Fuzzy Convexities via Overlap Functions

Overlap functions, as a typical kind of binary aggregation 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.