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E-raamat: Theory of Lattice-Ordered Groups

(Indiana University, South Bend)
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A reference for research mathematicians and a text for graduate students who want to become familiar with the standard theorems, constructions, and techniques of lattice-group theory. Assumes a knowledge of free groups, solvable and nilpotent groups, filters, and other topics encountered in graduate courses in group theory and topology. Begins with the general concepts common to all lattice-groups then proceeds through successively finer classes. Annotation copyright Book News, Inc. Portland, Or.

Provides a thorough discussion of the orderability of a group. The book details the major developments in the theory of lattice-ordered groups, delineating standard approaches to structural and permutation representations. A radically new presentation of the theory of varieties of lattice-ordered groups is offered.;This work is intended for pure and applied mathematicians and algebraists interested in topics such as group, order, number and lattice theory, universal algebra, and representation theory; and upper-level undergraduate and graduate students in these disciplines.;College or university bookstores may order five or more copies at a special student price which is available from Marcel Dekker Inc, upon request.

Arvustused

"The author. . .carries the reader to the frontiers of current research. . . . . .a useful reference. " ---Mathematical Reviews ". . .includes detailedproofs of the theorems and all the material needed from partially ordered setsand lattices. " ---Analele Stiintifice ". . .an up-to-date and systematic treatment of many topics in the theory of lattice-ordered groups. " ---Acta Scientifica Matematica

Preface iii
The Interplay of Algebra and Order
1(36)
Introduction
1(1)
Partially Ordered Sets, Directed Sets, and Lattices
2(4)
Partially Ordered Groups and Directed Groups
6(3)
Lattice-ordered Groups
9(7)
Absolute Values and the Triangle Inequalities
16(5)
l-permutations and l-homomorphisms
21(5)
l-subgroups and Saturated Subgroups
26(11)
Convex l-subgroups
37(32)
Convex Sets and Subgroups
37(7)
l-ideals and the Coset Ordering
44(4)
Prime Subgroups
48(5)
Regular Subgroups and Values
53(2)
Γ(G) and Plenary Subsets of Γ(G)
55(6)
Convex l-subgroups of l-subgroups
61(8)
Polars and Disjoint Elements
69(50)
The Polar Subgroups
70(6)
Polars and Prime Subgroups
76(6)
Z-subgroups
82(2)
Cardinal Sums and Summands
84(6)
Polar Subgroups of l-subgroups
90(3)
Projectable and Strongly Projectable l-groups
93(4)
Convex o-subgroups and Basic Elements
97(7)
Upper Extensions
104(15)
Complete Distributivity
119(18)
Closed Convex l-subgroups
120(9)
Completely Distributive l-groups
129(4)
Essential Subgroups and the Essential Radical
133(4)
Totally Ordered Structures
137(26)
o-groups
137(11)
o-rings
148(2)
Right-ordered Groups
150(4)
Lex Extensions and Lex Subgroups
154(9)
Order-preserving Permutations of a Chain
163(52)
Beginning Concepts
164(5)
The Cayley-Holland Theorem
169(4)
Convex l-subgroups of A(Omega;)
173(4)
Transitivity
177(7)
Congruences and Blocks
184(9)
Primitive l-group Actions and Their Classifications
193(9)
Cardinality Bounds on l-groups
202(3)
The Wreath Product
205(10)
Classes of l-groups
215(42)
Radical Classes
216(11)
Free l-groups
227(12)
A Brief Introduction to l-varieties
239(10)
The Lateral Completion of an l-group
249(8)
Normal-valued l-groups
257(44)
Normal Values
257(5)
The l-variety of Normal-valued l-groups
262(7)
c-groups and Free Normal-valued l-groups
269(3)
Above and Below Subgroups in Normal-valued l-groups
272(2)
Completely Distributive Normal-valued l-groups
274(1)
Special-valued l-groups
275(19)
Finite-valued l-groups
294(7)
Representable and Abelian l-groups
301(58)
Representable l-groups
301(8)
Completions of Representable l-groups
309(6)
Sheaf-theoretic Representations of Representable l-groups
315(9)
Weakly Abelian and Nilpotent l-groups
324(9)
Abelian l-groups and Vector Lattices
333(11)
Free Abelian l-groups and Vector Lattices
344(15)
Archimedean l-groups
359(60)
General Results on Archimedean l-groups
359(10)
Complete l-groups
369(11)
Hyperarchimedean l-groups
380(8)
p-endomorphisms of an Archimedean l-groups
388(16)
Representation by Almost-finite Continuous Functions
404(15)
l-varieties
419(54)
The l-semigroup of l-varieties
419(17)
Powers of the Representable l-variety
436(10)
l-varieties Based on Finite Conjugate Disjointness
446(7)
Covers
453(20)
Appendix One: The (G) Problem 473(4)
Appendix Two: Piecewise Linear Homogeneous Representations of Abelian l-groups 477(10)
Sources for the
Chapters
487(10)
Bibliography 497(32)
Symbols 529(2)
Index 531
Michael Darnel