An order induced by extended t-norms on convex normal functions

In this paper, we present an order ⪯⊙T induced by extended t-norms ⊙T on normal and convex functions. First, we investigate the relationship between the meet order (resp. join order) and ⪯⊙T. And then we define the set of incomparable convex normal functions with respect to ⪯⊙T and deeply study its properties. •An order induced by extended t-norms on convex normal functions of type-2 fuzzy set is studied.•The relationship between the meet order (resp. join order) and the order induced by extended t-norms is obtained.•The set of incomparable convex normal functions with respect to the order induced by extended t-norms is defined.•The set of convex normal functions may form a lattice under the order induced by extended t-norms.