Ebook: The Theory of Lattice-Ordered Groups
- Tags: Order Lattices Ordered Algebraic Structures, Group Theory and Generalizations, Mathematical Logic and Foundations
- Series: Mathematics and Its Applications 307
- Year: 1994
- Publisher: Springer Netherlands
- Edition: 1
- Language: English
- pdf
A partially ordered group is an algebraic object having the structure of a group and the structure of a partially ordered set which are connected in some natural way. These connections were established in the period between the end of 19th and beginning of 20th century. It was realized that ordered algebraic systems occur in various branches of mathemat ics bound up with its fundamentals. For example, the classification of infinitesimals resulted in discovery of non-archimedean ordered al gebraic systems, the formalization of the notion of real number led to the definition of ordered groups and ordered fields, the construc tion of non-archimedean geometries brought about the investigation of non-archimedean ordered groups and fields. The theory of partially ordered groups was developed by: R. Dedekind, a. Holder, D. Gilbert, B. Neumann, A. I. Mal'cev, P. Hall, G. Birkhoff. These connections between partial order and group operations allow us to investigate the properties of partially ordered groups. For exam ple, partially ordered groups with interpolation property were intro duced in F. Riesz's fundamental paper [1] as a key to his investigations of partially ordered real vector spaces, and the study of ordered vector spaces with interpolation properties were continued by many functional analysts since. The deepest and most developed part of the theory of partially ordered groups is the theory of lattice-ordered groups. In the 40s, following the publications of the works by G. Birkhoff, H. Nakano and P.
This volume makes both classical and new results of the theory of lattice-ordered groups available to a wide range of mathematicians in a comprehensive way, explaining the structure of the theory as well as indicating its applications.
The book contains the foundations of the theory of lattice-ordered groups, and the theory of ordered permutation groups. It describes totally-ordered and right-ordered groups, and highlights the theory of varieties and quasi-varieties of lattice-ordered groups. The distinguishing feature of this work is the group-theoretical and universal algebra attitude to the theory of lattice-ordered groups.
This volume will be of interest to graduate students and researchers with a basic knowledge of group theory. It serves as an excellent introduction to the theory of partially ordered groups, and as an overview of new ideas and results in this theory.
This volume makes both classical and new results of the theory of lattice-ordered groups available to a wide range of mathematicians in a comprehensive way, explaining the structure of the theory as well as indicating its applications.
The book contains the foundations of the theory of lattice-ordered groups, and the theory of ordered permutation groups. It describes totally-ordered and right-ordered groups, and highlights the theory of varieties and quasi-varieties of lattice-ordered groups. The distinguishing feature of this work is the group-theoretical and universal algebra attitude to the theory of lattice-ordered groups.
This volume will be of interest to graduate students and researchers with a basic knowledge of group theory. It serves as an excellent introduction to the theory of partially ordered groups, and as an overview of new ideas and results in this theory.
Content:
Front Matter....Pages i-xvi
Lattices....Pages 1-9
Lattice-ordered groups....Pages 11-29
Convex l-subgroups....Pages 31-50
Ordered permutation groups....Pages 51-90
Right-ordered groups....Pages 91-110
Totally ordered groups....Pages 111-131
Embeddings of lattice-ordered groups....Pages 133-160
Lattice properties in lattice-ordered groups....Pages 161-185
Varieties of lattice-ordered groups....Pages 187-236
Free l-groups....Pages 237-254
The semigroup of l-varieties....Pages 255-281
The lattice of l-varieties....Pages 283-334
Ordered permutation groups and l-varieties....Pages 335-343
Quasivarieties of lattice-ordered groups....Pages 345-377
Back Matter....Pages 379-400
This volume makes both classical and new results of the theory of lattice-ordered groups available to a wide range of mathematicians in a comprehensive way, explaining the structure of the theory as well as indicating its applications.
The book contains the foundations of the theory of lattice-ordered groups, and the theory of ordered permutation groups. It describes totally-ordered and right-ordered groups, and highlights the theory of varieties and quasi-varieties of lattice-ordered groups. The distinguishing feature of this work is the group-theoretical and universal algebra attitude to the theory of lattice-ordered groups.
This volume will be of interest to graduate students and researchers with a basic knowledge of group theory. It serves as an excellent introduction to the theory of partially ordered groups, and as an overview of new ideas and results in this theory.
Content:
Front Matter....Pages i-xvi
Lattices....Pages 1-9
Lattice-ordered groups....Pages 11-29
Convex l-subgroups....Pages 31-50
Ordered permutation groups....Pages 51-90
Right-ordered groups....Pages 91-110
Totally ordered groups....Pages 111-131
Embeddings of lattice-ordered groups....Pages 133-160
Lattice properties in lattice-ordered groups....Pages 161-185
Varieties of lattice-ordered groups....Pages 187-236
Free l-groups....Pages 237-254
The semigroup of l-varieties....Pages 255-281
The lattice of l-varieties....Pages 283-334
Ordered permutation groups and l-varieties....Pages 335-343
Quasivarieties of lattice-ordered groups....Pages 345-377
Back Matter....Pages 379-400
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