a poset is totally ordered if
DESCRIPTION
A poset is totally ordered if. A totally ordered poset is called a chain. The set. with the relation. Let be the set of all real numbers, and let have the usual meaning; then is a chain. if for all. - PowerPoint PPT PresentationTRANSCRIPT
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A poset is totally ordered if
YX,pair every for X,Yor YX .4
A totally ordered poset is called a chain.
) , (
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y x
• Let be the set of all real numbers, and let have theusual meaning; then is a chain.
• The set d with the relation )y ,...,y ,(y )x...,,x,x(" d21d21 if for all ii y x i" is a poset. It is a chain if and only if d = 1.
• Given a set E, the power set Ρ(E) comprising all subsets of Ebecomes a poset under the inclusion relation, that is, YX "if and only if ."YX The empty set is denoted by .
Examples:
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• Let be the set of all nonnegative integers, and put nmif m divides n. Then is a poset.
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• Inclusion relation and Hasse diagrams
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or
g)(f g)(f .,. ge
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and
)( pi is a segment (a subset of a partition) of X containing p
i
j
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The connected component labelling
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• Sequential algorithm
Initially: lbl (label image) 0; lval (current label) 0; t is the current flat zone
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• City-block metric:
• Chessboard metric
• Euclidean distance
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Distance functions
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= backward 4- or 8-connected neighbors
= forward 4- or 8-connected neighbors
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