A poset, or partially ordered setMathworldPlanetmath, consists of a set P and a binary relationMathworldPlanetmath on P which satisfies the following properties:

  • is reflexiveMathworldPlanetmathPlanetmath (http://planetmath.org/Reflexive), so aa always holds;

  • is antisymmetric, so if ab and ba hold, then a=b; and

  • is transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/Transitive3), so if ab and bc hold, then ac also holds.

The relationMathworldPlanetmath is called a partial orderMathworldPlanetmath on P. In practice, (P,) is usually conflated with P; if a distinction is needed, P is called the ground set or underlying set of (P,). The binary relation < defined by removing the diagonal from (i.e.  a<b iff ab and ab) satisfies the following properties:

  • < is irreflexiveMathworldPlanetmath, so if a<b holds, then b<a does not hold; and

  • < is transitive.

Since is reflexive, it can be uniquely recovered from < by adding the diagonal. For this reason, an irreflexive and transitive binary relation < (called a strict partial order) also defines a poset, by means of the associated relation described above (which is called weak partial order).

Since every partial order is reflexive and transitive, every poset is a preorderMathworldPlanetmath. The notion of partial order is stricter than that of preorder, Let Q be the structureMathworldPlanetmath with ground set Q={a,b} and binary relation ={(a,a),(a,b),(b,a),(b,b)}. A diagram of this structure, omitting loops, is displayed below.


Observe that the binary relation on Q is reflexive and transitive, so Q is a preorder. On the other hand, ab and ba, while ab. So the binary relation on Q is not antisymmetric, implying that Q is not a poset.

Since every total orderMathworldPlanetmath is reflexive, antisymmetric, and transitive, every total order is a poset. The notion of partial order is weaker than that of total order. A total order must obey the trichotomy law, which states that for any a and b in the order, either ab or ba. Let P be the structure with ground set {a,b,c} and binary relation ={(a,a),(a,b),(a,c),(b,b),(c,c)}. A diagram of this structure, omitting loops, is displayed below.


Observe that the binary relation on P is reflexive, antisymmetric, and transitive, so P is a poset. On the other hand, neither bc nor cb holds in P. Thus P fails to satisfy the trichotomy law and is not a total order.

The failure of the trichotomy law for posets motivates the following terminology. Let P be a poset. If ab or ba holds in P, we say that a and b are comparable; otherwise, we say they are incomparable. We use the notation ab to indicate that a and b are incomparable.

If (P,P) and (Q,Q) are posets, then a functionMathworldPlanetmath φ:PQ is said to be order-preserving, or monotone, provided that it preserves inequalities. That is, φ is order-preserving if whenever aPb holds, it follows that φ(a)Qφ(b) also holds. The identity functionMathworldPlanetmath on the ground set of a poset is order-preserving. If (P,P), (Q,Q), and (R,R) are posets and φ:PQ and ψ:QR are order-preserving functions, then the compositionMathworldPlanetmath ψφ:PR is also order-preserving.

Posets together with order-preserving functions form a category, which we denoted by 𝐏𝐨𝐬𝐞𝐭. Thus an order-preserving function between the ground sets of two posets is sometimes also called a morphism of posets. The category of posets has arbitrary productsPlanetmathPlanetmath (http://planetmath.org/ProductofPosets). Moreover, every poset can itself be viewed as a category, and it turns out that a morphism of posets is the same as a functor between the two posets.

Examples of posets

The two extreme posets are the chain, in which any two elements are comparable, and the antichainMathworldPlanetmath, in which no two elements are comparable. A poset with a singleton underlying set is necessarily both a chain and an antichain, but a poset with a larger underlying set cannot be both.

Example 1.

Let be the set of natural numbers. Inductively define a binary relation on by the following rules:

  • for any n, the relation 0n holds; and

  • whenever mn, the relation m+1n+1 also holds.

Then (,) is a chain, hence a poset. This structure can be naturally embedded in the larger chains of the integers, the rational numbers, and the real numbers.

The next example shows that nontrivial antichains exist.

Example 2.

Let P be a set with cardinality greater than 1. Let be the diagonal of P. Thus represents equality, which is trivially a partial order relation (which is also the intersectionDlmfMathworldPlanetmath of all partial orderings on P). By construction, ab in P if and only a=b. Thus no two elements of P are comparable.

So far the only posets we have seen are chains and antichains. Most posets are neither. The following construction gives many such examples.

Example 3.

If X is any set, the powerset P=P(X) of X is partially ordered by inclusion, that is, by the relation AB if and only if AB.

There are important structure theorems for posets concerning chains and antichains. One of the foundational results is Dilworth’s theorem. This theorem was massively generalized by Greene and Kleitman.

A final example shows that one can manufacture a poset from an existing one.

Example 4.

Let P be a poset ordered by . The dual poset of P is defined as follows: it has the same underlying set as P, whose order is defined by ab iff ba. It is easy to see that is a partial order. The dual of P is usually denoted by P.

Graph-theoretical view of posets

Let P be a poset with strict partial order <. Then P can be viewed as a directed graphMathworldPlanetmath with vertex set the ground set of P and edge set <. For example, the following diagram displays the Boolean algebraMathworldPlanetmath B2 as a directed graph.


If P is a sufficiently complicated poset, then drawing all of the edges of P can obscure rather than reveal the structure of P. For this reason it is convenient to restrict attention to a subrelation of < from which < can be uniquely recovered.

We describe a method of constructing a canonical subgraphMathworldPlanetmath of P from which the partial order can be recovered as long as every interval of P has finite height. If a and b are elements of P, then we say that b covers a if a<b and there are no elements of P strictly larger than a but strictly smaller than b, that is, if [a,b]={a,b}. Two elements are said to be consecutive if one covers another. Define a binary relation <: on P by

a<:b if and only if b covers a.

By construction, the binary relation <: is a subset of <. Since < is transitive, the transitive closureMathworldPlanetmath (http://planetmath.org/ClosureOfASetViaRelations) of <: is also contained in <.


Suppose every interval of P has finite height. Then < is the transitive closure of <:.


We prove this by inductionMathworldPlanetmath on height. By definition of <:, if a<b and the height of [a,b] is 1, then a<:b.

Assume for induction that whenever a<b and the height of [a,b] is at most n, then (a,b) is in the transitive closure of <:. Suppose that a<b and that the height of [a,b] is n+1. Since every chain in [a,b] is finite, it contains an element c which is strictly larger than a and minimalPlanetmathPlanetmath (http://planetmath.org/MaximalElement) with respect to this property. Therefore [a,c]={a,c}, from which we conclude that a<:c. Since the interval [c,b] is a proper subinterval of [a,b], it has height at most n, so by the induction assumptionPlanetmathPlanetmath we conclude that (c,b) is in the transitive closure of <:. Since (a,c) and (c,b) are in the transitive closure of <:, so is (a,b). Hence whenever a<b and the height of [a,b] is at most n+1, then (a,b) is in the transitive closure of <:.

This completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof. ∎

In the same way we associated a graph to < we can associate a graph to <:. The graph is usually called the Hasse diagramMathworldPlanetmath of the poset. Below we display the graph associated to the cover relation <: of B2.


For simplicity, the Hasse diagram of a poset is usually drawn as an undirected graph. Elements which are higher in the partial order relation are drawn physically higher. Since a strict partial order is acyclic, this can be done uniquely and the partial order can be recovered from the drawing.


Title poset
Canonical name Poset
Date of creation 2013-03-22 11:43:41
Last modified on 2013-03-22 11:43:41
Owner mps (409)
Last modified by mps (409)
Numerical id 22
Author mps (409)
Entry type Definition
Classification msc 06A99
Synonym partially ordered set
Related topic Relation
Related topic PartialOrder
Related topic Semilattice
Related topic StarProduct
Related topic HasseDiagram
Related topic GreatestLowerBound
Related topic NetsAndClosuresOfSubspaces
Related topic OrderPreservingMap
Related topic DisjunctionPropertyOfWallman
Defines comparable
Defines incomparable
Defines cover
Defines covering
Defines order-preserving function
Defines monotone
Defines monotonic
Defines order morphism
Defines morphism of posets
Defines dual poset
Defines consecutive