Digramatic Representation Of Partial Order Relations And Posets

Note : In many cases, when the graphical representation is so oriented that all the arrow heads point in one direction (upward, downward, left to right or right to
left). A graphical representation in which all the arrowheads point upwards, is known as Hasse diagram.
Example: `Let A = {1, 2, 3, 4, 6, 9} and relation R defined on A be “a divides b”. Hasse diagram for this relation is as follows:
Note : The reader is advised to verify that this relation is indeed a partial order relation. Further, arrive at the following Hasse diagram from the diagraph of a relation as per the rules defined earlier.

Example : Determine the Hasse diagram of the relation on A = {1,2,3,4,5} whose MR is given below :

Solution : Reflexivity is represented by 1 at diagonal place. So after removing reflexivity R is R = {(1,3), (1,4), (1,5), (2,3), (2,4), (2,5), (3,4), (3,5)}
Remove transitivity as

The Hasse Diagram is

Example : Determine matrix of partial order whose Hasse diagram is given as follow -

Solution : Here A = [1, 2, 3, 4, 5)
Write all ordered pairs (a, a) ∀ a ∈ A i.e. relation is reflexive. Then write all ordered pairs in upward direction. As (1, 2) ∈ R &
(2,4) ∈ R -> (1,4) ∈ R since R is transitive.
∴ R= { (1,1),(2,2),(3,3),(4,4),(5,5),(1,2),(2,4),(2,4),(1,4),(1,3),(3,5),(1,5)}

The matrix MR can be written as -

Now, we shall have a look at certain terms with reference to posets.