Preface ....................................................... iii
1 Preliminaries ................................................ 1
1.1 General Notions ......................................... 1
1.2 Ordered Sets and Lattices ............................... 2
1.3 Formal Concept Analysis ................................. 6
2 Approximations in Complete Lattices ......................... 15
2.1 Kernel-Closure Pairs ................................... 16
2.2 Approximations ......................................... 19
2.3 Containment Order ...................................... 24
2.4 Restraining Tolerances ................................. 27
2.5 Approximation Spaces ................................... 33
2.5.1 Connection to Kernel-Closure Pairs .............. 34
2.5.2 Constructions of Approximation Spaces ........... 37
2.5.3 Approximation Space Homomorphisms ............... 40
2.6 Coarsening and Refinement .............................. 44
2.7 Lattices with Additional Operations .................... 47
3 Concept Approximations ...................................... 51
3.1 Kernel-Closure Pairs in Concept Lattices ............... 52
3.2 Lattices of Preconcepts ................................ 54
3.3 Approximations as Conceptual Traces .................... 58
3.4 Sufficient Labellings .................................. 63
3.5 Contextual Representation .............................. 71
3.5.1 Basics about P-Products ......................... 71
3.5.2 Contextual Representation of Lattices of
Traces .......................................... 72
3.5.3 An Example ...................................... 77
3.6 Browsing the Big World Concept Lattice ................. 82
3.6.1 Diagram-aided Concept Exploration ............... 82
3.6.2 An Example ...................................... 85
3.7 Conceptual Approximation Spaces ........................ 91
3.8 Exploration of Lattices of Traces ...................... 95
3.8.1 Prospective Intermediate Contexts ............... 98
3.8.2 Exploring a Prospection ........................ 100
3.8.3 An Example ..................................... 101
3.8.4 How to Combine Lattices of Traces .............. 106
4 Rough Sets ................................................. 109
4.1 The Classical Case .................................... 110
4.2 Rough Sets as Approximations in Powerset Lattices ..... 111
4.2.1 Rough Sets ..................................... 112
4.2.2 Contextual Representation ...................... 115
4.2.3 Positive and Negative Granules ................. 117
4.3 Selfdual Kernel-Closure Pairs ......................... 119
4.3.1 The Negation of a Rough Set .................... 120
4.3.2 Cofinal Subsets of an Ordered Set .............. 123
4.3.3 The Distributive Case: Rough Sets Determined
by Quasiorders ................................. 124
4.4 Granules as Classifiers ............................... 129
4.4.1 Classifier Rough Sets .......................... 131
4.4.2 An Example ..................................... 132
4.4.3 The Distributive Case .......................... 136
List of Symbols ............................................... 141
Index ......................................................... 145
Bibliography .................................................. 149
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