| This thesis develops mathematical foundations of isotone Galois connections and the associated concept lattices. In particular, we study isotone fuzzy Galois connections and concept lattices parameterized by linguistic hedges. Isotone fuzzy Galois connections and concept lattices provide an alternative to antitone fuzzy Galois connections and concept lattices which are the foundational structures for formal concept analysis of data with fuzzy attributes. We demonstrate that hedges enable us to control the number of fixed points of Galois connections, i.e. collections of objects and attributes which represent interesting clusters in data. We present properties of isotone connections with hedges, including their axiomatization, and describe the structure of the associated concept lattices. In addition, we present a logic of if-then rules such as "if all attributes of an object are among those from A then they are among those from B." We provide basic syntactic and semantic notions, describe complete non-redundant sets of the if-then rules, and a logic for reasoning with such dependencies with its ordinary-style and graded-style completeness. |
