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Partially Ordered Set

In mathematics, there is an important branch called set theory in which we study about sets, relations and their applications. Particularly order theory (an area of set theory) deals with the order of sets and relations, especially binary relation which is defined as an ordered pair of elements of a set.
In order theory, the concept of partially ordered set is a very useful concept. A set which follows partial order relation, is known as a partially ordered set. The relation "$\leq$" defined on a set is known as partially ordered relation and the set is called partially ordered set.
Down in this page, we will learn about partially ordered set, its definition and related examples.

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Partially ordered set is also abbreviated as poset for convenience. It generalizes and formalizes the concept of ordering or arranging of the elements of a set. A partially ordered set or a poset is defined as a set along with a binary relation $\leq$ which satisfies three properties of sets - reflexivity, antisymmetry and transitivity.
According to more formal definition of partially ordered set :
Poset is an ordered pair of binary relation "$\leq$" defined over a set S, i.e. ($\leq$, S) that satisfies following properties (Lets say $a$, $b$, $c$ $\in$ $S$):
1) Reflexivity: i.e. $a$ $\leq$ $a$

2) Antisymmetry: i.e. if $a$ $\leq$ $b$ and $b$ $\leq$ $a$ $\Rightarrow$ $a$ = $b$

3) Transitivity: i.e. $a$ $\leq$ $b$ and $b$ $\leq$ $c$ $\Rightarrow$ $a$ $\leq$ $c$.

If the above conditions are satisfied, then the ordered pair ($\leq$, S) is termed as a partially ordered set.


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Partially ordered sets are very useful in higher-level mathematics. It plays an important role in modern algebra as well as abstract algebra. Posets are also studied under topology: an important branch of higher mathematics. In fact, poset topology is a separate field in maths. The concept of partially ordered set is also applied in probability theory. Not only in maths, it plays a vital tole in computer science also.

There are various different theorems and lemma that are based on the definition and application of partially ordered set. These theorems are utilized in many fields, problems and proofs.


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The few examples of partially ordered set are illustrated below :

Example 1: One of the most famous and obvious example of partially ordered set is a set of positive integers. There exists a natural ordering in the set of positive integers. For this set, we may write that every "$a \leq b$", in the sense if b is greater than a which is quite natural for positive integers. This set satisfies all the properties of a poset. In which every two elements can be compared.

Example 2: The set $A$ = {1, 2, 3} along with the binary operation $\leq$ satisfies the condition for partial order and hence is termed as a poset or partially ordered set. We may have the following relations :
{1} $\leq$ {1, 3}, {2} $\leq$ {1, 2}, {3} $\leq$ {2, 3}, {3} $\leq$ {1, 3}, {2} $\leq$ {2, 3}, {3} $\leq$ {1, 2}.

Example 3: The most popular example of partially ordered set is Hasse diagram. Hasse diagrams were introduced by a German mathematician Helmut Hasse. He showed a three-element set {$x, y, z$} whose poset diagram is shown below:
Partially Ordered Set

Example 4: One more example is of set {1, 2, 3, 5}. Its poset diagram is shown below:
Partially ordered Set Examples
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