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Unformatted text preview: MATH 681 Notes Combinatorics and Graph Theory I 1 Posets 1.1 Extreme elements Last week we defined maximal, minimal, greatest, and least elements of a poset. We will explicitly determine the useful properties of these elements. The proofs below will generally only present an argument for maximal and greatest elements; the same argument can be modified very slightly to work for minimal and least elements. Proposition 1. If ( S, ) is a nonempty finite poset, then S has at least one maximal element (and at least one minimal element). Proof. Let x S . If x is maximal, than S clearly has a maximal element; otherwise, there must be an x 1 6 = x such that x x 1 . Now let us subject x 1 to the same consideration; if x 1 is maximal, we are done, otherwise there is an x 2 6 = x 1 such that x 1 x 2 . Iterating this process must necessarily have one of two results: either there will be some x i which is maximal, or the sequence x x 1 x 2 x 3 is infinite. If some element is maximal, we will achieve our goal; we shall show that the second situation is impossible. Suppose we do have such an infinite sequence of elements of S , so that each x i x i +1 and x i 6 = x i +1 ....
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