Archangel Macsika
For divisibility relation on the set {1,2,3,6,8,12,24,36}, draw Hasse diagram. Then find minimal, maximal, greatest and least elements. Then give the topological sort using the divisibility relation
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Solution:
Given: A = {1,2,3,6,8,12,24,36, /}
Hasse diagram:
Maximal elements are 36 and 24 since they are succeeding all the elements.
Minimal element is 1 since it is preceding all the elements.
Greatest element does not exist since there is no any one element that succeeds all the elements.
Least element is 1 since there is no any one element that precedes 1.
If A is a poset with partial order ≤, we sometimes need to find a linear order < for the set A that will merely be an extension of the given partial order in the sense that if a ≤ b, then a < b. The process of constructing a linear order such as is called Toplogical Sorting .
Now, topological sorting:
Type 1 = {36,24,8,12,6,3,2,1}
Type 2 = {36,24,8,12,6,2,3,1}