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, S₄, partial order and linear order relations