Answer to Question #126880 in Functional Analysis for Amjad ali

Question

prove that a finite partially ordered set A has at least one maximal element

Answer 1

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 a₁ ≤ a₀ and a₁ ≠ a₀. 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 a₂ ≤ a₁ and a₂ ≠ a₁. We repeat this pattern until we obtain a maximal element or until we obtain a_n. For a_n we have the property:

a₀ ≤ a₁ ≤ a₂ ≤ … a_n-1 ≤ a_n

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