Characterization of homogeneous and quasi-homogeneous binary aggregation 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 is homogeneous of order if it satisfies for any . 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 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 is defined as ; the projections onto the first and the last coordinates are defined as and . By default, we assume that (unless stated otherwise).
Section snippets
Preliminaries
First, let us review essential prerequisites. A binary aggregation function is a mapping that is increasing in each argument and satisfies the boundary conditions and .
A t-norm [13] is a commutative and associative aggregation function with neutral element 1. Standard examples of t-norms are the minimum , the product , the Łukasiewicz t-norm given by , and the drastic product with , and 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 fulfilling that , and are decreasing on such that
Quasi-homogeneous aggregation functions
This section is devoted to characterizing quasi-homogeneous aggregation functions. If an aggregation function O is -quasi-homogeneous, then , otherwise impossible. Let . Direct checking verifies that and O is -quasi-homogeneous if and only if O is -quasi-homogeneous. In the following, we presuppose that .
Lemma 1 If an aggregation function O with neutral element e is -quasi-homogeneous, then
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 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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