| Preface | |
| Classical Methods | |
| Approximation Theorems for Valuations on Fields | p. 3 |
| Introduction to Valuation Theory | p. 3 |
| Partially ordered groups | p. 3 |
| Valuations on fields | p. 6 |
| Approximation Theorems for Krull Valuations | p. 11 |
| Weak approximation theorems | p. 11 |
| Approximation theorems | p. 15 |
| Approximation theorems and upper classes | p. 18 |
| Approximation theorems for Prufer domains | p. 25 |
| Approximation theorems for Prufer rings of Krull type | p. 29 |
| Extensions of approximation theorems | p. 34 |
| Applications in Topological Rings | p. 37 |
| Valuations on Commutative Rings | p. 49 |
| Basic Properties of the Manis Valuation | p. 50 |
| Rings of quotients | p. 50 |
| Large quotient rings | p. 51 |
| Definition of the Manis valuation | p. 52 |
| Comparison of valuations | p. 57 |
| Some useful inequalities | p. 64 |
| R-Prufer Rings | p. 66 |
| Definition of R-Prufer rings | p. 66 |
| R-Prufer valuation rings | p. 69 |
| Valuations with the Inverse Property | p. 72 |
| Definition and basic properties | p. 72 |
| Comparison of valuations with the inverse property | p. 75 |
| Inequalities for valuations with the inverse property | p. 77 |
| The independence of valuations | p. 80 |
| R-Prufer rings and the inverse property | p. 85 |
| Approximation Theorems | p. 89 |
| Compatibility conditions | p. 90 |
| Approximation theorem in the neighborhood of zero | p. 92 |
| General approximation theorem | p. 98 |
| R-Prufer rings and families of valuations | p. 110 |
| Ordered Groups and Homomorphisms | p. 127 |
| Groups of Divisibility | p. 127 |
| Lattice-ordered groups | p. 127 |
| Groups of divisibility | p. 135 |
| Groups with the Theory of Divisors | p. 146 |
| Approximation Theorems for Multistructures | p. 159 |
| Introduction to Multirings | p. 160 |
| Basic facts about m-rings | p. 160 |
| m-valuations | p. 167 |
| Approximation Theorem for Multirings | p. 175 |
| Introduction to d-Groups | p. 187 |
| Approximation Theorems for d-Groups | p. 200 |
| Categorical Methods | |
| Categorical Logic | p. 211 |
| Topoi and Sheaves | p. 211 |
| Interpretation of Logic in Categories | p. 230 |
| The syntax of L | p. 230 |
| The semantic of L | p. 231 |
| Interpretations as subobjects | p. 235 |
| Interpretations as morphisms | p. 237 |
| Relations between interpretations | p. 239 |
| Canonical language and its interpretation | p. 247 |
| Axioms Valid in Interpretations of Logic in Categories | p. 250 |
| Example of completeness | p. 250 |
| Examples of valid sequents | p. 253 |
| Approximation Theorems in Categories | p. 263 |
| Models of a Theory of Approximation Theorems | p. 263 |
| Approximation theorems in algebras | p. 263 |
| Approximation theorems in categories | p. 268 |
| Approximation Theorems and Sheaves | p. 277 |
| [epsilon]-sheaves | p. 277 |
| [epsilon]-sheaves over cofinal subsets | p. 281 |
| Relations between Approximation Theorems | p. 288 |
| Derived approximation theorems | p. 288 |
| Construction of derived approximation theorems | p. 301 |
| Examples | p. 304 |
| Bibliography | p. 307 |
| Index of Notation | p. 317 |
| Index | p. 327 |
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