Abelian Groups and Representations of Finite Partially Ordered Sets

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Springer Science & Business Media, Jun 16, 2000 - Mathematics - 244 pages
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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.
 

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Contents

Representations of Posets over a Field
1
12 Representations of Posets and Matrix Problems
10
13 Finite Representation Type
19
14 Tame and Wild Representation Type
31
15 Generic Representations
39
TorsionFree Abelian Groups
47
22 Nearisomorphism of Finite Rank Groups
56
23 Stable Range Conditions for Finite Rank Groups
62
Almost Completely Decomposable Groups
144
52 Isomorphism at p and Representation Type
154
53 Uniform Groups
164
54 Primary Regulating Quotient Groups
169
Representations over Fields and Exact Sequences
173
62 Coxeter Correspondences
178
63 Almost Split Sequences
187
64 A Torsion Theory and Localizations
191

24 SelfSmall Groups and Endomorphism Rings
69
Butler Groups
76
32 Characterizations of Finite Rank Groups
88
33 Quasiisomorphism and QRepresentations of Posets
100
34 Countable Groups
105
35 QuasiGeneric Groups
111
Representations over a Discrete Valuation Ring
126
42 Wild Modulo p Representation Type
135
43 Finite Rank Butler Groups and Isomorphism at p
140
Finite Rank Butler Groups
197
72 Endomorphism Rings
202
73 Bracket Groups
205
Applications of Representations and Butler Groups
211
82 Finite Valuated Groups
218
References
223
List of Symbols
235
Index of Terms
241
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About the author (2000)

David Arnold is Emeritus Professor of History at the University of Warwick, and previously taught at the University of Dar es Salaam, the University of Lancaster, and the School of Oriental and African Studies, University of London. A founder member of the Subaltern Studies group, he has been a visiting professor in Chicago and Zurich, and is a Fellow of the British Academy. His published work includes Colonizing the Body: State Medicine and Epidemic Disease in Nineteenth-Century India; Science, Technology and Medicine in Colonial India; Gandhi; The Tropics and the Traveling Gaze: India, Landscape, and Science, 1800 1856, and Everyday Technology: Machines and the Making of India's Modernity.

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