Lipschitz continuity of triangular subnorms

This paper deals with the Lipschitz property of triangular subnorms. Unlike the case of triangular norms, for these operations the problem is still open and presents an interesting variety of situations. We provide some characterization results by weakening the notion of convexity, introducing two generalized versions of convexity for real functions, called a-lower convexity and subconvexity. The a-lower convex and sub-convex real mappings present characteristics quite different from the usual convex real mappings. We will discuss the link between such kind of functions and the generators, and their pseudo-inverse, of continuous Archimedean triangular subnorms. (c) 2013 Elsevier B.V. All rights reserved.