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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.


Informal definition

A partial order defines a notion of comparison. 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, a Hasse diagram (; ) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ''(S, ≤)'' one represents ...
, which depicts the ordering relation.


Partial order relation

A partial order relation is a
homogeneous relation In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''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. 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, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
, 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 a
homogeneous relation In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''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 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 syntactic theory presented in Richard S. Kayne's 1994 monograph ''The Antisymmetry of Syntax''. It asserts that grammatical hierarchies in natural language follow a universal order, namely specifier-head-comple ...
: 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.


Strict partial order

An irreflexive, strong, or is a homogeneous relation < on a set P that is irreflexive, asymmetric, and transitive; that is, it satisfies the following conditions for all a, b, c \in P: # Irreflexivity: not a < a, i.e. no element is related to itself (also called anti-reflexive). # Asymmetry: if a < b then not b < a. # Transitivity: if a < b and b < c then a < c. 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 In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cas ...
.


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) over a set ''X'' is a binary relation over ''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. 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 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 ...
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''. For example, if ''X'' is a set of distinct numbers and ''x R y'' means "''x'' is less than ''y''", the ...
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 relation, or transpose, 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&n ...
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,\leq,<), where \leq is a non-strict partial order relation on P, < is the associated strict partial order relation on P (the
irreflexive kernel 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 ...
of \leq), \geq is the dual of \leq, and > is the dual of < . Any one of the four partial order relations \leq, <, \geq, \text > on a given set uniquely determines the other three. Hence, as a matter of notation, one may write (P,\leq) or (P,<), and assume that the other relations are defined appropriately. Defining via a non-strict partial order \leq is most common. 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 A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-clas ...
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 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 bicon ...
, \leq is a total order.


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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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 numbers \mathbb, the usual less than relation < is a strict partial order. 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 n ...
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 subsets 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 post ...
) ordered by
inclusion Inclusion or Include may refer to: Sociology * Social inclusion, aims to create an environment that supports equal opportunity for individuals and groups that form a society. ** Inclusion (disability rights), promotion of people with disabiliti ...
(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 calle ...
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 strings ordered by
substring In formal language theory and computer science, a substring is a contiguous sequence of characters within a string. For instance, "''the best of''" is a substring of "''It was the best of times''". In contrast, "''Itwastimes''" is a subsequenc ...
. * 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 ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
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 ...
. (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 v ...
ordered by
reachability In graph theory, reachability refers to the ability to get from one Vertex (graph theory), 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 Glossary of graph theory#Basics, ...
. * The set of subspaces of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
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 nu ...
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 calle ...
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 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 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 in not having a solid foundation along its whole length. ...
, 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, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws ...
and, in most cases,
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, 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 (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take thro ...
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 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 of two partially ordered sets are (see Fig.4): *the lexicographical order:   (''a'', ''b'') ≤ (''c'', ''d'') if ''a'' < ''c'' or (''a'' = ''c'' and ''b'' ≤ ''d''); *the
product order In mathematics, given two preordered sets A and B, the product order (also called the coordinatewise orderDavey & Priestley, ''Introduction to Lattices and Order'' (Second Edition), 2002, p. 18 or componentwise order) is a partial ordering o ...
:   (''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''. For example, if ''X'' is a set of distinct numbers and ''x R y'' means "''x'' is less than ''y''", the ...
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 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 pr ...
s 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), ''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 In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-or ...
, then so is their ordinal sum. Series-parallel partial orders are formed from the ordinal sum operation (in this context called series composition) and another operation called parallel composition. Parallel composition is the
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
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 (from grc, συμμετρία "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 precise definiti ...
. For example, \ is related to \, but not the reverse. * ''a'' and ''b'' are '' comparable'' if ''a'' ≤ ''b'' or ''b'' ≤ ''a''. Otherwise they are ''incomparable''. For example, \ and \ are comparable, while \ and \ are not. * A '' total order'' 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'' 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
singleton Singleton may refer to: Sciences, technology Mathematics * Singleton (mathematics), a set with exactly one element * Singleton field, used in conformal field theory Computing * Singleton pattern, a design pattern that allows only one instance ...
s \. * 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'' 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 "''a'' < ''b'' but not ''a'' < ''c'' < ''b'' 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 dually, that is, it is an elem ...
and least element: An element g \in P is a if for every element a \in P, a \leq g. An element m \in P is a if for every element a \in P, m \leq a. A poset can only have one greatest or least element. In our running example, the set \ is the greatest element, and \ is the least. * Maximal elements and minimal elements: An element 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: 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 (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s, 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 Fig.6). 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 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 because the only ways ...
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.


Mappings between partially ordered sets

Given two partially ordered sets (''S'', ≤) and (''T'', ≼), a function f : S \to T is called order-preserving, or monotone, or isotone, if for all x, y \in S, x \leq y implies ''f''(''x'') ≼ ''f''(''y''). If (''U'', ≲) 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 v ...
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, ''f''(''x'') ≼ ''f''(''y'') implies x \leq y. If f is both order-preserving and order-reflecting, then it is called an
order-embedding In order theory, a branch of mathematics, an order embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, order embeddings constitute a notion which is stric ...
of (''S'', ≤) into (''T'', ≼). In the latter case, f is necessarily injective, 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, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
, 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 cons ...
, and the partial orders (''S'', ≤) and (''T'', ≼) are said to be isomorphic. Isomorphic orders have structurally similar
Hasse diagram In order theory, a Hasse diagram (; ) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ''(S, ≤)'' one represents ...
s (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, un ...
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 post ...
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 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 because the only ways ...
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 \) 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, 17 ...
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 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 in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set ...
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 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 isomorphism, the sequence 1, 1, 2, 5, 16, 63, 318, ... is obtained.


Linear extension

A partial order \leq^* on a set X is 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, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...
, 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 (graph theory), 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 Glossary of graph theory#Basics, ...
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 v ...
s) are called topological sorting.


Directed acyclic graphs

Strict partial orders correspond directly to
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 v ...
s (DAGs). If a graph is constructed by taking each element of P to be a node and each element of \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 on a set is the smallest relation on that contains and is transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite ...
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 every preordered set) may be considered as a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
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) \circ (x, y) = (x, z). Such categories are sometimes called '' posetal''. Posets are
equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry * Equivalence class (music) *'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equiva ...
to one another if and only if they are isomorphic. 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 In category theory, a branch of mathematics, a subcategory \mathcal of a category \mathcal is said to be isomorphism closed or replete if every \mathcal-isomorphism h:A\to B with A\in\mathcal belongs to \mathcal. This implies that both B and h^:B\ ...
.


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 geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
, 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 2019 ...
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

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'' (that is, ''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, 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; 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 each element of S, if such an element exists. Consequently, the term ''greatest lo ...
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 \times \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 mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
= \. 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 is built up by including elements one at a time, and in which an element, once available for inclusion, remains available until it is included. Antimatroi ...
, 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 * * * * * 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 subs ...
– continuity of a function between two partial orders. * * * * Strict weak ordering – strict partial order "<" in which the relation is transitive. * *
Tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
– Data structure of set inclusion *


Notes


Citations


References

* * * * *


External links

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