Abelian Groups and Representations of Finite Partially Ordered Sets

by Arnold, David M.

• ISBN13: 9780387989822
• ISBN10: 038798982X
• Format: Hardcover
• Copyright: 2000-06-01
• Publisher: Springer Verlag

Summary

A recurring theme in a traditional introductory graduate algebra course is the existence and consequences of relationships between different algebraic structures. This is also the theme of this book, an exposition of connections between representations of finite partially ordered sets and abelian groups. Emphasis is placed throughout on classification, a description of the objects up to isomorphism, and computation of representation type, a measure of when classification is feasible. David M. Arnold is the Ralph and Jean Storm Professor of Mathematics at Baylor University. He is the author of "Finite Rank Torsion Free Abelian Groups and Rings" published in the Springer-Verlag Lecture Notes in Mathematics series, a co-editor for two volumes of conference proceedings, and the author of numerous articles in mathematical research journals. His research interests are in abelian group theory and related topics, such as representations of partially ordered sets and modules over discrete valuation rings, subrings of algebraic number fields, and pullback rings. He received his Ph. D. from the University of Illinois, Urbana and was a member of the faculty at New Mexico State University for many years.

Table of Contents

Preface	vii
Representations of Posets over a Field	1	(46)
Vector Spaces with Distinguished Subspaces	1	(9)
Representations of Posets and Matrix Problems	10	(9)
Finite Representation Type	19	(12)
Tame and Wild Representation Type	31	(8)
Generic Representations	39	(8)
Torsion-Free Abelian Groups	47	(29)
Quasi-isomorphism and Isomorphism at p	47	(9)
Near-isomorphism of Finite Rank Groups	56	(6)
Stable Range Conditions for Finite Rank Groups	62	(7)
Self-Small Groups and Endomorphism Rings	69	(7)
Butler Groups	76	(50)
Types and Completely Decomposable Groups	76	(12)
Characterizations of Finite Rank Groups	88	(12)
Quasi-isomorphism and Q-Representations of Posets	100	(5)
Countable Groups	105	(6)
Quasi-Generic Groups	111	(15)
Representations over a Discrete Valuation Ring	126	(18)
Finite and Rank-Finite Representation Type	126	(9)
Wild Modulo p Representation Type	135	(5)
Finite Rank Butler Groups and Isomorphism at p	140	(4)
Almost Completely Decomposable Groups	144	(29)
Characterizations and Properties	144	(10)
Isomorphism at p and Representation Type	154	(10)
Uniform Groups	164	(5)
Primary Regulating Quotient Groups	169	(4)
Representations over Fields and Exact Sequences	173	(24)
Projectives, Injectives, and Exact Sequences	173	(5)
Coxeter Correspondences	178	(9)
Almost Split Sequences	187	(4)
A Torsion Theory and Localizations	191	(6)
Finite Rank Butler Groups	197	(14)
Projectives, Injectives, and Exact Sequences	197	(5)
Endomorphism Rings	202	(3)
Bracket Groups	205	(6)
Applications of Representations and Butler Groups	211	(12)
Torsion-Free Modules over Discrete Valuation Rings	211	(7)
Finite Valuated Groups	218	(5)
References	223	(12)
List of Symbols	235	(6)
Index of Terms	241