Many-valued preorders I: The basis of many-valued mathematics

Abstract

Both parts of this paper form a survey on the close relationship between enriched category and fuzzy set theory and focuses on such fundamental axioms as reflexivity, transitivity, symmetry, antisymmetry. Part I (this is the present paper) deals with reflexivity and transitivity and develops the algebraic basis of manyvalued preordered sets including their Cauchy completion. Further the change of base is explained which plays a fundamental role in many-valued mathematics. Part II (Höhle, Many-valued preorders II: the symmetry axiom and probabilistic geometry (in this volume) [1]) will treat the symmetry axiom and its applications to probabilistic geometry—a theory which can be viewed as a predecessor of fuzzy set theory.