Partially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word partial in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable.
Formally, a partial order is any binary relation that is reflexive, antisymmetric, and transitive.
One familiar example of a partially ordered set is a collection of people ordered by genealogical descendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs of people are incomparable, with neither being a descendant of the other.
A poset can be visualized through its Hasse diagram, which depicts the ordering relation.
Formal definition
A partial order is a binary relation ≤ over a set P satisfying particular axioms which are discussed [|below]. When a ≤ b, we say that a is related to b.The axioms for a non-strict partial order state that the relation ≤ is reflexive, antisymmetric, and transitive. That is, for all a, b, and c in P, it must satisfy:
- a ≤ a.
- if a ≤ b and b ≤ a, then a = b.
- if a ≤ b and b ≤ c, then a ≤ c.
A set with a partial order is called a partially ordered set. The term ordered set is sometimes also used, as long as it is clear from the context that no other kind of order is meant. In particular, totally ordered sets can also be referred to as "ordered sets", especially in areas where these structures are more common than posets.
For a, b, elements of a partially ordered set P, if a ≤ b or b ≤ a, then a and b are comparable. Otherwise they are incomparable. In the figure on top-right, e.g. and are comparable, while and are not. A partial order under which every pair of elements is comparable is called a total order or linear order; a totally ordered set is also called a chain. A subset of a poset in which no two distinct elements are comparable is called an antichain. An element a is said to be strictly less than an element b, if a ≤ b and a≠b. An element a is said to be covered by another element b, written a⋖b, if a is strictly less than b and no third element c fits between them; formally: if both a≤b and a≠b are true, and a≤c≤b is false for each c with a≠c≠b. A more concise definition will be given below using the strict order corresponding to "≤". For example, is covered by in the top-right figure, but not by.
Examples
Standard examples of posets arising in mathematics include:- The real numbers ordered by the standard less-than-or-equal relation ≤.
- The set of subsets of a given set ordered by inclusion. Similarly, the set of sequences ordered by subsequence, and the set of strings ordered by substring.
- The set of natural numbers equipped with the relation of divisibility.
- The vertex set of a directed acyclic graph ordered by reachability.
- The set of subspaces of a vector space ordered by inclusion.
- For a partially ordered set P, the sequence space containing all sequences of elements from P, where sequence a precedes sequence b if every item in a precedes the corresponding item in b. Formally, if and only if for all n in ℕ, i.e. a componentwise order.
- For a set X and a partially ordered set P, the function space containing all functions from X to P, where f ≤ g if and only if f ≤ g for all x in X.
- A fence, a partially ordered set defined by an alternating sequence of order relations a < b > c < d...
- The set of events in special relativity and, in most cases, general relativity, where for two events X and Y, X ≤ Y if and only if Y is in the future light cone of X. An event Y can only be causally affected by X if X ≤ Y.
Extrema
- Greatest element and least element: An element g in P is a greatest element if for every element a in P, a ≤ g. An element m in P is a least element if for every element a in P, a ≥ m. A poset can only have one greatest or least element.
- Maximal elements and minimal elements: An element g in P is a maximal element if there is no element a in P such that a > g. Similarly, an element m in P is a minimal element if there is no element a in P such that a < m. If a poset has a greatest element, it must be the unique maximal element, but otherwise there can be more than one maximal element, and similarly for least elements and minimal elements.
- Upper and lower bounds: For a subset A of P, an element x in P is an upper bound of A if a ≤ x, for each element a in A. In particular, x need not be in A to be an upper bound of A. Similarly, an element x in P is a lower bound of A if a ≥ x, for each element a in A. A greatest element of P is an upper bound of P itself, and a least element is a lower bound of P.
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 :- the lexicographical order: ≤ if a < c or ;
- the product order: ≤ if a ≤ c and b ≤ d;
- the reflexive closure of the direct product of the corresponding strict orders: ≤ if or.
Applied to ordered vector spaces over the same field, the result is in each case also an ordered vector space.
See also orders on the Cartesian product of totally ordered sets.
Sums of partially ordered sets
Another way to combine two posets is the ordinal sum, 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.
The ordinal sum operation is one of two operations used to form series-parallel partial orders, and in this context is called series composition. The other operation used to form these orders, the disjoint union of two partially ordered sets is called in this context parallel composition.
Strict and non-strict partial orders
In some contexts, the partial order defined above is called a non-strict partial order. In these contexts, a strict partial order "<" is a binary relation that is irreflexive, transitive and asymmetric, i.e. which satisfies for all a, b, and c in P:- not a < a ,
- if a < b and b < c then a < c , and
- if a < b then not b < a .
Conversely, if "<" is a strict partial order, then the corresponding non-strict partial order "≤" is the reflexive closure given by:
This is the reason for using the notation "≤".
Using the strict order "<", the relation "a is covered by b" can be equivalently rephrased as "a<b, but not a<c<b for any c".
Strict partial orders are useful because they correspond more directly to directed acyclic graphs : every strict partial order is a dag, and the transitive closure of a dag is both a strict partial order and also a dag itself.
Inverse and order dual
The inverse of a partial order relation ≤ is the converse of ≤. Typically denoted ≥, it is the relation that satisfies x ≥ y if and only if y ≤ x. The inverse of a partial order relation is reflexive, transitive, and antisymmetric, and hence itself a partial order relation. The order dual of a partially ordered set is the same set with the partial order relation replaced by its inverse. The irreflexive relation > is to ≥ as < is to ≤.Any one of the four relations ≤, <, ≥, and > on a given set uniquely determines the other three.
In general two elements x and y of a partial order may stand in any of four mutually exclusive relationships to each other: either x < y, or x = y, or x > y, or x and y are incomparable. A totally ordered set is one that rules out this fourth possibility: all pairs of elements are comparable and we then say that trichotomy holds. The natural numbers, the integers, the rationals, and the reals are all totally ordered by their algebraic magnitude whereas the complex numbers are not. This is not to say that the complex numbers cannot be totally ordered; we could for example order them lexicographically via x+iy < u+iv if and only if x < u or, but this is not ordering by magnitude in any reasonable sense as it makes 1 greater than 100i. Ordering them by absolute magnitude yields a preorder in which all pairs are comparable, but this is not a partial order since 1 and i have the same absolute magnitude but are not equal, violating antisymmetry.
Mappings between partially ordered sets
Given two partially ordered sets and, a function f: S → T is called order-preserving, or monotone, or isotone, if for all x and y in S, x≤y implies f ≤ f.If is also a partially ordered set, and both f: S → T and g: T → U are order-preserving, their composition : S → U is order-preserving, too.
A function f: S → T is called order-reflecting if for all x and y in S, f ≤ f implies x≤y.
If f is both order-preserving and order-reflecting, then it is called an order-embedding of into.
In the latter case, f is necessarily injective, since f = f implies x ≤ y and y ≤ x. If an order-embedding between two posets S and T exists, one says that S can be embedded into T. If an order-embedding f: S → T is bijective, it is called an order isomorphism, and the partial orders and are said to be isomorphic. Isomorphic orders have structurally similar Hasse diagrams. It can be shown that if order-preserving maps f: S → T and g: T → S exist such that g∘f and f∘g yields the identity function on S and T, respectively, then S and T are order-isomorphic.
For example, a mapping f: ℕ → ℙ from the set of natural numbers to the power set of natural numbers can be defined by taking each number to the set of its prime divisors. It is order-preserving: if x divides y, then each prime divisor of x is also a prime divisor of y. However, it is neither injective nor order-reflecting. Taking instead each number to the set of its prime power divisors defines a map g: ℕ → ℙ that is order-preserving, order-reflecting, and hence an order-embedding. It is not an order-isomorphism, but it can be made one by restricting its codomain to g. The right picture shows a subset of ℕ and its isomorphic image under g. The construction of such an order-isomorphism into a power set can be generalized to a wide class of partial orders, called distributive lattices, see "Birkhoff's representation theorem".
Number of partial orders
Sequence in OEIS gives the number of partial orders on a set of n labeled elements:The number of strict partial orders is the same as that of partial orders.
If the count is made only up to isomorphism, the sequence 1, 1, 2, 5, 16, 63, 318, … is obtained.
Linear extension
A partial order ≤^{*} on a set X is an extension of another partial order ≤ on X provided that for all elements x and y of X, whenever x ≤ y, it is also the case that x ≤^{*} y. A linear extension is an extension that is also a linear order. Every partial order can be extended to a total order.In computer science, algorithms for finding linear extensions of partial orders are called topological sorting.
In category theory
Every poset may be considered as a category where, for objects x and y, there is at most one morphism from x to y. More explicitly, let hom = if x ≤ y and ∘ =. Such categories are sometimes called posetal.Posets are equivalent to one another if and only if they are isomorphic. In a poset, the smallest element, if it exists, is an initial object, and the largest element, if it exists, is a terminal object. Also, every preordered set is equivalent to a poset. Finally, every subcategory of a poset is isomorphism-closed.
Partial orders in topological spaces
If P is a partially ordered set that has also been given the structure of a topological space, then it is customary to assume that is a closed subset of the topological product space. Under this assumption partial order relations are well behaved at limits in the sense that if, and a_{i} ≤ b_{i} for all i, then a ≤ 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.For a ≤ b, the closed interval is the set of elements x satisfying a ≤ 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. An open interval may be empty even if a < b. For example, the open interval on the integers is empty since there are no integers such that.
The half-open intervals and are defined similarly.
Sometimes the definitions are extended to allow a > b, in which case the interval is empty.
An interval is bounded if there exist elements a and b of P such that. Every interval that can be represented in interval notation is obviously bounded, but the converse is not true. For example, let as a subposet of the real numbers. The subset is a bounded interval, but it has no infimum or supremum in P, so it cannot be written in interval notation using elements of P.
A poset is called locally finite if every bounded interval is finite. For example, the integers are locally finite under their natural ordering. The lexicographical order on the cartesian product ℕ×ℕ is not locally finite, since.
Using the interval notation, the property "a is covered by b" can be rephrased equivalently as =.
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.