 | Casals-Ruiz M. On systems of equations over free partially commutative groups / M.Casals-Ruiz, I.Kazachkov. - Providence: American Mathematical Society, 2011. - vii, 153 p. - (Memoirs of the American Mathematical Society; vol.212, N 999). - Bibliogr.: p.139-143. - Ind.: p.145-147. - ISBN 978-0-8218-5258-3; ISSN 0065-9266
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List of Figures ............................................... vii
Chapter 1. Introduction ......................................... 1
Chapter 2. Preliminaries ....................................... 13
2.1. Graphs and relations ...................................... 13
2.2. Elements of algebraic geometry over groups ................ 13
2.3. Formulas in the languages  A and G .......................
2.4. First order logic and algebraic geometry .................. 17
2.5. Partially commutative groups .............................. 17
2.6. Partially commutative monoids and DM-normal forms ......... 22
Chapter 3. Reducing systems of equations over  to
constrained generalised equations over  ............ 25
3.1. Definition of (constrained) generalised equations ......... 27
3.2. Reduction to generalised equations: from partially
commutative groups to monoids ............................. 32
3.3. Reduction to generalised equations: from partially
commutative monoids to free monoids ....................... 37
3.4. Example ................................................... 41
Chapter 4. The process: construction of the tree T ............. 49
4.1. Preliminary definitions ................................... 50
4.2. Elementary transformations ................................ 51
4.3. Derived transformations ................................... 56
4.4. Construction of the tree Т(Ω) ............................. 63
Chapter 5. Minimal solutions ................................... 71
Chapter 6. Periodic structures ................................. 77
6.1. Periodic structures ....................................... 77
6.2. Example ................................................... 86
6.3. Strongly singular and singular cases ...................... 88
6.4. Example ................................................... 91
6.5. Regular case .............................................. 92
Chapter 7. The finite tree Т0(Ω) and minimal solutions ......... 99
7.1. Automorphisms ............................................ 100
7.2. The finite subtree Т0(Ω) ................................. 103
7.3. Paths p(H) are in Т0(Ω) .................................. 110
Chapter 8. Prom the coordinate group GR(Ω*) to proper
quotients: the decomposition tree Тdес and the
extension tree Text ................................. 121
8.1. The decomposition tree Тdес(Ω) ............................ 121
8.2. Example .................................................. 127
8.3. The extension tree Text(Ω) ................................ 127
Chapter 9. The solution tree Tsol(Ω) and the main theorem ...... 131
9.1. Example .................................................. 136
Bibliography .................................................. 139
Index ......................................................... 145
Glossary of notation .......................................... 149
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