Abstract
Overlap and grouping functions, as two novel continuous binary aggregation functions, have been discussed in the literature for applications in image processing, decision making, classification and so on. Since Walker et al. proved that the algebraic structure of fuzzy truth values has the same property as the algebraic structure of type-2 fuzzy sets, the study of extension operators for fuzzy truth values has become a hot research topic. This article considers this research topic and discusses mainly the so-called O-extension operators based on overlap functions. Firstly, the O-extension operators of general binary operators for fuzzy truth values on a linearly ordered set are introduced, and some of their basic properties are discussed. Secondly, the properties of O-extended maximum operators, O-extended minimum operators, O-extended overlap functions and O-extended grouping functions are studied. Thirdly, O-extension operators on normal fuzzy sets and O-convex fuzzy sets are investigated. Fourthly, the O-extension operators for fuzzy truth values are extended to type-2 fuzzy sets. Finally, we provide a brief comparison of O-extension operators with other common operators for fuzzy truth values.
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