On Algebraic Foundations of Information Granulation

Abstract

In his paper Towards a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic L.A. ZADEH writes There are three basic concepts that underline human cognition: granulation, organisation and causation. Informally, granulation involves decomposition of whole into parts; organisation involves integration of parts into whole; and causation involves association of causes with effects.Granulation of an object A leads to a collection of granules of A, with a granule being a clump of points (objects) drawn together by indistinguishability, similarity, proximity or functionality?Furthermore, he writes: Modes of information granulation (IG) in which the granules are crisp (c-granular) play important roles in a wide variety of methods, approaches and techniques. Crisp IG, however, does not reflect the fact that in almost all of human reasoning and concept formation the granules are fuzzy (f-granular).The contribution presented contains first steps to develop a systematic mathematical theory fitting ZADEH's approach. Because of lacking space we restrict our considerations to the generation of granulations by equivalence, tolerance, S4, partial order, and linear order relations, however, in the crisp as well as in the fuzzy case.