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

Abstract

We give an answer to Problem 9.3 stated by Mesiar and Stupňanová [8]. We show that the class of semicopulas solving this problem contains any associative semicopula S such that for each a ∈ [0, 1] the function x ↦ S(a, x) is continuous and increasing on a countable number of intervals.

Introduction

Let $(X,\mathcal{A})$ be a measurable space, where $\mathcal{A}$ is a σ-algebra of subsets of a non-empty set X, and let $\mathcal{S}$ be the family of all measurable spaces. The class of all $\mathcal{A}$-measurable functions f : X → [0, 1] is denoted by ${\mathcal{F}}_{(X,\mathcal{A})}.$ A capacity on $\mathcal{A}$ is a non-decreasing set function $\mu :\mathcal{A}\to [0,1]$ with $\mu \left(\varnothing \right)=0$ and $\mu \left(X\right)=1.$ We denote by ${\mathcal{M}}_{(X,\mathcal{A})}$ the class of all capacities on $\mathcal{A}.$

Suppose that S : [0, 1]² → [0, 1] is a non-decreasing function in both coordinates with neutral element 1, called a semicopula, a conjunctor or a t-seminorm (see [2], [3]). It is clear that S(x, y) ≤ x ∧ y and $\mathrm{S}(x,0)=0=\mathrm{S}(0,x)$ for all x, y ∈ [0, 1]. We denote the class of all semicopulas by $\mathfrak{S}.$ Typical examples of semicopulas include: $\mathrm{M}(a,b)=a\wedge b,$$\Pi (a,b)=ab,$$\mathrm{S}(x,y)=xy(x\vee y)$ and ${\mathrm{S}}_L(a,b)=(a+b-1)\vee 0;$S_L is called the Łukasiewicz t-norm [6]. Hereafter, $a\wedge b=\min (a,b)$ and $a\vee b=\max (a,b)$.

The generalized Sugeno integral is defined by

$${\mathbf{I}}_{\mathrm{S}}(\mu ,f):=\underset{t\in [0,1]}{\sup }\mathrm{S}(t,\mu \left(\{f⩾t\}\right)),$$

where $\{f⩾t\}=\{x\in X:f(x)⩾t\},$$(X,\mathcal{A})\in \mathcal{S}$ and $(\mu ,f)\in {\mathcal{M}}_{(X,\mathcal{A})}\times {\mathcal{F}}_{(X,\mathcal{A})}.$ In the literature, I_S is also called the seminormed fuzzy integral [4], [7], [9]. Replacing semicopula S with M, we get the Sugeno integral [11]. Moreover, if $\mathrm{S}=\Pi ,$ then I_Π is called the Shilkret integral [10].

Below we present Problem 9.3 from [8], which was posed by Hutník during The Twelfth International Conference on Fuzzy Set Theory and Applications held from January 26 to January 31, 2014 in Liptovský Ján, Slovakia.

Problem 9.3 To characterize a class of semicopulas S for which the property

$$\left({\forall }_{a\in [0,1]}\right){\mathbf{I}}_{\mathrm{S}}(\mu ,\mathrm{S}(a,f))=\mathrm{S}(a,{\mathbf{I}}_{\mathrm{S}}(\mu ,f))$$

holds for all$(X,\mathcal{A})\in \mathcal{S}$ and all$(\mu ,f)\in {\mathcal{M}}_{(X,\mathcal{A})}\times {\mathcal{F}}_{(X,\mathcal{A})}.$

Hutník et al. [5], [8] conjectured that (1) characterizes the two element class {M, Π}. We show that (1) holds for any associative semicopula with continuous selections satisfing some mild conditions.