Type-$(2,k)$ Overlap Indices

Automatic image detection is one of the most important areas in computing due to its potential application in numerous real-world scenarios. One important tool to deal with that is called overlap indices . They were introduced as a procedure to provide the maximum lack of knowledge when comparing two fuzzy objects. They have been successfully applied in the following fields: image processing, fuzzy rule-based systems, decision making, and computational brain interfaces. This notion of overlap indices is also necessary for applications in which type-2 fuzzy sets are required. In this article, we introduce the notion of type-$(2,k)$ overlap index ( $k \in \lbrace 0,1,2\rbrace$ ) in the setting of type-2 fuzzy sets. We describe both the reasons that have led to this notion and the relationships that naturally arise among the algebraic underlying structures. Finally, we illustrate how type- $(2,k)$ overlap indices can be employed in the setting of fuzzy rule-based systems when the involved objects are type-2 fuzzy sets.