Distributivity equations and Mayor’s aggregation operators

Abstract

The focus of this paper are distributivity equations involving the binary aggregation operators on the unit interval [0, 1] with either absorbing or neutral element from the open interval (0, 1), and the Mayor’s aggregation operators from [28]. In the second part of this paper, problem is extended to aggregation operators that have neither neutral nor absorbing element.

Introduction

Aggregation operators play an important role in many different theoretical and practical fields (fuzzy sets theory, theory of optimization, operations research, information theory, engineering design, game theory, voting theory, integration theory, etc.), particularly in decision making theory (see [2], [15], [17], [21], [23]). Lately, a high level of attention is directed towards characterizations of pairs of aggregation operators that are satisfying the distributivity law. Investigation of this problem has roots in [1] and, in recent years, it has been focused on t-norms and t-conorms [15], aggregation operators, quasi-arithmetic means [5], pseudo-arithmetical operations [3], fuzzy implications [30], [31], uninorms and nullnorms [8], [14], [25], [26], [33]. Additionally, many authors are considering distributivity inequalities [9], [10], as well as distributivity equations on a restricted domain [4], [12], [13], [18], [20], [21], [22], [29], [32]. An interesting application of this restricted setting on two Borel-Cantelli lemmas and independence of events for decomposable measures is given in [7].

The aim of this paper is to extend research from [5] towards binary aggregation operators that have either an absorbing element or a neutral element from (0, 1). In [5] the previous problem is solved for one special case, i.e., when neutral elements are limited to 1 and 0 (t-norms and t-conorms). Furthermore, the presented research also extends results form [5] towards non-commutative and non-associative operators. This line of research presents a contemporary topic (see [20], [33]) that is highly interesting since it opens some new possibilities in the utility theory (see [19]). Therefore, the main concern of this paper is how to solve functional equations

$$F(x\text{,}G(y\text{,}z))=G(F(x\text{,}y)\text{,}F(x\text{,}z))\text{,}x\text{,}y\text{,}z\in [0\text{,}1]$$

and

$$F(G(y\text{,}z)\text{,}x)=G(F(y\text{,}x)\text{,}F(z\text{,}x))\text{,}x\text{,}y\text{,}z\in [0\text{,}1]$$

where one of unknown functions is an aggregation operator defined in the sense of G. Mayor (see [28]), and another one is either a relaxed uninorm or a relaxed nullnorm [8]. The second part of this paper contains even further extension of this problem involving aggregation operators that have neither neutral nor absorbing element. Also, results presented in this paper are additionally clarifying structure of the observed GM-operators. Since paper [5] has considered only min and max as options for the GM-operators, stricture of the GM-operators for other cases was not investigated, presented results present a step forward for this investigation.

This paper is organized as follows. Section 2 contains preliminary notions concerning aggregation operators defined in the sense of G. Mayor, aggregation operators with neutral and absorbing elements and distributivity equations. Results on distributivity between aggregation operator given in the sense of G. Mayor and relaxed nullnorm are given in the third section. Section 4 consists of results on distributivity between aggregation operator in the sense of G. Mayor and relaxed uninorm from the classes $N_e^{\mathit{max}}$ and $N_e^{\mathit{min}}$. Topic of the fifth section is distributivity when one of the aggregation operators has neither neutral nor absorbing element. Some concluding remarks are given in the sixth section.