44 - immediate predecessors; the other pairs in the partial...

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5.1 Hasse Diagrams Since partial orders are special binary relations on a set A , we can repre- sent them by directed graphs. However, these graphs get rather cluttered if every arrow is drawn. We therefore introduce Hasse diagrams , which pro- vide a compact way of representing the partial order. First we require some de±nitions. D EFINITION 5.3 If R is a partial order on a set A and a R b for a ± = b , we call a a predecessor of b , and similarly b a successor of a . If a is a predecessor of b and there is no c with a R c and c R b , then a is the immediate predecessor of b . Hasse diagrams are like directed graphs, except that they just record the
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Unformatted text preview: immediate predecessors; the other pairs in the partial order can be inferred. Also the direction of the lines is usually omitted, with the convention that all lines are directed up the page. We give two examples of Hasse diagrams. E XAMPLE 5.4 For example, the Hasse diagram for the relation is a divisor of for the set { 1 , 2 , 3 , 6 , 12 , 18 } is 1 2 3 6 12 18 E XAMPLE 5.5 The Hasse diagram for the binary relation on P ( { 1 , 2 , 3 } ) is { 3 } { 1 , 3 } { 1 , 2 , 3 } { 2 , 3 } { 1 , 2 } { 1 } { 2 } 44...
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