The paper discusses some core topics which present open problems in the field of mathematical fuzzy logic and in the foundations of fuzzy set theory. It may reasonably be assumed that these problems shall have great influence for the future development of fuzzy logic within the next decade.

A complex fuzzy set is a set whose membership values are vectors in the unit circle in the complex plane. This paper establishes the orthogonality relation of complex fuzzy sets. Two complex fuzzy sets are said to be orthogonal if their membership vectors are perpendicular. We present the basic properties of orthogonality of complex fuzzy sets and various results on orthogonality with respect t...

Based on fuzzy set theory, flexible querying enables users to express preferences inside requirements. Quantifications and preferences on data have led us to define the notion of fuzzy integers ( f). This framework has been extended (to f and f ) in order to dealing with queries based on difference or division operations. In this paper we study how to define fuzzy order relations between such f...

In this manuscript, we introduce  a new class of fuzzy problems, namely ``fuzzy inclusion linear systems" and   propose a fuzzy solution set for it. Then, we present a theoretical discussion about the relationship between  the fuzzy solution set of a  fuzzy inclusion linear system and the algebraic solution of a fuzzy linear system. New necessary and sufficient conditions are derived for obtain...

It is well known fact that binary relations are generalized mathematical functions. Contrary to functions from domain to range, binary relations may assign to each element of domain two or more elements of range. Some basic operations on functions such as the inverse and composition are applicable to binary relations as well. Depending on the domain or range or both are fuzzy value fuzzy set, i...

Granular Computing (GrC), a knowledge-oriented computing which covers the theory of fuzzy information granularity, rough set theory, the theory of quotient space and interval computing etc, is a way of dealing with incomplete, unreliable, uncertain fuzzy knowledge. In recent years, it is becoming one of the main study streams in Artificial Intelligence (AI). With selecting the size structure fl...