On algebraic foundations of information granulation. I. The crisp case

Zadeh (1997) writes "There are three basic concepts that underline human cognition: granulation, organisation and causation. 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 of 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 paper is a first step to develop a systematic mathematical theory fitting Zadeh's approach. We restrict our considerations to the "crisp case", i.e., we consider only crisp granules. Furthermore, we study only the generation of granulation by binary relations, i.e., by equivalence, tolerance, S/sub 4/, partial order and linear order relations.