DHARMLATTICE THEORY 151- Similarly the relation “less than” or relation “greater than” are also partial ordered
relation over set of real numbers i.e., (R, <) or (R, >). - The relation “⊆” (inclusion) defined over power set of X is a partial ordered relation
i.e., (P(X), ⊆). Let X = {a, b}; then power set of X is P(X) = {Ø, {a}, {b}, {a, b}}. Since,
every element of P(X) is subset of itself so relation “⊆” over P(X) is reflexive. It is also
antisymmetric and transitive i.e. for the element Ø ⊆ {b} and {b} ⊆ {a, b} then, Ø ⊆ {a, b}
and similarly true for all elements of P(X). - The relation “perfect division” or “integral multiple” defined over set of positive inte-
ger (I+) are partial ordered relation. For example let X = {2, 3, 4, 6} ∈ I+; then partial
ordered relation ‘≤’ is “perfect division” (i.e., for any x and y ∈ X the relation “x
divides y”) over X will be given as,
‘≤’ = {(2, 2), (3, 3), (4, 4), (6, 6), (2, 4), (2, 6), (3, 6)}
Similarly, partial ordered relation “integer multiple” (i.e., for any x and y ∈ X, and any
integer k; y = x k; ‘y is an integer multiple of x’) will be given as,
≥ = {(2, 2), (3, 3), (4, 4), (6, 6), (4, 2), (6, 2), (6, 3)}
Here, we used partial ordered relation symbol ‘≥’ because; last poset is dual of previous
poset.
Comparability and Noncomparability
Let (X, ≤) be a poset, then elements x and y ∈ X are said to be comparable if x = y or y = x. If x
and y are not related i.e., x (^) /≤ y or y (^) /≤ x then they are called noncomparable. For example, the
elements of power set of X say P(X) are noncomparable with respect to partial ordered relation
“⊆”.
6.3 Representation of a Poset (Hasse Diagram)..................................................................
A poset (X, ≤) is represented by a diagram called Hasse diagram. Hasse diagram is a directed
graph, where ordered between the elements are preserved. Since, it is a directed graph which
consists of vertices and edges where, all elements of X are in set of vertices that are repre-
sented by a circle or dot and the connections between vertices (elements of X) called edges that
will be drawn as follows,
l For the element x, y ∈ X if x > y and there is no element z ∈ X i.e., x ≥ z ≥ y then circle
for y is putted below the circle for x and are connected by a direct edge.
l Otherwise, if x > y and there exist at least an element z ∈ X i.e., x ≥ z ≥ y then they are
not connected by a direct edge, however they are connected by other elements of set
X.Example 6.1. Consider set X = {2, 3, 4, 6, 8, 24} and the partial ordered relation ‘≤’ be i.e.,
x ≤ y ⇒ “x divides y” (perfect division)
then, poset (X, ≤) will be given as,
≤ = {(2, 2), (2, 4), (2, 6), (2, 8), (2, 24), (3, 3), (3, 6), (3, 24), (4, 4), (4, 8), (4, 24),
(6, 6), (6, 24), (8, 8), (8, 24)}
Since, relation is understood to be reflexive, so we can leave the pairs of similar elements.
Since, relation is understood to be transitive, so we can leave the pairs that come in sequence
of pairs i.e. it need not to write the pair (2, 24), if the sequence of pairs (2, 6) and (6, 24) are
there.