Partial order

In mathematics, a partial order ≤ on a set X is a binary relation that is reflexive, antisymmetric and transitive, i.e., it holds for all a, b and c in X that:

a ≤ a (reflexivity)

if a ≤ b and b ≤ c then a ≤ c (transitivity)

if a ≤ b and b ≤ a then a = b (antisymmetry)

A set with a partial order on it is called a partially ordered set, poset, or, often, simply an ordered set.

Examples of posets include the integers and real numbers with their ordinary ordering, subsets of a given set ordered by inclusion, strings ordered lexicographically (as in a phone book), and natural numbers ordered by divisibility.

Finite posets are most easily visualized as "Hasse diagrams", that is, graphss where the vertices are the elements of the poset and the ordering relation is indicated by edges and the relative positioning of the vertices. The element x is smaller than y if and only if there exists a path from x to y always going upwards. This can be generalized: any poset can be represented by a directed acyclic graph, where the nodes are the elements of the poset and there is a directed path from a to b if and only if a≤b.

If S is a subset of the poset X, we say that

• the element m of S is a maximal element of S if the only element s of S with m≤s is s=m.
• the element u of X is an upper bound for S if s≤u for all s in S
• the element l is the largest (or greatest, or top) element in S if l is an upper bound for S and an element of S.

There can be many maximal elements but at most one largest element of S; if a largest element exists, then it is the unique maximal element of S. Minimal elements, lower bounds and smallest (least, bottom) elements are defined analogously.

A subset of a partially ordered set inherits a partial order. New partially ordered sets can also be constructed by cartesian products, disjoint unions and other set-theoretic operations. Since the intersection of partial orders on a given set X is again a partial order on X, every relation R on X generates a unique partial order on X, the smallest partial order containing R. Every poset (X,≤) has a unique dual poset (X,≥), where we define a >= b if and only if b ≤ a. Every poset also gives rise to an irreflexive relation <, where a < b if and only if a ≤ b and a ≠ b.

Special cases of partially ordered sets are

• totally ordered sets, where for any pair of elements a,b, either a≤b or b≤a. For example the real numbers with the usual order relation ≤ form a totally ordered set. Another name for totally ordered set is "linearly ordered set". A chain is a linearly ordered subset of a poset.
• well-ordered sets, where all non-empty subsets have smallest elements.
• lattices, where any two elements have both a greatest lower bound (infimum) and a least upper bound (supremum). Lattices are considered algebraic structures with the operations "sup" and "inf".
• boolean algebras, which are lattices with additional properties that allow for the definition of a logical negation.
• bounded posets, which have a largest and a smallest element.

A related concept is that of a directed set, where every finite subset has an upper bound. Directed sets are not required to have the antisymmetric property, so they are not necessarily posets.

A partially ordered set is complete if every one of its subsets has a least upper bound and a greatest lower bound. Various types of complete partially ordered sets are used in, for example, program semantics. The best-known type of complete partially ordered sets are the Scott-Ershov domainss. These structures are important in that they constitute a cartesian closed category and in that they provide a natural theory of approximations. That the class of Scott-Ershov domains is cartesian closed category enables the solution of so-called domain equations, e.g., D = [D -> D], where the right-hand side denotes the space of all continuous functions on D.

Partially ordered sets can be given a topology, for example, the Alexandrov topology, consisting of all upwards closed subsets. A subset U of a partially ordered set is upwards closed if x in U and x ≤ y implies that y belongs to U. For special types of partially ordered sets other topologies may be more interesting. For example, the natural topology on Scott-Ershov domains is the Scott topology.

A poset is locally finite if every closed interval [a, b] in it is finite. Locally finite posets give rise to incidence algebras which in turn can be used to define the Euler characteristic of finite bounded posets.