A note on two open problems of Alsina, Frank and Schweizer

Abstract

An open problem of Alsina, Frank and Schweizer on k-Lipschitz t-norms is solved. k-convexity of monotone functions is introduced. Archimedean k-Lipschitz t-norms are shown to be generated by k-convex additive generators. Moreover, an example of a strict superadditive t-norm which is not a copula is given thus solving another open problem posed by Alsina, Frank and Schweizer.

Overlap functions

“…On the contrary, although in general, the mapping min(kx, ky, 1) for k > 1 and x, y ∈ [0, 1] such that kx, ky ∈ [0, 1] does not define an overlap function (since by taking x = y = 1 k we see that it does not fulfill condition (G S 3)), min(x, y) is an overlap function, so we have the following corollary. For associative k-Lipschitz overlap functions we have the next result which can be derived from [24,25].…”

Lipschitzian De Morgan triplets of fuzzy connectives

“…Note that if a strictly decreasing function f is k-convex, then k P 1, and moreover, f is continuous on ]0, 1] and p-convex for all p P k. On the other hand, if a strictly increasing function f is k-convex, then k 6 1, and moreover, f is continuous on [0,1[ and p-convex for all 0 < p 6 k. We recall the characterization of Archimedean k-Lipschitzian t-norms proved by Mesiarová [15]. In general, a t-norm T is k-Lipschitzian if and only if it is an ordinal sum of t-norms with k-Lipschitzian summands [15]. As the Lipschitzian property (and the Lipschitzian constant) is preserved with respect to the standard duality, we omit the results concerning k-Lipschitzian t-conorms.…”

“…A complete characterization of Archimedean k-Lipschitzian t-norms based on the k-convexity of their additive generators has been given by Mesiarová [15][16][17]: Á . Note that if a strictly decreasing function f is k-convex, then k P 1, and moreover, f is continuous on ]0, 1] and p-convex for all p P k. On the other hand, if a strictly increasing function f is k-convex, then k 6 1, and moreover, f is continuous on [0,1[ and p-convex for all 0 < p 6 k. We recall the characterization of Archimedean k-Lipschitzian t-norms proved by Mesiarová [15]. In general, a t-norm T is k-Lipschitzian if and only if it is an ordinal sum of t-norms with k-Lipschitzian summands [15].…”

“…We will focus on the study of Lipschitzian (strong) negations and relationships between the best Lipschitzian constants of connectives T, S and n. Note that k-Lipschitzian t-norms and t-conorms for k = 1 have been characterized by Moynihan [18], and for k > 1 by Mesiarová [15][16][17], see also [3]. For completeness, these characterizations are briefly recalled in Section 3.…”

T-norms in subtractive clustering and backpropagation

“…The choice of a t-norm will result in a fuzzy system with a set of properties; for example, to ensure the stability of a fuzzy system, the constituent t-norm will be required to satisfy certain Lipschitz condition. 23,24 There are four basic tnorms (more details on t-norms can be found in Ref. 19…”