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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
List Price: $54.00 MAA Member Price:$48.60
AMS Member Price: $32.40 Click above image for expanded view The Structure of$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  List Price:$54.00 MAA Member Price: $48.60 AMS Member Price:$32.40
• Book Details

Memoirs of the American Mathematical Society
Volume: 1291997; 166 pp
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

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)
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 1291997; 166 pp
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

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)
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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