This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full
Document
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.
 Fall '09
 WILDSTROM
 Sets

Click to edit the document details