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The Structure of $k$-$CS$- Transitive Cycle-Free Partial Orders
 
Richard Warren University of Leeds, England
Front Cover for The Structure of k-CS- Transitive Cycle-Free Partial Orders
Available Formats:
Electronic ISBN: 978-1-4704-0199-3
Product Code: MEMO/129/614.E
166 pp 
List Price: $54.00
MAA Member Price: $48.60
AMS Member Price: $32.40
Front Cover for The Structure of k-CS- Transitive Cycle-Free Partial Orders
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  • Front Cover for The Structure of k-CS- Transitive Cycle-Free Partial Orders
  • Back Cover for The Structure of k-CS- Transitive Cycle-Free Partial Orders
The Structure of $k$-$CS$- Transitive Cycle-Free Partial Orders
Richard Warren University of Leeds, England
Available Formats:
Electronic ISBN:  978-1-4704-0199-3
Product Code:  MEMO/129/614.E
166 pp 
List Price: $54.00
MAA Member Price: $48.60
AMS Member Price: $32.40
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1291997
    MSC: Primary 06;



    The class of cycle-free partial orders (CFPOs) is defined, and the CFPOs fulfilling a natural transitivity assumption, called \(k\)-connected set transitivity (\(k\)-\(CS\)-transitivity), are analyzed in some detail. Classification in many of the interesting cases is given. This work generalizes Droste's classification of the countable \(k\)-transitive trees (\(k \geq 2\)). In a CFPO, the structure can branch downwards as well as upwards, and can do so repeatedly (though it never returns to the starting point by a cycle). Mostly it is assumed that \(k \geq 3\) and that all maximal chains are finite. The main classification splits into the sporadic and skeletal cases. The former is complete in all cardinalities. The latter is performed only in the countable case. The classification is considerably more complicated than for trees, and skeletal CFPOs exhibit rich, elaborate and rather surprising behavior.

    Features:

    • Lucid exposition of an important generalization of Droste's work
    • Extended introduction clearly explaining the scope of the memoir
    • Visually attractive topic with copious illustrations
    • Self-contained material, requiring few prerequisites

    Readership

    Undergraduate students, graduate students, research mathematicians and physicists interested in elliptic functions.

  • Table of Contents
     
     
    • Chapters
    • 1. Extended introduction
    • 2. Preliminaries
    • 3. Properties of $k$-$CS$-transitive CFPOs
    • 4. Constructing CFPOs
    • 5. Characterization and isomorphism theorems
    • 6. Classification of skeletal CFPOs (Part 1)
    • 7. Classification of skeletal CFPOs (Part 2)
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Volume: 1291997
MSC: Primary 06;



The class of cycle-free partial orders (CFPOs) is defined, and the CFPOs fulfilling a natural transitivity assumption, called \(k\)-connected set transitivity (\(k\)-\(CS\)-transitivity), are analyzed in some detail. Classification in many of the interesting cases is given. This work generalizes Droste's classification of the countable \(k\)-transitive trees (\(k \geq 2\)). In a CFPO, the structure can branch downwards as well as upwards, and can do so repeatedly (though it never returns to the starting point by a cycle). Mostly it is assumed that \(k \geq 3\) and that all maximal chains are finite. The main classification splits into the sporadic and skeletal cases. The former is complete in all cardinalities. The latter is performed only in the countable case. The classification is considerably more complicated than for trees, and skeletal CFPOs exhibit rich, elaborate and rather surprising behavior.

Features:

  • Lucid exposition of an important generalization of Droste's work
  • Extended introduction clearly explaining the scope of the memoir
  • Visually attractive topic with copious illustrations
  • Self-contained material, requiring few prerequisites

Readership

Undergraduate students, graduate students, research mathematicians and physicists interested in elliptic functions.

  • Chapters
  • 1. Extended introduction
  • 2. Preliminaries
  • 3. Properties of $k$-$CS$-transitive CFPOs
  • 4. Constructing CFPOs
  • 5. Characterization and isomorphism theorems
  • 6. Classification of skeletal CFPOs (Part 1)
  • 7. Classification of skeletal CFPOs (Part 2)
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