Elsevier

Fuzzy Sets and Systems

Volume 433, 15 April 2022, Pages 96-107
Fuzzy Sets and Systems

Characterization of homogeneous and quasi-homogeneous binary aggregation functions

https://doi.org/10.1016/j.fss.2021.04.020Get rights and content

Abstract

Homogeneity, which plays an essential role in decision making, economics and image processing, reflects the regularity of aggregation functions with respect to the inputs with the same ratio. Quasi-homogeneity is a relaxed homogeneity that reflects the original output as well as the same ratio. This paper is devoted to the characterization of all homogeneous/quasi-homogeneous binary aggregation functions in terms of single-argument functions.

Introduction

Homogeneity, reflecting the regularity of aggregation of binary inputs with the same ratio, plays a fundamental role in a large variety of fields, such as decision making [5], economics [6], [9] and image processing [7], [11], [12], [14], [18]. A function O:[0,1]2[0,1] is homogeneous of order k>0 if it satisfies O(λx,λy)=λkO(x,y) for any x,y,λ[0,1]. The studies of the homogeneity mainly focus on specific aggregation functions such as t-norms/t-conorms [2, Theorems 3.4.1 and 3.4.4] and copulas [16, Theorem 3.4.2]. From these results, one can state that the homogeneity is a rather restrictive property for associative functions and there are only trivial homogeneous t-norms/t-conorms. Thus, we need a more relaxed homogeneity, one that reflects the original value O(x,y) as well as the concentration factor λ. A relaxed homogeneity, called quasi-homogeneity, was introduced in [8]. Once again, the studies of quasi-homogeneity also pay close attention to certain aggregation functions, such as t-norms [8, Theorem 2.3] and copulas [15, Corollary 1]. In [14], the notion of homogeneity was extended to the interval-valued setting and interval homogeneous t-norms and t-conorms were investigated. Observe that only specific aggregation functions, e.g., t-norms, t-conorms and copulas, were considered for homogeneity and quasi-homogeneity. For general aggregation functions, the homogeneity of order 1 was studied in [17], see also [10, Chapter 7]. In this paper, homogeneity/quasi-homogeneity is considered in the most general setting, that is, we characterize all homogeneous/quasi-homogeneous aggregation functions in terms of single-argument functions. We also present the necessary and sufficient conditions for homogeneous/quasi-homogeneous aggregation functions that are commutative, continuous, associative or have a neutral element.

The paper is organized as follows. In Section 2 we present the preliminary notions and results that are necessary in the remainder of this paper. In Section 3 the characterizations of homogeneous aggregation functions are presented, and Section 4 is devoted to characterizing quasi-homogeneous aggregation functions. Finally, Section 5 includes some conclusions and future work.

Hereof, we follow the two-dimensional functions; the symbols ∨ and ∧ stand for maximum and minimum, respectively; the identity function id:[0,1][0,1] is defined as id(x)=x; the projections onto the first and the last coordinates are defined as P1(x,y)=x and P2(x,y)=y. By default, we assume that 00=0 (unless stated otherwise).

Section snippets

Preliminaries

First, let us review essential prerequisites. A binary aggregation function is a mapping O:[0,1]2[0,1] that is increasing in each argument and satisfies the boundary conditions O(0,0)=0 and O(1,1)=1.

A t-norm [13] is a commutative and associative aggregation function with neutral element 1. Standard examples of t-norms are the minimum TM, the product TP, the Łukasiewicz t-norm TL given by TL(x,y)=max(x+y1,0), and the drastic product TD with TD(1,x)=TD(x,1)=x, and TD(x,y)=0 otherwise. A

Homogeneous aggregation functions

In this section, we provide a characterization of homogeneous of order k aggregation functions, and also present the necessary and sufficient conditions for homogeneous of order k aggregation functions that are commutative, continuous, associative or have a neutral element.

Theorem 5

An aggregation function O is homogeneous of order k if and only if there exist increasing functions h,g:[0,1][0,1] fulfilling that h(1)=g(1)=1, h(x)/xk and g(x)/xk are decreasing on ]0,1] such thatO(x,y)={0if(x,y)=(0,0),ykh(x

Quasi-homogeneous aggregation functions

This section is devoted to characterizing quasi-homogeneous aggregation functions. If an aggregation function O is (φ,f)-quasi-homogeneous, then φ(1)0, otherwiseO(λ,λ)=φ1(f(λ)φ(O(1,1)))=φ1(0)=1for each λ[0,1], impossible. Let φ˜(x)=φ(x)/φ(1). Direct checking verifies that φ˜(1)=1 and O is (φ,f)-quasi-homogeneous if and only if O is (φ˜,f)-quasi-homogeneous. In the following, we presuppose that φ(1)=1.

Lemma 1

If an aggregation function O with neutral element e is (φ,f)-quasi-homogeneous, then e{0,1}

Concluding remarks and future work

In this paper, all homogeneous/quasi-homogeneous binary aggregation functions have been characterized in terms of single-argument functions. We also have presented the necessary and sufficient conditions for homogeneous/quasi-homogeneous aggregation functions that are commutative, continuous, associative or have a neutral element. In the future, we will discuss an extended quasi-homogeneity, e.g., pseudo-homogeneity introduced in [19]: A function O:[0,1]2[0,1] is pseudo-homogeneous if there

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

Support from the National Natural Science Foundation of China (nos. 11801220, 11901239, 11971417 and 12071259), the Natural Science Foundation of Jiangsu Province (no. BK20180590) and the Natural Science Foundation of Shandong Province (no. ZR2019BA005) is fully acknowledged. R. Mesiar was supported by the project APVV-18-0052.

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