Answer to Question #126880 in Functional Analysis for Amjad ali

Question #126880
prove that a finite partially ordered set A has at least one maximal element
1
Expert's answer
2020-07-20T18:25:18-0400

Let A be a finite partially ordered set. If A is nonempty, then A has at least one maximal element.

PROOF

Since A is nonempty finite set, then there exist a positive integer n such that A has n elements. We can choose an element a0 ∈ A. If a0 is maximal, then we are done as there exist at least one maximal element. Otherwise, there exists an element a1 such that a1a0 and a1a0. If a1 is maximal, then we are done as there exist at least one maximal element. Otherwise we can choose an element a2 such that a2a1 and a2a1. We repeat this pattern until we obtain a maximal element or until we obtain an. For an we have the property:

a0 ≤ a1 ≤ a2 ≤ … an-1 ≤ an

However, there then do not exist any other element in A as A contained n distinct elements. We then note that an needs to be a maximal element as an is smaller than all other elements in A and thus at least one maximal element exists.


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