Logical and Algebraic Foundations of Orderings in Fuzzy and Many-Valued Logics

Within the past thirty years, fuzzy relations have turned out to as rich and meaningful concepts - not only only from a purely theoretical point of view. Fruitful applications have emerged in decision analysis, business intelligence, fuzzy control, computational linguistics, and several other disciplines. While fuzzy equivalence relations are nowadays established as fundamental concepts, fuzzy orderings have attracted almost no interest. As pointed out in the applicants doctoral thesis and post-doc research work, there is a fundamental problem in the original conception of fuzzy orderings. Consequently, a generalized approach has been introduced which solves these problems by taking an underlying fuzzy equivalence relation into account, thereby, complying with the classical factorization construction. The present project is concerned with a deeper investigation of the new framework of fuzzy orderings with the intention to establish a well-founded order theory based on fuzzy and many-valued logics. In order to achieve this goal, it is necessary to study fuzzy orderings from a deeper logical and algebraic level. This includes the issues how linearity and lexicographic composition can be defined in a meaningful way, how the theory of fuzzy orderings can be reformulated in formal logical frameworks, and more application-oriented research, e.g. aggregation for flexible query systems and ordering-based modifiers.