Partially Ordered Sets (POSets) Let R be a relation on a set S.
Then R is called a partial order if it is Reflexive a R a, a S
Antisymmetric If a R b and b R a a = b Transitive If a R b and b R
c a R c The set S with partial order is called partially ordered
set or poset.
Slide 3
Ex. The relation on the real numbers, is a partial order. Sol.
Reflexive : a a for all real numbers Antisymmetric : If a b, b a
then a = b Transitive : If a b, b c then a c This order relation on
N or R is called usual order Ex. (Z +, | ), the relation divides on
+ve integers. Ex. (Z, | ), the relation divides on integers. Ex. (2
S, ), the relation subset on set of all subsets of S.
Slide 4
Comparability Let a and b be the elements in a partially
ordered set (S, ). Then a and b are called comparable if a b or b
a. They are incomparable or non-comparable, written as a b if
neither a b nor b a. Ex. In poset (Z +, |), 3 and 6 are comparable,
6 and 3 are comparable, 3 and 5 are not, 8 and 12 are not. Dual
Order Let be any partial ordering of set S. If the relation is also
a partial ordering of S, then it is called dual order.
Slide 5
Let A be any subset of an ordered set S Suppose a, b A. Define
a b as elements of A whenever a b as elements of S. This defines a
partial ordering of A called the induced order on A. The subset A
with the induced order is called an ordered subset of S. Ordered
Subsets
Slide 6
If (S, ) is a poset and every two elements of S are comparable,
then S is called totally ordered or linearly ordered. A totally
ordered set is also called a chain. Ex. The poset (Z, ), is totally
ordered, because either a b or b a when a and b are integers. Ex.
The poset (Z +, |), is not totally ordered because it contains
elements that are incomparable such as 5 and 7. Totally Ordered
Set
Slide 7
Well-Ordered Set A poset (S, ) is called a well-ordered set if
the order relation is a total-ordering and every non-empty subset
of S has a least element. Ex. The set (Z, ) is not well-ordered
because the set of ve integers, which is a subset of Z, has no
least element.
Slide 8
Product Order Suppose S and T are ordered sets. Then is an
order relation on the product set S T, called the product order
such that (a, b) (a, b) if a a and b b Lexicographical Order
Suppose S and T are linearly ordered sets. Then the order relation
on the product set S T, called the lexicographical order such that
(a, b) (a, b) if a a or if a = a and b b It is also called
dictionary order.
Slide 9
Ex. Determine whether (3, 5) (4, 8), whether (3, 8) (4, 5)
whether (4, 9) (4, 11) in the poset (Z Z, ) is the lexicographic
ordering Ex. (1, 2, 3, 5) (1, 2, 4, 3) is the lexicographic
ordering.
Slide 10
Lexicographic Ordering on the Set of Strings A string is less
than a second string if the letter in the first string in the first
position where the strings differ comes before the letter in the
second string in this position, or if the first string and the
second string agree in all positions, but the second string has
more letters. This ordering is the same as that used in
dictionaries. Ex. discreet discrete discreet discreetness discrete
discretion
Slide 11
Hasse Diagram Let S be a partially ordered set let a, b S If a
b, then a is called an immediate predecessor of b, or b is known an
immediate successor of a, or b is a cover of a, written as a b but
no element in S lies between a and b, i.e., there exists no element
c in S such that a c b The set of pairs (a, b) such that b covers a
is called the covering relation of the poset S.
Slide 12
Hasse Diagram Let S be a finite partially ordered set. The
Hasse diagram of S is the directed graph whose vertices are the
elements of S and there is a directed edge from a to b whenever a b
in S. (At place of an arrow from a to b, we can place b higher than
a and draw a line between them)
Slide 13
Ex. Hasse diagram of poset ( {1, 2, 3, 4}, ) 4 3 2 1 Also find
the covering relation
Slide 14
Ex. Draw the Hasse diagram representing the partial ordering {
(a, b) | a divides b } on {1, 2, 3, 4, 6, 8, 12} 8 4 2 3 1 12 6
Also find the covering relation
Slide 15
Ex. Draw the Hasse diagram for the partial ordering { (A,B) | A
B } on the power set P(S) where S = {a, b, c} {a,b,c} {b,c} {c} {}
or {b} {a,c} {a} {a,b} Also find the covering relation