On S-homogeneity property of seminormed fuzzy integral: An answer to an open problem

On comonotone commuting and weak subadditivity properties of seminormed fuzzy integrals

“…Obviously, • = PL and • = PF fulfills (2). By Proposition 2, • = ∨ also fulfills (2). Setting • = ∧ in (2), we obtain that for comonotone functions f = a1 A and g = b1 B…”

“…Assume that S ∈ S e 2 ∩ S e 3 Thus, the integral I S possesses the comonotone •-commuting property for any S ∈ S e 1 ∩ S e 3 . If S(e 2 , x) < S(e 3 , x) for all x > 0 and S ∈ S e 2 ∩ S < e 3 , we have…”

“…The set S * consists of all strict t -norms like the product , Aczel-Alsina t-norm, Dombi t -norm, Frank t -norm for the parameter λ ∈ (0, ∞) (see [7], Theorem 2.43 and Appendix A). with numbers (e i ) being defined as in Cases 1,2,3,6 and 7. In the next proposition, we give an answer to modified Problem 1. The modification was inspired by the results of [6].…”

“…It is clear that S(x, y) x ∧ y and S(x, 0) = 0 = S(0, x) for all x, y ∈ [0, 1], where x ∧ y = min(x, y) (see [1,3,7]). We denote the class of all semicopulas by S. There are three important examples of semicopulas: M, and S L , where M(a, b) = a ∧ b, (a, b) = ab and S L (a, b) = (a + b − 1) ∨ 0 usually called the Łukasiewicz t-norm [7].…”

“…where a = S(e3 , α), b = S(e 3 , β) and d = μ(B). Observe that a and b are arbitrary numbers from [0, e 3 ].…”