Discrete Mathematics..

Hasse Diagrams 195Figure 4.94. Show that neither of the configurations in figure 4.10 can occur anywherein the Hasse diagram of a poset.SolutionFigure 4.10In (a) the line joining a to c should not occur because a R b and b R cwhich means that c does not cover a. Of course, a is related to c by thetransitive property.The configuration in figure 4.10(b) cannot occur for a similar reason. Theright-hand part of the diagram implies that a R b and b R d (assumingtransitivity), so d does not cover a. The line joining a to d shouldtherefore be deleted.5. Let A ={a 1 , a 2 ,...,a n } be a finite set with a total order R. Theneverypair of elements are related, so given x, y ∈ A either we can get from xto y or we can get from y to x by a sequence of rising lines in the Hassediagram.This means that, in the Hasse diagram, the elements are arranged in asingle vertical line as in figure 4.11. This diagram explains why a totalorder is sometimes called a linear order.