Knowledge Representation and Rough Sets
Jouni Järvinen · 2007
Contents 1 Introduction 1 1.1 Complete Congruences and Morphisms of Semilattices . . . . . . 2 1.2 Information Systems and Preimage Relations . . . . . . . . . . . 4 1.3 Dependence Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Rough Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Preliminaries 8 2.1 General Notation and Conventions . . . . . . . . . . . . . . . . . 8 2.2 Orders and Lattices . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 Complete Congruences and Morphisms of Semilattices 22 3.1 Closure Operators on Ordered Sets . . . . . . . . . . . . . . . . . 22 3.2 Complete Congruences on Semilattices . . . . . . . . . . . . . . 33 3.3 Complete Morphisms of Semilattices . . . . . . . . . . . . . . . . 42 3.4 Dense Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.5 Dual Galois Connections . .