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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.

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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.

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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.

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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

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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

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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.

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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.

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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.

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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

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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.

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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)

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Ex. Hasse diagram of poset ( {1, 2, 3, 4}, ) 4 3 2 1 Also find the covering relation

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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

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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