Fuzzy Convexities via Overlap Functions

Overlap functions, as a typical kind of binary ag-gregation functions, have been widely investigated from both of theoretical and applied viewpoints. In this article, we will continue on the theoretical research on overlap functions. First, we will define O-inclusion subsethoods to describe the inclusion degrees between fuzzy sets based on the residuum induced from an overlap function O. Second, by means of O-inclusion subsethoods, we will propose the concepts of O-convexities, algebraic O-closure operators and O-hull operators, and show that they are one-to-one corresponding. Finally, we will establish the relationship between O-convexities and fuzzy interval operators. These results will not only present a new perspective for theoretical research on overlap functions, but also provide a new approach to fuzzifications of convexities.

Automated summary

This article continues the theoretical research on overlap functions. Firstly, O-inclusion subsethoods are defined to describe the inclusion degrees between fuzzy sets. Secondly, using O-inclusion subsethoods, the concepts of O-convexities, algebraic O-closure operators, and O-hull operators are proposed and shown to have one-to-one correspondence. Finally, the relationship between O-convexities and fuzzy interval operators is established. These results not only provide a new perspective for theoretical research on overlap functions but also offer a new approach to fuzzifications of convexities.