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Relations Related to Betweenness: Their Structure and Automorphisms

S. A. Adeleke Western Illinois University, Macomb, IL
Peter M. Neumann Queen’s College, Oxford University, Oxford, England
Available Formats:
Electronic ISBN: 978-1-4704-0212-9
Product Code: MEMO/131/623.E
List Price: $52.00 MAA Member Price:$46.80
AMS Member Price: $31.20 Click above image for expanded view Relations Related to Betweenness: Their Structure and Automorphisms S. A. Adeleke Western Illinois University, Macomb, IL Peter M. Neumann Queen’s College, Oxford University, Oxford, England Available Formats:  Electronic ISBN: 978-1-4704-0212-9 Product Code: MEMO/131/623.E  List Price:$52.00 MAA Member Price: $46.80 AMS Member Price:$31.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 1311998; 125 pp
MSC: Primary 08; 06; 03; 20; 54;

This volume is about tree-like structures, namely semilinear ordering, general betweenness relations, $C$-relations and $D$-relations. It contains a systematic study of betweenness and introduces $C$- and $D$-relations to describe the behavior of points at infinity (“leaves” or “ends” or “directions”) of trees. The focus is on structure theorems and on automorphism groups, with applications to the theory of infinite permutation groups.

Features:

• offers the first systematic treatment of betweenness relations
• introduces important new concepts of $C$-relations and $D$-relations
• elucidates the close relationships between semilinear orderings, betweenness relations, $C$- and $D$-relations
• considers their automorphism groups as important examples of Jordan permutation groups

Graduate students and research mathematicians interested in relational systems and laws of composition.

• Chapters
• Part I. Preparation
• 1. Introduction
• 2. Terminology and notation
• 3. Linear relational structures
• Part II. Semilinear order relations
• 4. Semilinearly ordered sets
• 5. Examples of semilinear orderings
• 6. Automorphism groups of semilinear orderings
• 7. Maximal chains in semilinear orderings
• 8. Piecewise linear maximal chains in semilinear orderings
• 9. Enriching a semilinear ordering
• Part III. Abstract chain sets
• 10. $C$-relations
• 11. Examples of $C$-sets
• 12. The classification of $C$-sets
• 13. A topology for $C$-sets
• 14. Automorphism groups of $C$-sets
• Part IV. General betweenness relations
• 15. $B$-relations and general betweenness relations
• 16. Lines, half-lines and directions in a $B$-set
• 17. The relationship between $B$-sets and semilinear orderings
• 18. Components of $B$-sets
• 19. Branch points and sectors of $B$-sets
• 20. Automorphism groups of $B$-sets
• 21. Improving a $B$-relation to a betweenness relation
• Part V. Abstract direction sets
• 22. $D$-relations
• 23. Examples of $D$-sets
• 24. Structural partitions
• 25. Linking of structural partitions
• 26. The betweenness relation derived from a $D$-relation
• 27. The topology on a $D$-set
• 28. Automorphism groups of $D$-sets
• Part VI. Applications and commentary
• 29. Combinatorial trees and discrete $B$-sets
• 30. Arboreal group theory
• 31. $B$-relations and topological spaces
• 32. Cameron’s tree-like objects
• 33. $B$-relations and partially ordered sets
• 34. Configurations of sets, with applications to permutation groups
• 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: 1311998; 125 pp
MSC: Primary 08; 06; 03; 20; 54;

This volume is about tree-like structures, namely semilinear ordering, general betweenness relations, $C$-relations and $D$-relations. It contains a systematic study of betweenness and introduces $C$- and $D$-relations to describe the behavior of points at infinity (“leaves” or “ends” or “directions”) of trees. The focus is on structure theorems and on automorphism groups, with applications to the theory of infinite permutation groups.

Features:

• offers the first systematic treatment of betweenness relations
• introduces important new concepts of $C$-relations and $D$-relations
• elucidates the close relationships between semilinear orderings, betweenness relations, $C$- and $D$-relations
• considers their automorphism groups as important examples of Jordan permutation groups

Graduate students and research mathematicians interested in relational systems and laws of composition.

• Chapters
• Part I. Preparation
• 1. Introduction
• 2. Terminology and notation
• 3. Linear relational structures
• Part II. Semilinear order relations
• 4. Semilinearly ordered sets
• 5. Examples of semilinear orderings
• 6. Automorphism groups of semilinear orderings
• 7. Maximal chains in semilinear orderings
• 8. Piecewise linear maximal chains in semilinear orderings
• 9. Enriching a semilinear ordering
• Part III. Abstract chain sets
• 10. $C$-relations
• 11. Examples of $C$-sets
• 12. The classification of $C$-sets
• 13. A topology for $C$-sets
• 14. Automorphism groups of $C$-sets
• Part IV. General betweenness relations
• 15. $B$-relations and general betweenness relations
• 16. Lines, half-lines and directions in a $B$-set
• 17. The relationship between $B$-sets and semilinear orderings
• 18. Components of $B$-sets
• 19. Branch points and sectors of $B$-sets
• 20. Automorphism groups of $B$-sets
• 21. Improving a $B$-relation to a betweenness relation
• Part V. Abstract direction sets
• 22. $D$-relations
• 23. Examples of $D$-sets
• 24. Structural partitions
• 25. Linking of structural partitions
• 26. The betweenness relation derived from a $D$-relation
• 27. The topology on a $D$-set
• 28. Automorphism groups of $D$-sets
• Part VI. Applications and commentary
• 29. Combinatorial trees and discrete $B$-sets
• 30. Arboreal group theory
• 31. $B$-relations and topological spaces
• 32. Cameron’s tree-like objects
• 33. $B$-relations and partially ordered sets
• 34. Configurations of sets, with applications to permutation groups
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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