Electronic ISBN:  9781470402129 
Product Code:  MEMO/131/623.E 
List Price:  $52.00 
MAA Member Price:  $46.80 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 131; 1998; 125 ppMSC: Primary 08; 06; 03; 20; 54;
This volume is about treelike 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
ReadershipGraduate students and research mathematicians interested in relational systems and laws of composition.

Table of Contents

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, halflines 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 treelike objects

33. $B$relations and partially ordered sets

34. Configurations of sets, with applications to permutation groups


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This volume is about treelike 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, halflines 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 treelike objects

33. $B$relations and partially ordered sets

34. Configurations of sets, with applications to permutation groups