posetnotes - NOTES ON POSETS 1. Lecture, scribed by Sara...

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Unformatted text preview: NOTES ON POSETS 1. Lecture, scribed by Sara Evans Definition. Posets are irreflexive, transitive relations. Previously, • ≤- reflexive, antisymmetric, transitive • <- irreflexive, transitive Definition. Graphs are irreflexive, symmetric relations. Think of ground sets as points. In the case of graphs, connect pairs of points. Figure 1. Point a is in relation with point b . Let X be the ground set of the posets. Recall that if a,b ∈ X , and a || b then there exists a linear extension of ( X, ≤ ) such that a ≤ b . Notations. P = ( X, ≤ ) • a ≤ b : ( a,b ) ∈≤ • a < b : ( a,b ) ∈≤ ,a 6 = b • a || b : a 6≤ b and b 6≤ a • a ⊥ b : a ≤ b or b ≤ a Corollary. P = ( X, ≤ ) and { L 1 ,L 2 ,...,L k } are the set of all linear extensions. Then L 1 ∩ L 2 ∩ ··· ∩ L k = P Proof. If ( a,b ) ∈ P = ⇒ ( a,b ) ∈ L i ∀ i = ⇒ ( a,b ) ∈ L 1 ∩ ··· ∩ L k . If ( a,b ) / ∈ P = ⇒ a || b = ⇒ ∃ L i 1 ,L i 2 such that a ≤ b in L i 1 and b ≤ a in L i 2 = ⇒ ( a,b ) / ∈ L i 1 ∩ L i 2 = ⇒ ( a,b ) / ∈ L 1 ∩ ··· ∩ L k . Definition. Let P = ( X, ≤ ). A set { L 1 ,L 2 ,...,L t } of linear extensions of P is called a realizer if L 1 ∩ L 2 ∩ ··· ∩ L t = P. 1 2 NOTES ON POSETS Definition. The dimension of a poset P (denoted dim( P )) is the minimum cardinality of a realizer....
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This note was uploaded on 01/12/2012 for the course MATH 681 taught by Professor Wildstrom during the Fall '09 term at University of Louisville.

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posetnotes - NOTES ON POSETS 1. Lecture, scribed by Sara...

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