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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, especially
order theory Order theory is a branch of mathematics that 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". This article intr ...
, a partial order on a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' 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 generalize
total order In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( re ...
s, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an
ordered pair In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unord ...
P=(X,\leq) consisting of a set X (called the ''ground set'' of P) and a partial order \leq on X. When the meaning is clear from context and there is no ambiguity about the partial order, the set X itself is sometimes called a poset.


Partial order relations

The term ''partial order'' usually refers to the reflexive partial order relations, referred to in this article as ''non-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 a
one-to-one correspondence In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivale ...
, so for every strict partial order there is a unique corresponding non-strict partial order, and vice versa.


Partial orders

A reflexive, weak, or , commonly referred to simply as a partial order, is a
homogeneous relation In mathematics, a homogeneous relation (also called endorelation) on a set ''X'' is a binary relation between ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation ...
≤ on a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
P that is reflexive, antisymmetric, and transitive. That is, for all a, b, c \in P, it must satisfy: # Reflexivity: a \leq a, i.e. every element is related to itself. #
Antisymmetry In linguistics, antisymmetry, is a theory of syntax described in Richard S. Kayne's 1994 book ''The Antisymmetry of Syntax''. Building upon X-bar theory, it proposes a universal, fundamental word order for phrases (Branching (linguistics), branchin ...
: if a \leq b and b \leq a then a = b, i.e. no two distinct elements precede each other. # Transitivity: if a \leq b and b \leq c then a \leq c. A non-strict partial order is also known as an antisymmetric
preorder In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive relation, reflexive and Transitive relation, transitive. The name is meant to suggest that preorders are ''almost'' partial orders, ...
.


Strict partial orders

An irreflexive, strong, or is a homogeneous relation < on a set P that is
irreflexive In mathematics, a binary relation R on a set X is reflexive if it relates every element of X to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal to itself. A ...
, asymmetric and transitive; that is, it satisfies the following conditions for all a, b, c \in P: # Irreflexivity: \neg\left( a < a \right), i.e. no element is related to itself (also called anti-reflexive). #
Asymmetry Asymmetry is the absence of, or a violation of, symmetry (the property of an object being invariant to a transformation, such as reflection). Symmetry is an important property of both physical and abstract systems and it may be displayed in pre ...
: if a < b then not b < a. # Transitivity: if a < b and b < c then a < c. 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 an asymmetric 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 \leq may be converted to a strict partial order by removing all relationships of the form a \leq a; that is, the strict partial order is the set < \; := \ \leq\ \setminus \ \Delta_P where \Delta_P := \ is the
identity relation In mathematics, a homogeneous relation (also called endorelation) on a set ''X'' is a binary relation between ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation ...
on P \times P and \;\setminus\; denotes
set subtraction In set theory, the complement of a set , often denoted by A^c (or ), is the set of elements not in . When all elements in the universe, i.e. all elements under consideration, are considered to be members of a given set , the absolute complement ...
. Conversely, a strict partial order < on P may be converted to a non-strict partial order by adjoining all relationships of that form; that is, \leq\; := \;\Delta_P\; \cup \;<\; is a non-strict partial order. Thus, if \leq is a non-strict partial order, then the corresponding strict partial order < is the irreflexive kernel given by a < b \text a \leq b \text a \neq b. Conversely, if < is a strict partial order, then the corresponding non-strict partial order \leq is the
reflexive closure In mathematics, the reflexive closure of a binary relation R on a set X is the smallest reflexive relation on X that contains R, i.e. the set R \cup \. For example, if X is a set of distinct numbers and x R y means "x is less than y", then the refl ...
given by: a \leq b \text a < b \text a = b.


Dual orders

The ''dual'' (or ''opposite'') R^ of a partial order relation R is defined by letting R^ be the
converse relation In mathematics, the converse of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent of'. In formal terms ...
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

Given a set P and a partial order relation, typically the non-strict partial order \leq, we may uniquely extend our notation to define four partial order relations \leq, <, \geq, and >, where \leq is a non-strict partial order relation on P, < is the associated strict partial order relation on P (the irreflexive kernel of \leq), \geq is the dual of \leq, and > is the dual of < . Strictly speaking, the term ''partially ordered set'' refers to a set with all of these relations defined appropriately. But practically, one need only consider a single relation, (P,\leq) or (P,<), or, in rare instances, the non-strict and strict relations together, (P,\leq,<). The term ''ordered set'' is sometimes used as a shorthand for ''partially 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 \leq such as \sqsubseteq or \preceq to distinguish partial orders from total orders. When referring to partial orders, \leq should not be taken as the complement of > . The relation > is the converse of the irreflexive kernel of \leq, which is always a subset of the complement of \leq, but > is equal to the complement of \leq if, and only if, \leq is a total order.


Alternative definitions

Another way of defining a partial order, found in
computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
, is via a notion of
comparison 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 ...
. Specifically, given \leq, <, \geq, \text > as defined previously, it can be observed that 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 , or , or , or ''x'' and ''y'' are ''incomparable''. This can be represented by a function \text: P \times P \to \ that returns one of four codes when given two elements. This definition is equivalent to a ''partial order on a setoid'', where equality is taken to be a defined
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...
rather than set equality. Wallis defines a more general notion of a ''partial order relation'' as any
homogeneous relation In mathematics, a homogeneous relation (also called endorelation) on a set ''X'' is a binary relation between ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation ...
that is transitive and antisymmetric. This includes both reflexive and irreflexive partial orders as subtypes. A finite poset can be visualized through its Hasse diagram. Specifically, taking a strict partial order relation (P,<), a
directed acyclic graph In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one ...
(DAG) may be constructed by taking each element of P to be a node and each element of < to be an edge. The transitive reduction of this DAG is then the Hasse diagram. Similarly this process can be reversed to construct strict partial orders from certain DAGs. In contrast, the graph associated to a non-strict partial order has self-loops at every node and therefore is not a DAG; when a non-strict order is said to be depicted by a Hasse diagram, actually the corresponding strict order is shown.


Examples

Standard examples of posets arising in mathematics include: * The
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s, or in general any totally ordered set, ordered by the standard ''less-than-or-equal'' relation ≤, is a partial order. * On the real numbers \mathbb, 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. The main types of inequality ar ...
relation < is a strict partial order. The same is also true of the usual greater than relation > on \R. * By definition, every
strict weak order In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other. Weak orders are a generalization of totally ordered set ...
is a strict partial order. * The set of
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
s of a given set (its
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
) ordered by
inclusion Inclusion or Include may refer to: Sociology * Social inclusion, action taken to support people of different backgrounds sharing life together. ** Inclusion (disability rights), promotion of people with disabilities sharing various aspects of lif ...
(see Fig. 1). Similarly, the set of
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
s ordered by
subsequence In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...
, 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), a Canadian anim ...
s ordered by substring. * The set of
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
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 (mathematics), multiple'' of m. An integer n is divis ...
. (see Fig. 3 and Fig. 6) * The vertex set of a
directed acyclic graph In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one ...
ordered by
reachability In graph theory, reachability refers to the ability to get from one vertex to another within a graph. A vertex s can reach a vertex t (and t is reachable from s) if there exists a sequence of adjacent vertices (i.e. a walk) which starts with s a ...
. * The set of subspaces of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
ordered by inclusion. * For a partially ordered set ''P'', the
sequence space In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural num ...
containing all
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
s of elements from ''P'', where sequence ''a'' precedes sequence ''b'' if every item in ''a'' precedes the corresponding item in ''b''. Formally, \left(a_n\right)_ \leq \left(b_n\right)_ if and only if a_n \leq b_n for all n \in \N; that is, a componentwise order. * For a set ''X'' and a partially ordered set ''P'', the
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...
containing all functions from ''X'' to ''P'', where if and only if for all x \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 net (textile), netting. A fence differs from a wall in not having a solid foundation along its ...
, a partially ordered set defined by an alternating sequence of order relations * The set of events in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
and, in most cases,
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, where for two events ''X'' and ''Y'', if and only if ''Y'' is in the future
light cone In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single Event (relativity), event (localized to a single point in space and a single moment in time) and traveling in all direct ...
of ''X''. An event ''Y'' can be causally affected by ''X'' only if . One familiar example of a partially ordered set is a collection of people ordered by
genealogical Genealogy () 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 obtain information about a family and to demonstrate kin ...
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 the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...
of two partially ordered sets are (see Fig. 4): * the
lexicographical order In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a ...
:   if or ( and ); * the
product order In mathematics, given partial orders \preceq and \sqsubseteq on sets A and B, respectively, the product order (also called the coordinatewise order or componentwise order) is a partial order \leq on the Cartesian product A \times B. Given two pa ...
:   (''a'', ''b'') ≤ (''c'', ''d'') if ''a'' ≤ ''c'' and ''b'' ≤ ''d''; * the
reflexive closure In mathematics, the reflexive closure of a binary relation R on a set X is the smallest reflexive relation on X that contains R, i.e. the set R \cup \. For example, if X is a set of distinct numbers and x R y means "x is less than y", then the refl ...
of the
direct product In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abs ...
of the corresponding strict orders:   if ( and ) or ( and ). 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 (disjoint) posets is the ordinal sum (or linear sum), , defined on the union of the underlying sets ''X'' and ''Y'' by the order 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.
Series-parallel partial order In order theory, order-theoretic mathematics, a series-parallel partial order is a partially ordered set built up from smaller series-parallel partial orders by two simple composition operations... The series-parallel partial orders may be charact ...
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, the disjoint union (or discriminated union) A \sqcup B of the sets and is the set formed from the elements of and labelled (indexed) with the name of the set from which they come. So, an element belonging to both and appe ...
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 (\mathcal(\),\subseteq) consisting of the set of all subsets of a three-element set \, 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 be
symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. For example, \ is related to \, but not the reverse. * ''a'' and ''b'' are '' comparable'' if or . Otherwise they are ''incomparable''. For example, \ and \ are comparable, while \ and \ are not. * A ''
total order In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( re ...
'' 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 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 but linear, rigid, and load-bearing in tension. A ...
'' is a subset of a poset that is a totally ordered set. For example, \ is a chain. * An '' antichain'' is a subset of a poset in which no two distinct elements are comparable. For example, the set of singletons \. * An element ''a'' is said to be ''strictly less than'' an element ''b'', if ''a'' ≤ ''b'' and a \neq b. For example, \ is strictly less than \. * 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 ** Book design ** Back cover copy, part of ...
'' 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 \neq b are true, and ''a'' ≤ ''c'' ≤ ''b'' is false for each ''c'' with a \neq c \neq b. Using the strict order <, the relation ''a'' ⋖ ''b'' can be equivalently rephrased as " but not for any ''c''". For example, \ is covered by \, but is not covered by \.


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 \in P is a if a \leq g for every element a \in P. An element m \in P is a if m \leq a for every element a \in P. A poset can only have one greatest or least element. In our running example, the set \ is the greatest element, and \ 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 defined dually as an ...
s 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. 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 In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is every element of . Dually, a lower bound or minorant of is defined to be an element of that is less th ...
: 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 \ is an for the collection of elements \. As another example, consider the positive
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s, ordered by divisibility: 1 is a least element, as it
divides 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 divisibl ...
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 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 \, which does not have any lower bound (since 1 is not in the poset); on the other hand 2 is a lower bound of the subset of powers of 2, which does not have any upper bound. If the number 0 is included, this will be the greatest element, since this is a multiple of every integer (see Fig. 6).


Mappings between partially ordered sets

Given two partially ordered sets and , a function f : S \to T is called
order-preserving In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
, or monotone, or isotone, if for all x, y \in S, x \leq y implies . If is also a partially ordered set, and both f : S \to T and g : T \to U are order-preserving, their
composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography * Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...
g \circ f : S \to U is order-preserving, too. A function f : S \to T is called order-reflecting if for all x, y \in S, implies x \leq y. If is both order-preserving and order-reflecting, then it is called an order-embedding of into . In the latter case, is necessarily
injective In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
, since f(x) = f(y) implies x \leq y \text y \leq x and in turn x = y according to the antisymmetry of \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 \to T is
bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
, it is called an
order isomorphism In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be co ...
, and the partial orders and are said to be isomorphic. Isomorphic orders have structurally similar Hasse diagrams (see Fig. 7a). It can be shown that if order-preserving maps f : S \to T and g : T \to U exist such that g \circ f and f \circ g yields 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 that always returns the value that was used as its argument, unc ...
on ''S'' and ''T'', respectively, then ''S'' and ''T'' are order-isomorphic. For example, a mapping f : \N \to \mathbb(\N) from the set of natural numbers (ordered by divisibility) to the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
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 divides , then each prime divisor of is also a prime divisor of . 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 its
prime power In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 1 ...
divisors defines a map g : \N \to \mathbb(\N) that is order-preserving, order-reflecting, and hence an order-embedding. It is not an order-isomorphism (since it, for instance, does not map any number to the set \), but it can be made one by restricting its codomain to g(\N). Fig. 7b shows a subset of \N and its isomorphic image under . 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 o ...
s; see '' Birkhoff's representation theorem''.


Number of partial orders

Sequence A001035in
OEIS The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to th ...
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 object, mathematical objects and are called "equal up to an equivalence relation " * if and are related by , that is, * if holds, that is, * if the equivalence classes of and with respect to are equal. This figure of speech ...
isomorphism, the sequence 1, 1, 2, 5, 16, 63, 318, ... is obtained.


Subposets

A poset P^*=(X^*, \leq^*) is called a subposet of another poset P=(X, \leq) provided that X^* is a
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of X and \leq^* is a subset of \leq. The latter condition is equivalent to the requirement that for any x and y in X^* (and thus also in X), if x \leq^* y then x \leq y. If P^* is a subposet of P and furthermore, for all x and y in X^*, whenever x \leq y we also have x \leq^* y, then we call P^* the subposet of P induced by X^*, and write P^* = P ^*/math>.


Linear extension

A partial order \leq^* on a set X is called an extension of another partial order \leq on X provided that for all elements x, y \in X, whenever x \leq y, it is also the case that x \leq^* y. A linear 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). In
computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
, algorithms for finding linear extensions of partial orders (represented as the
reachability In graph theory, reachability refers to the ability to get from one vertex to another within a graph. A vertex s can reach a vertex t (and t is reachable from s) if there exists a sequence of adjacent vertices (i.e. a walk) which starts with s a ...
orders of
directed acyclic graph In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one ...
s) are called topological sorting.


In category theory

Every poset (and every preordered set) may be considered as a
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
where, for objects x and y, there is at most one
morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
from x to y. More explicitly, let if (and otherwise the
empty set In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
) and (y, z) \circ (x, y) = (x, z). Such categories are sometimes called '' posetal''. Posets are equivalent to one another if and only if they are
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
. In a poset, the smallest element, if it exists, is an
initial object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element) ...
, and the largest element, if it exists, is a
terminal object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...
. 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 In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, then it is customary to assume that \ is a closed subset of the topological
product space In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...
P \times P. Under this assumption partial order relations are well behaved at
limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2009 ...
in the sense that if \lim_ a_i = a, and \lim_ b_i = b, and for all i, a_i \leq b_i, then a \leq b.


Intervals

A convex set 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 definition of intervals of
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s. When there is possible confusion with
convex set In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...
s of
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, one uses order-convex instead of "convex". A convex sublattice of a lattice ''L'' is a sublattice of ''L'' that is also a convex set of ''L''. Every nonempty convex sublattice can be uniquely represented as the intersection of a filter and an ideal of ''L''. An interval in a poset ''P'' is a subset that can be defined with interval notation: * For ''a'' ≤ ''b'', the ''closed interval'' is the set of elements ''x'' satisfying (that is, and ). It contains at least the elements ''a'' and ''b''. * Using the corresponding strict relation "<", the ''open interval'' is the set of elements ''x'' satisfying (i.e. and ). An open interval may be empty even if . For example, the open interval on the integers is empty since there is no integer such that . * The ''half-open intervals'' and are defined similarly. Whenever 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 is convex, but not an interval. An interval is bounded if there exist elements a, b \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. The subset is a bounded interval, but it has no
infimum In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique ...
or
supremum In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
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 \times \N is not locally finite, since . Using the interval notation, the property "''a'' is covered by ''b''" can be rephrased equivalently as , 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 * * * * * * * * *
Nested set collection A nested set collection or nested set family is a collection of sets that consists of chains of subsets forming a hierarchical structure, like Matryoshka doll, Russian dolls. It is used as reference concept in hierarchy, scientific hierarchy def ...
* * * * * Poset topology, a kind of topological space that can be defined from any poset *
Scott continuity In mathematics, given two partially ordered sets ''P'' and ''Q'', a function ''f'': ''P'' → ''Q'' between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema. That is, for every directed su ...
– continuity of a function between two partial orders. * * * Szpilrajn extension theorem – every partial order is contained in some total order. * *
Strict weak ordering In mathematics, especially order theory, a weak ordering is a mathematical formalization of the intuitive notion of a ranking of a set, some of whose members may be tied with each other. Weak orders are a generalization of totally ordered s ...
– strict partial order "<" in which the relation is transitive. * *


Notes


Citations


References

* * * * * * *


External links

; each of which shows an example for a partial order * * {{Authority control Order theory Binary relations