Fuzzy "Between" Operators in the Framework of Fuzzy Orderings

This chapter focuses on the applications in fuzzy rule-based systems along with the understanding of the construction of two binary ordering-based modifiers that model a concept of fuzzy between, both in an inclusive and a non-inclusive setting. Fuzzy systems are regarded as appropriate methodologies for controlling complex systems and for carrying out complicated decision processes. Almost all fuzzy systems involving numerical variables implicitly use orderings. It is standard to decompose the universe of a linearly ordered system variable into a certain number of fuzzy sets by means of the ordering of the universe-typically resulting in labels like "small", "medium", or "large". The chapter also concerns with an extension of this framework by two binary ordering based modifiers named BTW and SET and both represent fuzzy "between" operators, where BTW stands for the inclusive and SET ("strictly between") stands for the noninclusive interpretation. The operator BTW is designed for computing the fuzzy set of all objects lying between two fuzzy sets including both boundaries. The purpose of SBT is to extract those objects that are lying strictly between two fuzzy sets-not including the two boundaries. The chapter concludes that two operators are appropriate as modifiers for fuzzy systems applications and for rule interpolation.