On algebraic foundations of information granulation

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, S-4, partial order, and linear order relations, however, in the crisp as well as in the fuzzy case.