Electronic ISBN:  9781470401993 
Product Code:  MEMO/129/614.E 
List Price:  $54.00 
MAA Member Price:  $48.60 
AMS Member Price:  $32.40 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 129; 1997; 166 ppMSC: Primary 06;
The class of cyclefree 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
 Selfcontained material, requiring few prerequisites
ReadershipUndergraduate 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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The class of cyclefree 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
 Selfcontained material, requiring few prerequisites
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)