Starting with some well-known concepts from algebra, i.e., the concepts of universal algebra and homomorphism, the author introduces a general notion of model and modelling. Then he describes the construction method I generating a universal algebra ALG which is a model of a given universal algebra ALG'. The described construction method II solves the "inverse" problem, i.e., to a given universal algebra ALG a universal algebra ALG' is generated such that ALG is a model of ALG'. Furthermore, the concept of linguistic variable is updated to the notion of linguistic (universal) algebra. Finally, the construction method I is applied to generate a linguistic model for a given fuzzy (universal) algebra, and vice versa, and a fuzzy algebra such that a given linguistic algebra is a linguistic model for the generated fuzzy algebra (by applying the construction method II). A decisive means in this considerations is the concept of context dependent fuzzy set introduced previously by the author (1999).

For pt.II, 31st IEEE International Symposium on Multiple-Valued Logic (2001). In three previous papers we have developed axiomatic characterizations of approximation operators which: firstly, are defined by classical diamond and box operator of the modal logic (Thiele, 2000); secondly, are defined by the "fuzzified" diamond and box operator applied to crisp sets, i.e. by using the concept of fuzzy rough set (Thiele, 2000); thirdly, are defined by the classical diamond and box operator applied to fuzzy sets, i.e. by using the concepts of rough fuzzy sets (Thiele, 2001). The paper presented is the last one concerning these ideas, i.e. we take the "fuzzified" diamond and box operator and apply these to fuzzy sets. So we define a further type of fuzzy approximation operators and develop an axiomatic characterization of such operators.

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.

The paper deals with the well-known defuzzification procedures on the one hand and approaches to axiomatize the concept of defuzzification, on the other hand. We present a new attempt to establish an axiomatic foundation for defuzzification theory using the theory of groups and the theory of partially ordered sets, and in particular, the theory of GALOIS connections.

We investigate the input-output behaviour of the inference machine of a fuzzy controller, which is interpreted as a functional operator operating on the set of all fuzzy sets defined on a given universe. The concepts of fixed point, eigen function, eigen value and linearity are suitably translated into the fuzzy approach presented. Classical fixed point theorems (Birkhoff, Zorn, Banach) are applied in order to investigate the behaviour of inference machines. The concept of linearity is studied for Mamdani interpretations.