
# ''a'' ≤ ''a'' (
reflexivity: every element is related to itself).
# if ''a'' ≤ ''b'' and ''b'' ≤ ''a'', then ''a'' = ''b'' (
antisymmetry: two distinct elements cannot be related in both directions).
# if ''a'' ≤ ''b'' and ''b'' ≤ ''c'', then ''a'' ≤ ''c'' (
transitivity: if a first element is related to a second element, and, in turn, that element is related to a third element, then the first element is related to the third element).
In other words, a partial order is an antisymmetric
preorder.
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.
For ''a'', ''b'', elements of a partially ordered set ''P'', if ''a'' ≤ ''b'' or ''b'' ≤ ''a'', then ''a'' and ''b'' are
comparable. Otherwise they are incomparable. In the figure on top-right, e.g. and are comparable, while and are not. A partial order under which every pair of elements is comparable is called a
total order or linear order; a totally ordered set is also called a chain (e.g., the natural numbers with their standard order). A subset of a poset in which no two distinct elements are comparable is called an
antichain (e.g. the set of
singletons in the top-right figure). An element ''a'' is said to be strictly less than an element ''b'', if ''a'' ≤ ''b'' and ''a'' ≠ ''b''. An element ''a'' is said to be
covered 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'' ≠ ''b'' are true, and ''a'' ≤ ''c'' ≤ ''b'' is false for each ''c'' with ''a'' ≠ ''c'' ≠ ''b''. A more concise definition will be given
below using the strict order corresponding to "≤". For example, is covered by in the top-right figure, but not by .
Examples
Standard examples of posets arising in mathematics include:
* The
real numbers ordered by the standard ''less-than-or-equal'' relation ≤ (a totally ordered set as well).
* The set of
subsets of a given set (its
power set) ordered by
inclusion (see the figure on top-right). Similarly, the set of
sequences ordered by
subsequence, and the set of
strings ordered by
substring.
* The set of
natural numbers equipped with the relation of
divisibility.
* The vertex set of a
directed acyclic graph ordered by
reachability.
* The set of
subspaces of a
vector space ordered by inclusion.
* For a partially ordered set ''P'', the
sequence space containing all
sequences of elements from ''P'', where sequence ''a'' precedes sequence ''b'' if every item in ''a'' precedes the corresponding item in ''b''. Formally, if and only if for all ''n'' in ℕ, i.e. a
componentwise order.
* For a set ''X'' and a partially ordered set ''P'', the
function space containing all functions from ''X'' to ''P'', where ''f'' ≤ ''g'' if and only if ''f''(''x'') ≤ ''g''(''x'') for all ''x'' in ''X''.
* A
fence, a partially ordered set defined by an alternating sequence of order relations ''a'' < ''b'' > ''c'' < ''d'' ...
* The set of events in
special relativity and, in most cases,
general relativity, where for two events ''X'' and ''Y'', ''X'' ≤ ''Y'' if and only if ''Y'' is in the future
light cone of ''X''. An event ''Y'' can only be causally affected by ''X'' if ''X'' ≤ ''Y''.
Extrema
There are several notions of "greatest" and "least" element in a poset ''P'', notably:
*
Greatest element and least element: An element ''g'' in ''P'' is a greatest element if for every element ''a'' in ''P'', ''a'' ≤ ''g''. An element ''m'' in ''P'' is a least element if for every element ''a'' in ''P'', ''a'' ≥ ''m''. A poset can only have one greatest or least element.
*
Maximal elements and minimal elements: An element ''g'' in P is a maximal element if there is no element ''a'' in ''P'' such that ''a'' > ''g''. Similarly, an element ''m'' in ''P'' is a minimal element if there is no element ''a'' in P such that ''a'' < ''m''. If a poset has a greatest element, it must be the unique maximal element, but otherwise there can be more than one maximal element, and similarly for least elements and minimal elements.
*
Upper and lower bounds: 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''.
For example, consider the
positive integers, ordered by divisibility: 1 is a least element, as it
divides 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 figure). 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 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.
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 the
Cartesian product of two partially ordered sets are (see figures):
*the
lexicographical order: (''a'',''b'') ≤ (''c'',''d'') if ''a'' < ''c'' or (''a'' = ''c'' and ''b'' ≤ ''d'');
*the
product order: (''a'',''b'') ≤ (''c'',''d'') if ''a'' ≤ ''c'' and ''b'' ≤ ''d'';
*the
reflexive closure of the
direct product 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 spaces over the same
field, 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 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'' ≤
''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
well-ordered, then so is their ordinal sum.
The ordinal sum operation is one of two operations used to form
series-parallel partial orders, and in this context is called series composition. The other operation used to form these orders, the
disjoint union of two partially ordered sets (with no order relation between elements of one set and elements of the other set) is called in this context parallel composition.
Strict and non-strict partial orders
A partial order may be referred to as a non-strict (or reflexive) partial order for emphasis or contrast to
strict partial orders.
A strict (or irreflexive) partial order — also known as a
strict preorder — on
is a
homogeneous binary relation
on
that is
irreflexive,
transitive and
asymmetric; that is, it satisfies the following conditions for all
:
#
Irreflexivity: not ,
#
Transitivity: if and then , and
#
Asymmetry: if then not .
Irreflexivity and transitivity together imply asymmetry,
so a homogeneous relation is a strict partial order if and only if it is transitive and irreflexive. Also, asymmetry implies irreflexivity
[ Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric".] so a homogeneous relation is a strict partial order if and only if it is transitive and asymmetric. In other words, a transitive relation is asymmetric if and only if it is irreflexive.
By definition, every
strict weak order is a strict partial order. On the
real numbers
the usual
less than relation
is a strict partial order and the same is also true of the usual
greater than relation
on
Equivalence
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 the diagonal of
and
denotes
set 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 the
irreflexive kernel given by
:
if
and
Conversely, if
is a strict partial order, then the corresponding non-strict partial order
is the
reflexive closure given by:
:
if
or
This is the reason for using the notation
Using the strict order
the relation "
is
covered by
" can be equivalently rephrased as "
but not
for any
".
Directed acyclic graphs
Strict partial orders correspond more directly to
directed acyclic graphs (DAGs). If a graph is constructed by taking each element of
to be a node and each element of
to be an edge, then every strict partial order is a DAG, and the
transitive closure 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.
Inverse and order dual
The inverse (or converse) of a partial order relation ≤ is the
converse of ≤. Typically denoted ≥, it is the relation that satisfies ''x'' ≥ ''y'' if and only if ''y'' ≤ ''x''. The inverse of a partial order relation is reflexive, transitive, and antisymmetric, and hence itself a partial order relation. The
order dual of a partially ordered set is the same set with the partial order relation replaced by its inverse. The irreflexive relation > is to ≥ as < is to ≤.
Any one of the four relations ≤, <, ≥, and > on a given set uniquely determines the other three.
In general two elements ''x'' and ''y'' of a partial order may stand in any of four mutually exclusive relationships to each other: either ''x'' < ''y'', or ''x'' = ''y'', or ''x'' > ''y'', or ''x'' and ''y'' are ''
incomparable'' (none of the other three, the definition of incomparability here is different from that in
weak ordering). A
totally ordered set is one that rules out this fourth possibility: all pairs of elements are comparable and we then say that
trichotomy holds. The
natural numbers, the
integers, the
rationals, and the
reals are all totally ordered by their algebraic (signed) magnitude whereas the
complex numbers are not. This is not to say that the complex numbers cannot be totally ordered; we could for example order them lexicographically via ''x'' + i''y'' < ''u'' + i''v'' if and only if ''x'' < ''u'' or (''x'' = ''u'' and ''y'' < ''v''), but this is not ordering by magnitude in any reasonable sense as it makes 1 greater than 100i. Ordering them by absolute magnitude yields a preorder in which all pairs are comparable, but this is not a partial order since 1 and i have the same absolute magnitude but are not equal, violating antisymmetry.
Mappings between partially ordered sets
Given two partially ordered sets (''S'', ≤) and (''T'', ≼), a function ''f'': ''S'' → ''T'' is called
order-preserving, or
monotone, or isotone, if for all ''x'' and ''y'' in ''S'', ''x'' ≤ ''y'' implies ''f''(''x'') ≼ ''f''(''y'').
If (''U'', ≲) is also a partially ordered set, and both ''f'': ''S'' → ''T'' and ''g'': ''T'' → ''U'' are order-preserving, their
composition ''g''∘''f'' : ''S'' → ''U'' is order-preserving, too.
A function ''f'': ''S'' → ''T'' is called order-reflecting if for all ''x'' and ''y'' in ''S'', ''f''(''x'') ≼ ''f''(''y'') implies ''x'' ≤ ''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, since ''f''(''x'') = ''f''(''y'') implies ''x'' ≤ ''y'' and ''y'' ≤ ''x'' and in turn ''x''=''y'' according to the antisymmetry of ≤. 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'' → ''T'' is
bijective, it is called an
order isomorphism, and the partial orders (''S'', ≤) and (''T'', ≼) are said to be isomorphic. Isomorphic orders have structurally similar
Hasse diagrams (cf. right picture). It can be shown that if order-preserving maps ''f'': ''S'' → ''T'' and ''g'': ''T'' → ''S'' exist such that ''g''∘''f'' and ''f''∘''g'' yields the
identity function on ''S'' and ''T'', respectively, then ''S'' and ''T'' are order-isomorphic.
For example, a mapping ''f'': ℕ → ℙ(ℕ) from the set of natural numbers (ordered by divisibility) to the
power set of natural numbers (ordered by set inclusion) can be defined by taking each number to the set of its
prime divisors. 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 ) nor order-reflecting (since 12 doesn't divide 6). Taking instead each number to the set of its
prime power divisors defines a map ''g'': ℕ → ℙ(ℕ) that is order-preserving, order-reflecting, and hence an order-embedding. It is not an order-isomorphism (since it e.g. doesn't map any number to the set ), but it can be made one by
restricting its codomain to ''g''(ℕ). The right picture shows a subset of 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 lattices, see "
Birkhoff's representation theorem".
Number of partial orders
Sequence
A001035in
OEIS 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 isomorphism, the sequence 1, 1, 2, 5, 16, 63, 318, … is obtained.
Linear extension
A partial order ≤
* on a set ''X'' is an extension of another partial order ≤ on ''X'' provided that for all elements ''x'' and ''y'' of ''X'', whenever ''x'' ≤ ''y'', it is also the case that ''x'' ≤
* ''y''. A
linear extension is an extension that is also a linear (i.e., 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, algorithms for finding linear extensions of partial orders (represented as the
reachability orders of
directed acyclic graphs) are called
topological sorting.
In category theory
Every poset (and every
preordered set) may be considered as a
category where, for objects ''x'' and ''y'', there is at most one
morphism from ''x'' to ''y''. More explicitly, let hom(''x'', ''y'') = if ''x'' ≤ ''y'' (and otherwise the empty set) and (''y'', ''z'')∘(''x'', ''y'') = (''x'', ''z''). Such categories are sometimes called ''
posetal''.
Posets are
equivalent to one another if and only if they are
isomorphic. In a poset, the smallest element, if it exists, is an
initial object, and the largest element, if it exists, is a
terminal object. Also, every preordered set is equivalent to a poset. Finally, every subcategory of a poset is
isomorphism-closed.
Partial orders in topological spaces
If ''P'' is a partially ordered set that has also been given the structure of a
topological space, then it is customary to assume that
is a
closed subset of the topological
product space . Under this assumption partial order relations are well behaved at
limits in the sense that if
, and
, and for all
, then
.
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'', the
closed interval is the set of elements ''x'' satisfying ''a'' ≤ ''x'' ≤ ''b'' (i.e. ''a'' ≤ ''x'' and ''x'' ≤ ''b''). It contains at least the elements ''a'' and ''b''.
Using the corresponding strict relation "<", the
open interval 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'' and ''b'' of ''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. The subset is a bounded interval, but it has no
infimum or
supremum 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 are locally finite under their natural ordering. The lexicographical order on the cartesian product ℕ×ℕ is not locally finite, since .
Using the interval notation, the property "''a'' is covered by ''b''" can be rephrased equivalently as
'a'', ''b''= .
This concept of an interval in a partial order should not be confused with the particular class of partial orders known as the
interval orders.
See also
*
Antimatroid, a formalization of orderings on a set that allows more general families of orderings than posets
*
Causal set, a poset-based approach to quantum gravity
*
Comparability graph
*
Complete partial order
*
Directed set
*
Graded poset
*
Incidence algebra
*
lattice
*
Locally finite poset
*
Möbius function on posets
*
Nested Set Collection
*
Order polytope
*
Ordered group
*
Poset topology, a kind of topological space that can be defined from any poset
*
Scott continuity – continuity of a function between two partial orders.
*
Semilattice
*
Semiorder
*
Stochastic dominance
*
Strict weak ordering – strict partial order "<" in which the relation is transitive.
*
Total order
*
Tree (data structure of set inclusion)
*
Zorn's lemma
Notes
References
*
*
*
*
External links
*
* {{OEIS el|1=A000112|2=Number of partially ordered sets ("posets") with n unlabeled elements.
Category:Order theory
Category:Binary relations
de:Ordnungsrelation#Halbordnung