## Abelian Groups and Representations of Finite Partially Ordered SetsA 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 |

223 | |

List of Symbols | 235 |

241 | |

### Other editions - View all

Abelian Groups and Representations of Finite Partially Ordered Sets David Arnold Limited preview - 2012 |

Abelian Groups and Representations of Finite Partially Ordered Sets Springer No preview available - 2012 |

Abelian Groups and Representations of Finite Partially Ordered Sets David Arnold No preview available - 2012 |

### Common terms and phrases

algebra completely decomposable group consequence Corollary countable critical typeset Define direct sum discrete valuation ring elements End G endomorphism ring epimorphism exact sequence Example faithful functor finite direct sum finite lattice finite poset finite rank Butler finite rank torsion-free finite representation type finite valuated finite-dimensional frA/p free module G and H G is isomorphic group G groups of finite Hence Hom(G homomorphism idempotent indecomposable representations induces isomorphic to H lattice of types Lemma Let G matrix modulo p representation monomorphism morphism nearly isomorphic nonzero integer p-group partially ordered sets prime principal ideal domain PROOF Proposition pure rank-l subgroup pure subgroup pure submodule QEnd quasi-isomorphic r-homogeneous completely decomposable rank Butler group rank-l group regulating subgroup representation morphism split sequence subgroup of G subposet subring subset summand of G Theorem torsion-free abelian group typeset G w(Sr wild modulo wild representation type