In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of 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 ...
. A poset consists of a set together with a
binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order."
The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize
total order
In mathematics, a total 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 ( reflexive) ...
s, in which every pair is comparable.
Informal definition
A partial order defines 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 ...
. 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 ea ...
, 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 s ...
, 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
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 ...
that is reflexive, antisymmetric, and transitive. That is, for all it must satisfy:
# Reflexivity: , 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-compl ...
: if and then , i.e. no two distinct elements precede each other.
# Transitivity: if and then .
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 and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special c ...
.
Strict partial order
An irreflexive, strong,[ or is a homogeneous relation < on a set 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 ...
, asymmetric, and transitive; that is, it satisfies the following conditions for all
# Irreflexivity: not , 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 then not .
# Transitivity: if and then .
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 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 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 and denotes set subtraction
In set theory, the complement of a set , often denoted by (or ), is the set of elements not in .
When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is the ...
. 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
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
Conversely, if < is a strict partial order, then the corresponding non-strict partial order 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:
Dual orders
The ''dual'' (or ''opposite'') of a partial order relation is defined by letting 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& ...
of , i.e. if and only if . 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 , where is a non-strict partial order relation on , is the associated strict partial order relation on (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 ), is the dual of , and is the dual of .
Any one of the four partial order relations on a given set uniquely determines the other three. Hence, as a matter of notation, one may write or , and assume that the other relations are defined appropriately. Defining via a non-strict partial order is most common. Some authors use different symbols than such as or to distinguish partial orders from total orders.
When referring to partial orders, 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-class ...
of . The relation is the converse of the irreflexive kernel of , which is always a subset of the complement of , but is equal to the complement of if, and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
, 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 real ...
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 , the usual less than
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different n ...
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 .
* 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, 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 are ...
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, 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 subsequence ...
.
* 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 n ...
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 ve ...
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 an ...
.
* 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 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 calle ...
s 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 ; that is, a componentwise 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 ...
.
* 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 vect ...
containing all functions from ''X'' to ''P'', where ''f'' ≤ ''g'' if and only if ''f''(''x'') ≤ ''g''(''x'') for all
* 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 o ...
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
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 kinsh ...
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 ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
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 ...
: (''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 ...
: (''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
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, 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-orde ...
, 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, 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 (th ...
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 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
Comparable may refer to:
* Comparability, in mathematics
* Comparative
general linguistics, the comparative is a syntactic construction that serves to express a comparison between two (or more) entities or groups of entities in quality or degr ...
'' if ''a'' ≤ ''b'' or ''b'' ≤ ''a''. Otherwise they are ''incomparable''. For example, and are comparable, while and are not.
* A ''total order
In mathematics, a total 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 ( reflexive) ...
'' 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 c ...
'' is a subset of a poset that is a totally ordered set. For example, is a chain.
* An ''antichain
In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable.
The size of the largest antichain in a partially ordered set is known as its wid ...
'' 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 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 co ...
'' 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 are true, and ''a'' ≤ ''c'' ≤ ''b'' is false for each ''c'' with 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 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 eleme ...
and least element: An element is a if for every element An element is a if for every element A poset can only have one greatest or least element. In our running example, the set is the greatest element, and is the least.
* Maximal 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 defin ...
s and minimal elements: An element is a maximal element if there is no element such that Similarly, an element is a minimal element if there is no element such that If a poset has a greatest element, it must be the unique maximal element, but otherwise there can be more than one maximal element, and similarly for least elements and minimal elements. In our running example, and are the maximal and minimal elements. Removing these, there are 3 maximal elements and 3 minimal elements (see Fig.5).
* Upper and lower bounds
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of .
Dually, a lower bound or minorant of is defined to be an element ...
: 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 language ...
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 divisible by ...
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 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 order ...
, or monotone
Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony.
Monotone or monotonicity may also refer to:
In economics
*Monotone preferences, a property of a consumer's preference ordering.
*Monotonic ...
, or isotone, if for all implies ''f''(''x'') ≼ ''f''(''y'').
If (''U'', ≲) is also a partially ordered set, and both and 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 ...
is order-preserving, too.
A function is called order-reflecting if for all ''f''(''x'') ≼ ''f''(''y'') implies
If 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, 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; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
, since implies and in turn 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 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 s ...
, 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 ea ...
s (see Fig.7a). It can be shown that if order-preserving maps and exist such that and 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 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 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 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, 17 ...
divisors defines a map that is order-preserving, order-reflecting, and hence an order-embedding. It is not an order-isomorphism (since it, for instance, does not map any number to the set ), but it can be made one by restricting its codomain to Fig.7b shows a subset of 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 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 uni ...
s, see "Birkhoff's representation theorem
:''This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).''
In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice ...
".
Number of partial orders
Sequence A001035
A, or a, is the first letter and the first vowel of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''a'' (pronounced ), plural ''aes'' ...
in 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 Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' wi ...
isomorphism, the sequence 1, 1, 2, 5, 16, 63, 318, ... is obtained.
Linear extension
A partial order on a set is an extension of another partial order on provided that for all elements whenever it is also the case that A linear extension
In order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered sets is a linear extens ...
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 order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered sets is a linear extens ...
).
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 Applied science, practical discipli ...
, 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 an ...
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 ve ...
s) are called topological sorting
In computer science, a topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge ''uv'' from vertex ''u'' to vertex ''v'', ''u'' comes before ''v'' in the ordering. For ins ...
.
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 ve ...
s (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
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 s ...
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
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive relation, reflexive and Transitive relation, transitive. Preorders are more general than equivalence relations and (non-strict) partia ...
) 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 and there is at most one morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
from to More explicitly, let hom(''x'', ''y'') = if ''x'' ≤ ''y'' (and otherwise the empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
) and 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 mathematics, an isomorphism is a structure-preserving mapping 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 them. The word is ...
. 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 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 points ...
, then it is customary to assume that is a closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
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-seemin ...
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 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'' (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 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 low ...
or supremum
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 l ...
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 is not locally finite, since .
Using the interval notation, the property "''a'' is covered by ''b''" can be rephrased equivalently as
This concept of an interval in a partial order should not be confused with the particular class of partial orders known as 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. Antimatroids ...
, 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 In mathematics, the poset topology associated to a poset (''S'', ≤) is the Alexandrov topology (open sets are upper sets) on the poset of finite chains of (''S'', ≤), ordered by inclusion.
Let ''V'' be a set of vertices. An abstract simplicia ...
, 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 subse ...
– 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
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External links
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{{Authority control
Order theory
Binary relations
de:Ordnungsrelation#Halbordnung