Representation of quasi-overlap functions for normal convex fuzzy truth values based on generalized extended overlap functions

At present, (quasi-)overlap functions have been extended to various universes of discourse and become a hot research topic. Meanwhile, the investigation of extended aggregation operations for normal convex fuzzy truth values has also attracted much attention. This paper mainly studies the representation of quasi-overlap functions for normal convex fuzzy truth values based on generalized extended overlap functions, which is the fundamental problem in the whole study of overlap functions for normal convex fuzzy truth values. Firstly, we present the definitions of (restrictive-)quasi-overlap functions and lattice-ordered-(restrictive-)quasi-overlap functions for normal convex fuzzy truth values and generalized extended overlap functions, respectively. Secondly, we present the (equivalent) characterizations for the closure properties of generalized extended overlap functions for various fuzzy truth values. Thirdly, we characterize the basic properties of generalized extended overlap functions for normal convex fuzzy truth values. Finally, by an equivalent characterization with a prerequisite, we successfully represent quasi-overlap functions for normal convex fuzzy truth values based on generalized extended overlap functions. Notably, we can quickly obtain (restrictive-)quasi-overlap functions for normal convex fuzzy truth values using the left-continuous quasi-overlap functions on interval [0,1]. Moreover, regarding the relationships between four types of quasi-overlap functions for normal convex fuzzy truth values, the details implication relations are that lattice-ordered-(restrictive-)quasi-overlap functions are strictly stronger than (restrictive-)quasi-overlap functions for normal convex fuzzy truth values even if all of them are constructed by generalized extended overlap functions.