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Fuzzy models of first order languages. (English) Zbl 0583.03043
The concepts of generalized algebra and realization introduced by H. Rasiowa and R. Sikorski [The mathematics of metamathematics (1963; Zbl 0122.243)] are utilized to propose a general foundation of fuzzy set theory. The fuzzy models of a first order language are treated in a suitable categorial setting. Moreover, the concepts of direct product, and quotient for fuzzy models are introduced and examined. Finally, the question of associating to a fuzzy model a suitable crisp version with the same first order properties is examined. To give some examples of applications of such results, we examine the case of the fuzzy algebras. Then the concepts of direct product, congruence, morphism, quotient for fuzzy algebra are defined.

MSC:
03E72 Theory of fuzzy sets, etc.
03C90 Nonclassical models (Boolean-valued, sheaf, etc.)
03G30 Categorical logic, topoi
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