n-K-Increasing Aggregation Functions

Abstract

We introduce and discuss the concept of n-ary K-increasing fusion functions and n-ary K-increasing aggregation functions, K being a subset of the index set $\{1,\dots ,n\}$ indicating in which variables a considered function is increasing. It is also assumed that this function is decreasing in all other variables. We show that each n-ary K-increasing aggregation function is generated by some aggregation function which enables us to introduce and study the properties of n-ary K-increasing aggregation functions related to the properties of their generating aggregation functions. In particular, we also discuss binary K-increasing aggregation functions, including fuzzy implication and complication functions, among others.

1. Introduction

Most functions used in information fusion processes are functions of the type ${\mathbb{I}}^n\to \mathbb{I}$, $\mathbb{I}$ being a real interval, which are often constrained by some boundary or monotonicity conditions. Prominent functions in this framework are aggregation functions (usually acting on the real unit interval $[0,1]$) which are required to be monotone increasing (and to satisfy certain additional boundary conditions). This property is equivalent to the monotone increasingness of the considered functions in each variable. More information on aggregation functions can be found, e.g., in Refs. [1,2,3]. However, in real world applications we can also meet situations in which the considered fusion functions are required to be monotone in each variable but not in the same way—in certain variables they should be monotone increasing, and monotone decreasing in the rest of them. Such functions will be referred to as hybrid monotone functions. As an example, we can mention utility functions in multi-criteria decision making [4,5,6] when positive and negative criteria are processed simultaneously. For example, in the case of car buying, when our decision is based on the three criteria—prize, average fuel consumption (negative criteria), and safety (a positive criterion), a hybrid monotone utility function is needed to make a decision, with monotone decreasing in the first two variables and increasing in the third one. As a possible example of such utility function $F:{[0,1]}^3\to [0,1]$ one can consider F given by

$$F(x_1,x_2,x_3)=0.5(1-x_1)+0.3(1-x_2)+0.2x_3=0.8-0.5x_1-0.3x_2+0.2x_3.$$

As another example of hybrid monotone functions we can mention fuzzy implication functions, i.e., functions $I:{[0,1]}^2\to [0,1]$ which are monotone decreasing in the first variable and increasing in the second one (and satisfying certain boundary conditions), see Ref. [7]. As a particular example, we recall the Reichenbach fuzzy implication function given by $I_R(x,y)=1-x+xy$, $(x,y)\in {[0,1]}^2$. Aggregation theory and its generalizations have significantly advanced in the last three decades, see, e.g., the overviews Refs. [1,2,3,8,9]. However, surprisingly, a deeper study of hybrid monotone functions is still missing. The attention has mostly been focused on some particular classes of hybrid monotone functions only, especially on fuzzy implication functions.

It is worth noticing that the classical aggregation theory (based on monotone increasingness in each variable) is closely related to the so-called clone theory. We can recall that clones are, roughly said, sets of functions closed under projections and their natural compositions. In Ref. [10], we have observed that the clones of aggregation functions on any finite lattice L are finitely generated, i.e., they can be composed of a finite set of unary and binary functions. On the other hand, logical operations such as negations and implications are not order-preserving in all arguments. Consequently, the collection of all such operations on a poset is not a clone since it is not closed under usual composition of operations. Hence there was a natural question how the clone theory could be generalized for functions that do not fulfill the same monotonicity in each argument.

In a recent paper [11], the authors introduced and studied the so-called S-preclones. Here S denotes a set of different properties that the operations in consideration could have (such as order-preserving and order-reversing via partial order relations). The set S of “properties” (called signa) itself has an algebraic structure which reflects the composition of the operations (e.g., order-reversing composed with order-reversing gives order-preserving), i.e., S has to be a monoid. The collection of all S-operations with prescribed properties for their signed arguments leads to the notion of S-preclone. These preclones are closed under a special composition described in the paper. This approach seems to give a very promising new tool for the study of generalized aggregation functions.

The aim of this paper is to fill the above mentioned gap by introducing the concepts of n-ary K-increasing fusion functions, and n-ary K-increasing aggregation functions. This approach enables us to build a common framework for all monotone functions (i.e., monotone increasing, decreasing or hybrid monotone) which are used in information fusion processes.

For this purpose, we first briefly recall the definitions of several basic notions that will be needed throughout the paper.

Let $n\in \mathbb{N}$. Recall that

• Any function $F:{[0,1]}^n\to [0,1]$ is called a fusion function [12]. In algebra, it corresponds to an n-ary groupoid. The values of a fusion function F will usually be written as $F\left(\mathbf{x}\right)=F(x_1,\dots ,x_n)$;
• A fusion function $F:{[0,1]}^n\to [0,1]$ which satisfies the boundary conditions $F(0,\dots ,0)=0$ and $F(1,\dots ,1)=1$ is called a semi-aggregation function [13];
• A semi-aggregation function $F:{[0,1]}^n\to [0,1]$ which is monotone increasing in each variable is called an (n-ary) aggregation function [1,3,14].

Note that a fusion function $F:{[0,1]}^n\to [0,1]$ is monotone increasing (decreasing) in the ith variable, $i\in \{1,\dots ,n\}$, whenever $F\left(\mathbf{a}\right)\le F\left(\mathbf{b}\right)$ ($F\left(\mathbf{a}\right)\ge F\left(\mathbf{b}\right)$) for all inputs $\mathbf{a}=(a_1,\dots ,a_n),\mathbf{b}=(b_1,\dots ,b_n)\in {[0,1]}^n$ such that $a_i\le b_i$ and $a_j=b_j$ for all $j\ne i$, and that F is increasing (decreasing) if it is increasing (decreasing) in each variable. We emphasize that a fusion function $F:{[0,1]}^n\to [0,1]$ is an aggregation function if and only if it is monotone increasing in each variable and satisfies the boundary conditions $F(0,\dots ,0)=0$ and $F(1,\dots ,1)=1$.

Further, consider any vector $\overrightarrow{r}\in {\mathbb{R}}^n\setminus \left\{(0,\dots ,0)\right\}$. Recall that

• A fusion function $F:{[0,1]}^n\to [0,1]$ is $\overrightarrow{r}$-increasing (decreasing) [12] if

  $$F(\mathbf{x}+c\overrightarrow{r})\ge F\left(\mathbf{x}\right)$$

  $$(F(\mathbf{x}+c\overrightarrow{r})\le F\left(\mathbf{x}\right))$$

  for all $\mathbf{x}\in {[0,1]}^n$ and $c>0$ such that $\mathbf{x}+c\overrightarrow{r}\in {[0,1]}^n$.
  Such fusion functions are called directionally monotone, see Ref. [12]. In particular, if a fusion function F is directionally monotone in the direction $\overrightarrow{r}=(1,\dots ,1)$, it is called weakly monotone [15];
• A semi-aggregation function $F:{[0,1]}^n\to [0,1]$, which is $\overrightarrow{r}$-increasing for some direction $\overrightarrow{r}\in {\left({\mathbb{R}}_0^{+}\right)}^n\setminus \left\{(0,\dots ,0)\right\}$, is called a pre-aggregation function [16].

There are also known some other concepts of monotonicity, for example, ordered directional monotonicity [17], cone monotonicity [18], or curve monotonicity [19]. However, except for the standard monotonicity of real n-ary functions, none of the mentioned types of monotonicity can be examined by means of the monotonicity in individual variables.

The rest of the paper is structured as follows. In Section 2, we introduce and discuss n-ary K-increasing fusion functions and n-ary K-increasing aggregation functions and their basic properties. It is also shown that any n-ary K-increasing aggregation function is generated by some n-ary aggregation function, which is a very important property for obtaining the results in Section 3 where we investigate several properties of n-ary K-increasing aggregation functions closely related to the distinguished properties of their generating functions. Section 4 is devoted to binary K-increasing aggregation functions and the last section contains several concluding remarks.

2. $\mathit{n}$-$\mathit{K}$-Increasing Fusion Functions and $\mathit{n}$-$\mathit{K}$-Increasing Aggregation Functions

In this section, we first introduce the notion of n-ary K-increasing real functions defined on an interval $\mathbb{I}\subseteq \mathbb{R}$ whose basic characteristic is that they are in each variable either monotone increasing or monotone decreasing. Then we focus our attention on n-ary K-increasing fusion functions and finally, on their particular subclass, the class of all n-ary K-increasing aggregation functions.

Definition 1.

Let $n\in \mathbb{N}$, $K\subseteq \{1,\dots ,n\}$, and let $\mathbb{I}\subseteq \mathbb{R}$ be an interval. An n-ary function $F:{\mathbb{I}}^n\to \mathbb{R}$ is called K-increasing if it is increasing in each variable $x_i$ with $i\in K$, and decreasing in each variable $x_i$ with $i\in K^c=\{1,\dots ,n\}\setminus K$.

As for determining the complement $K^c$ of a given set K, the information about the basic set $\{1,\dots ,n\}$ is necessary, and the arity n of the considered functions will always be stressed. Therefore, the functions introduced in the previous definition will be called either n-ary K-increasing functions or n-K-increasing functions for short.

Obviously, $F:{\mathbb{I}}^n\to \mathbb{R}$ is an increasing function if and only if it is n-K-increasing with $K=\{1,\dots ,n\}$, and F is a decreasing function if and only if it is n-K-increasing with $K=\varnothing$. Thus, for any proper subset $K\subset \{1,\dots ,n\}$, n-K-increasing functions are hybrid monotone whenever they are neither n-∅- nor n-$\{1,\dots ,n\}$-increasing.

Although the presented concept could be built in a more general framework (for fusion and aggregation functions $F:{\mathbb{I}}^n\to \mathbb{I}$ on any real interval $\mathbb{I}$), in what follows, we restrict ourselves to the unit real interval mostly used in aggregation theory. We start our investigation with n-K-increasing fusion functions, i.e., functions $F:{[0,1]}^n\to [0,1]$ which are increasing in each variable $x_i$ with $i\in K$, and decreasing in each variable $x_i$ with $i\in K^c$.

Throughout the whole paper, the set K always indicates the set of those indices $i\in \{1,\dots ,n\}$ of variables in which the considered fusion function is increasing. Thus, without any confusion, we will call n-K-increasing fusion functions simply n-K-fusion functions.

The following proposition characterizes constant fusion functions.

Proposition 1.

A fusion function $F:{[0,1]}^n\to [0,1]$ is constant if and only if there is a subset $K\subseteq \{1,\dots ,n\}$ such that F is both an n-K- and n-$K^c$-fusion function.

Proof. 

If F is a constant function then it is an n-K-fusion function for any $K\subseteq \{1,\dots ,n\}$. On the other hand, if F is both an n-K- and n-$K^c$-fusion function for some $K\subseteq \{1,\dots ,n\}$, then it is increasing and also decreasing in each variable, and thus a constant function. □

Similarly, one can show that if F is an n-$K_1$- and n-$K_2^c$-fusion function, and $i\in K_1\Delta K_2$, where $\Delta$ denotes the symmetric difference of the sets $K_1$ and $K_2$, then F does not depend on the ith variable.

It follows from Definition 1 that, for any n-K-fusion function $F:{[0,1]}^n\to [0,1]$, the maximal value of F is given by $maxF=F\left({\mathbf{1}}_K\right)$, where ${\mathbf{1}}_K$ is the characteristic vector of K, i.e.,

$${\left({\mathbf{1}}_K\right)}_i=\left\{\begin{array}{cc}1\hfill & \mathrm{if}i\in K,\hfill \\ 0\hfill & \mathrm{if}i\in K^c.\hfill \end{array}\right.$$

Similarly, its minimal value is $minF=F\left({\mathbf{1}}_{K^c}\right)$.

The following results will be useful for the construction and characterization of special n-K-fusion functions. Their proofs are trivial and therefore omitted (except for the proof of item (v)).

Proposition 2.

Let $K\subseteq \{1,\dots ,n\}$ and let $F:{[0,1]}^n\to [0,1]$ be an n-K-fusion function. Then we have:

(i) 

Suppose $i\in K$ is fixed. Consider a function $h:[0,1]\to [0,1]$ and define a function $F_{i,h}:{[0,1]}^n\to [0,1]$ by

$$F_{i,h}\left(\mathbf{x}\right)=F\left({\mathbf{x}}_{i,h}\right),$$

where ${\mathbf{x}}_{i,h}$ is an n-tuple with components

$${\left({\mathbf{x}}_{i,h}\right)}_j=\left\{\begin{array}{cc}h\left(x_i\right)\hfill & ifj=i,\hfill \\ x_j\hfill & ifj\ne i,\hfill \end{array}\right.j\in \{1,\dots ,n\}.$$

Then, $F_{i,h}$ is an n-K-fusion function whenever h is increasing, and it is an n-$(K\setminus \{i\left\}\right)$-fusion function whenever h is decreasing.

(ii) 

Let $f:[0,1]\to [0,1]$ be an increasing function and $g:[0,1]\to [0,1]$ a decreasing function. Then the composite $f\circ F$ is an n-K-fusion function and $g\circ F$ is an n-$K^c$-fusion function.

(iii) 

If F is not constant then the function $G:{[0,1]}^n\to [0,1]$ defined by

$$G\left(\mathbf{x}\right)=\frac{F\left(\mathbf{x}\right)-F\left({\mathbf{1}}_{K^c}\right)}{F\left({\mathbf{1}}_K\right)-F\left({\mathbf{1}}_{K^c}\right)}$$

is an n-K-fusion function which is a semi-aggregation function.

(iv) 

The dual fusion function $F^d:{[0,1]}^n\to [0,1]$ to an n-K-fusion function F, defined by $F^d\left(\mathbf{x}\right)=1-F(1-x_1,\dots ,1-x_n)$, is also an n-K-fusion function.

(v) 

Let $L\subseteq K^c$ and let ${\mathbf{x}}^L$ be an n-tuple whose components ${\left({\mathbf{x}}^L\right)}_j$, $j=1,\dots ,n$, are given by

$${\left({\mathbf{x}}^L\right)}_j=\left\{\begin{array}{cc}1-x_j\hfill & ifj\in L,\hfill \\ x_j\hfill & ifj\notin L.\hfill \end{array}\right.$$

Then the function $F^L:{[0,1]}^n\to [0,1]$ given by $F^L\left(\mathbf{x}\right)=F\left({\mathbf{x}}^L\right)$ is an n-$(K\cup L)$-fusion function. In particular, if $L=K^c$ then $F^{K^c}$ is an increasing fusion function.

(vi) 

Let $L\subseteq \{1,\dots ,n\}$. Then $F^L:{[0,1]}^n\to [0,1]$ given as in (v) is an n-$(K\Delta L)$-fusion function.

Proof. 

As mentioned above, we only prove claim (v): It is clear that due to the disjointness of K and L, the function $F^L$ is increasing in the ith variable for each $i\in K\cup L$. This claim follows from the n-K-increasingness of F whenever $i\in K$. If $i\in L$, F is decreasing in the ith variable, but then ${\left({\mathbf{x}}^L\right)}_i=1-x_i$, and hence $F^L$ is increasing in this variable. In a similar way, we can show that if $i\notin K\cup L$ then $F^L$ is decreasing in the ith variable. To summarize, we see that $F^L$ is an n-$(K\cup L)$-fusion function. The claim for $L=K^c$ follows immediately. □

Recall that the standard aggregation functions are just increasing fusion functions for which $maxF=1$ and $minF=0$. Following this observation, we define the notion of n-K-increasing aggregation function.

Definition 2.

Let $K\subseteq \{1,\dots ,n\}$ and let $F:{[0,1]}^n\to [0,1]$ be an n-K-increasing fusion function. Then F is an n-K-increasing aggregation function if it satisfies the boundary properties $maxF=F\left({\mathbf{1}}_K\right)=1$ and $minF=F\left({\mathbf{1}}_{K^c}\right)=0$.

Again, to simplify notations, n-K-increasing aggregation functions will be called simply n-K-aggregation functions.

Proposition 2 (v) enables us to formulate the following important properties. We keep the notations introduced in Proposition 2 (v).

Lemma 1.

Let $K\subseteq \{1,\dots ,n\}$. A function $F:{[0,1]}^n\to [0,1]$ is an n-K-fusion function if and only if $F^{K^c}$ is an increasing n-ary fusion function.

Proof. 

If F is an n-K-fusion function, the increasingness of $F^{K^c}$ follows from Proposition 2 (v). Conversely, suppose that $F^{K^c}$ is an increasing n-ary fusion function. For any $\mathbf{x}\in {[0,1]}^n$, $F^{K^c}\left(\mathbf{x}\right)=F\left({\mathbf{x}}^{K^c}\right)$ and thus, $F\left(\mathbf{x}\right)=F^{K^c}\left({\mathbf{x}}^{K^c}\right)$. Then, since the components of ${\mathbf{x}}^{K^c}$ are given by

$${\left({\mathbf{x}}^{K^c}\right)}_i=\left\{\begin{array}{cc}1-x_i\hfill & \mathrm{if}i\in K^c,\hfill \\ x_i\hfill & \mathrm{if}i\in K,\hfill \end{array}\right.i=1,\dots ,n,$$

F is decreasing in each ith variable with $i\in K^c$ and increasing if $i\in K$, which proves that F is an n-K-fusion function. □

Lemma 2.

Let L be an arbitrary fixed subset of $\{1,\dots ,n\}$. Then a fusion function $F:{[0,1]}^n\to [0,1]$ satisfies the boundary conditions $maxF=1$ and $minF=0$ if and only if $F^L$ satisfies the conditions $max{F}^L=1$ and $min{F}^L=0$.

Proof. 

The proof follows from the facts that for each $\mathbf{x}\in {[0,1]}^n$ there is a unique element $\mathbf{y}\in {[0,1]}^n$ such that $\mathbf{x}={\mathbf{y}}^L$ and that for each $\mathbf{x}\in {[0,1]}^n$, ${\left({\mathbf{x}}^L\right)}^L=\mathbf{x}$. □

Theorem 1.

A function $F:{[0,1]}^n\to [0,1]$ is an n-K-aggregation function if and only if the function $A=F^{K^c}$ is an n-ary aggregation function.

Proof. 

The claim follows from Lemma 1, Lemma 2 for $L=K^c$ and Definition 2. □

Remark 1.

(i) Due to the above theorem, each n-K-aggregation function F is generated by some (unique) n-ary aggregation function A. We have $A\left(\mathbf{x}\right)=F^{K^c}\left(\mathbf{x}\right)$ and $F\left(\mathbf{x}\right)=A^{K^c}\left(\mathbf{x}\right)$.

(ii) Fix a set $K\subseteq \{1,\dots ,n\}$. Let $\mathcal{F}$ denote the set of all n-ary fusion functions, i.e.,

$$\mathcal{F}=\{F:{[0,1]}^n\to [0,1]\}.$$

Define a mapping $d_K:\mathcal{F}\to \mathcal{F}$ by $d_K\left(F\right)=F^K$, i.e., $d_K\left(F\right)\left(\mathbf{x}\right)=F\left({\mathbf{x}}^K\right)$, see Proposition 2 (v). Then the mapping $d_K$ is a duality, i.e., $d_K\left(d_K\left(F\right)\right)=F$. This fact allows us to introduce and discuss several properties of n-K-aggregation functions inherited from the properties of n-ary aggregation functions (as will be done in Section 3). Observe that this approach is typical, e.g., for the study of t-conorms which is either based on the properties of t-norms or independent of t-norms, see Ref. [20].

Example 1.

(i) Consider a real n-ary affine function G on $[0,1]$, i.e., a function $G:{[0,1]}^n\to \mathbb{R}$ given by

$$G\left(\mathbf{x}\right)=a+{\displaystyle \sum _{i=1}^n}{b}_i{x}_i.$$

In each variable, G is either increasing or decreasing (or both increasing and decreasing). So, it is an n-K-fusion function for some $K\subseteq \{1,\dots ,n\}$ if and only if $G\left({\mathbf{1}}_K\right)=a+{\displaystyle \sum _{i\in K}}{b}_i\le 1$ and $a+{\displaystyle \sum _{i\in K^c}}{b}_i\ge 0$, and $b_i\ge 0$ for all $i\in K$, $b_i\le 0$ for all $i\in K^c$.

For example, let $B:{[0,1]}^3\to \mathbb{R}$, $B(x_1,x_2,x_3)=1+x_1-x_2+x_3$. Then, B is a ternary affine function which is $\{1,3\}$-increasing but it is not a fusion function. The function $C=\frac{1}{4}G$ is a ternary $\{1,3\}$-fusion function, but not a $\{1,3\}$-aggregation function. Finally, $D=\frac{1}{3}G$ is a ternary $\{1,3\}$-aggregation function.

(ii) A general affine function G considered in (i) is an n-K-aggregation function if and only if $b_i\ge 0$ for any $i\in K$, $b_i\le 0$ for any $i\in K^c$, and it is constrained by the boundary conditions $maxG=1$ and $minG=0$ which hold if and only if $a+{\displaystyle \sum _{i\in K}}{b}_i=1$ and $a+{\displaystyle \sum _{i\in K^c}}{b}_i=0$. Then ${\displaystyle \sum _{i\in K}}{b}_i-{\displaystyle \sum _{i\in K^c}}{b}_i=1$, i.e., ${\displaystyle \sum _{i=1}^n}\left|b_i\right|=1$. Putting $|b_i|=w_i$, we get $w_i\ge 0$, $i=1,\dots ,n$, and ${\displaystyle \sum _{i=1}^n}{w}_i=1$, thus $\mathbf{w}=(w_1,\dots ,w_n)$ is a normed weighting vector. We can write $a=1-{\displaystyle \sum _{i\in K}}{w}_i={\displaystyle \sum _{i\in K^c}}{w}_i$, and for any K such that $\{i\mid b_i>0\}\subseteq K\subseteq \{i\mid b_i\ge 0\}$ we have

$$\begin{array}{c}\hfill G\left(\mathbf{x}\right)={\displaystyle \sum _{i\in K^c}}{w}_i+{\displaystyle \sum _{i\in K}}{w}_i{x}_i-{\displaystyle \sum _{i\in K^c}}{w}_i{x}_i={\displaystyle \sum _{i\in K}}{w}_i{x}_i+{\displaystyle \sum _{i\in K^c}}{w}_i(1-x_i)=W_{\mathbf{w}}^{K^c}\left(\mathbf{x}\right),\end{array}$$

where $W_{\mathbf{w}}$ is a weighted arithmetic mean, which is given by $W_{\mathbf{w}}\left(\mathbf{x}\right)={\displaystyle \sum _{i=1}^n}{w}_i{x}_i$.

We see that G is an affine n-K-aggregation function if and only if it is derived from a weighted arithmetic mean $W_{\mathbf{w}}$, and then $G=W_{\mathbf{w}}^{K^c}$.

(iii) Considering instead of a general weighted arithmetic mean $W_{\mathbf{w}}$ the standard arithmetic mean $AM$ whose weighting vector $\mathbf{w}$ equals $(1/n,\dots ,1/n)$, we get the corresponding n-K-aggregation function G given by

$$G\left(\mathbf{x}\right)=A{M}^{K^c}\left(\mathbf{x}\right)=\frac{1}{n}\left({\displaystyle \sum _{i\in K}}{x}_i-{\displaystyle \sum _{i\in K^c}}{x}_i+\mathrm{card}\left(K^c\right)\right).$$

(1)

For example, if $n=2$ and $K=\left\{1\right\}$, then G given by $G(x_1,x_2)=\frac{1}{2}(1+x_1-x_2)$ is a binary $\left\{1\right\}$-aggregation function.

3. Properties of $\mathit{n}$-$\mathit{K}$-Increasing Aggregation Functions

There are plenty of properties considered for aggregation functions. For any property P of an aggregation function A, one can introduce a related property for the corresponding n-K-aggregation function $F=A^{K^c}$. In general:

Definition 3.

An n-K-aggregation function $F:{[0,1]}^n\to [0,1]$ has a property n-K-P whenever its generating aggregation function $A=F^{K^c}$ (see Theorem 1) has the property P.

So, for n-K-aggregation functions one can study n-K-symmetry, n-K-neutral element, n-K-annihilator, n-K-idempotency, etc.

For example, an n-K-aggregation function F generated by an aggregation function A (i.e., $F=A^{K^c}$) is n-K-symmetric if and only if A is symmetric. Recall that A is symmetric whenever for any n-tuple $\mathbf{x}\in {[0,1]}^n$ and for any permutation $\sigma :\{1,\dots ,n\}\to \{1,\dots ,n\}$, $A\left(\mathbf{x}\right)=A\left({\mathbf{x}}_{\sigma }\right)$, where ${\mathbf{x}}_{\sigma }=(x_{\sigma \left(1\right)},\dots ,x_{\sigma \left(n\right)})$.

In what follows, we give a direct characterization of the n-K-symmetry of F.

Proposition 3.

An n-K-aggregation function $F:{[0,1]}^n\to [0,1]$ is n-K-symmetric if and only if for any n-tuple $\mathbf{x}\in {[0,1]}^n$ and any permutation $\sigma :\{1,\dots ,n\}$$\to \{1,\dots ,n\}$ we have

$$F\left(\mathbf{x}\right)=F\left({\mathbf{x}}_{\sigma }^K\right),$$

where ${\mathbf{x}}_{\sigma }^K$ is an n-tuple with components ${\left({\mathbf{x}}_{\sigma }^K\right)}_i$, $i=1,\dots ,n$, given by

$${\left({\mathbf{x}}_{\sigma }^K\right)}_i=\left\{\begin{array}{cc}{x}_{\sigma \left(i\right)}\hfill & if\{i,\sigma \left(i\right)\}\subseteq Kor\{i,\sigma \left(i\right)\}\subseteq K^c,\hfill \\ 1-x_{\sigma \left(i\right)}\hfill & otherwise.\hfill \end{array}\right.$$

(2)

Proof. 

By Theorem 1, F is an n-K-aggregation function if and only if it is generated by an aggregation function A via $F=A^{K^c}$, and F is n-K-symmetric if and only if A is a symmetric aggregation function. It means that F is an n-K-symmetric n-K-aggregation function if and only if for any $\mathbf{x}\in {[0,1]}^n$ and any permutation $\sigma :\{1,\dots ,n\}\to \{1,\dots ,n\}$,

$$\begin{array}{ccc}F\left(\mathbf{x}\right)\hfill & =& A^{K^c}\left(\mathbf{x}\right)=A\left({\mathbf{x}}^{K^c}\right)=A\left({\left({\mathbf{x}}^{K^c}\right)}_{\sigma }\right)=F^{K^c}\left({\left({\mathbf{x}}^{K^c}\right)}_{\sigma }\right)\hfill \\ & =& F\left({\left({\left({\mathbf{x}}^{K^c}\right)}_{\sigma }\right)}^{K^c}\right)=F\left({\mathbf{y}}^{K^c}\right),\hfill \end{array}$$

(3)

where we have put $\mathbf{y}={\left({\mathbf{x}}^{K^c}\right)}_{\sigma }$. Then for any $i\in \{1,\dots ,n\}$,

$$y_i={\left({\left({\mathbf{x}}^{K^c}\right)}_{\sigma }\right)}_i={\left({\mathbf{x}}^{K^c}\right)}_{\sigma \left(i\right)}=\left\{\begin{array}{cc}1-x_{\sigma \left(i\right)}\hfill & \mathrm{if}\sigma \left(i\right)\in K^c,\hfill \\ x_{\sigma \left(i\right)}\hfill & \mathrm{if}\sigma \left(i\right)\in K,\hfill \end{array}\right.$$

and

$${\left({\mathbf{y}}^{K^c}\right)}_i=\left\{\begin{array}{cc}1-y_i\hfill & \mathrm{if}i\in K^c,\hfill \\ y_i\hfill & \mathrm{if}i\in K.\hfill \end{array}\right.$$

Therefore,

$${\left({\mathbf{y}}^{K^c}\right)}_i=\left\{\begin{array}{cc}1-(1-x_{\sigma \left(i\right)})=x_{\sigma \left(i\right)}\hfill & \mathrm{if}i\in K^c\mathrm{and}\sigma \left(i\right)\in K^c,\hfill \\ 1-x_{\sigma \left(i\right)}\hfill & \mathrm{if}i\in K^c\mathrm{and}\sigma \left(i\right)\in K,\hfill \\ 1-x_{\sigma \left(i\right)}\hfill & \mathrm{if}i\in K\mathrm{and}\sigma \left(i\right)\in K^c,\hfill \\ x_{\sigma \left(i\right)}\hfill & \mathrm{if}i\in K\mathrm{and}\sigma \left(i\right)\in K,\hfill \end{array}\right.$$

which means that for each $i\in \{1,\dots ,n\}$, ${\left({\mathbf{y}}^{K^c}\right)}_i={\left({\mathbf{x}}_{\sigma }^K\right)}_i$, and thus ${\mathbf{y}}^{K^c}={\mathbf{x}}_{\sigma }^K$ (see (2)). Due to (3), we have shown that F is an n-K-symmetric n-K-aggregation function if and only if for each $\mathbf{x}\in {[0,1]}^n$ and any permutation $\sigma$ on $\{1,\dots ,n\}$, $F\left(\mathbf{x}\right)=F\left({\mathbf{x}}_{\sigma }^K\right)$. □

Remark 2.

(i) Note that ${\mathbf{x}}_{\sigma }^K[:={\left({\left({\mathbf{x}}^K\right)}_{\sigma }\right)}^K]$ differs, in general, from both ${\left({\mathbf{x}}_{\sigma }\right)}^K$ and ${\left({\mathbf{x}}^K\right)}_{\sigma }$. (ii) In Definition 3, based on the properties of aggregation functions, we have introduced the related properties of K-aggregation functions. Proposition 3 has shown how, in some particular cases, one can introduce properties of K-aggregation functions, independently of the related properties of standard aggregation functions, and, similarly, the following Propositions 4 and 5.

Example 2.

(i) Let $n=3$. Consider the function $F:{[0,1]}^3\to [0,1]$,

$$F(x_1,x_2,x_3)=x_1(1-x_2)x_3.$$

Obviously, F is a ternary K-aggregation function with $K=\{1,3\}$. In this case, it is easy to see that F is related to the standard ternary product, $\Pi :{[0,1]}^3\to [0,1]$, $\Pi (x_1,x_2,x_3)=x_1{x}_2{x}_3$, i.e., $F={\Pi }^{\left\{2\right\}}$. Following Definition 3, as Π is a symmetric aggregation function, F is 3-$\{1,3\}$-symmetric.

To illustrate, consider a permutation σ given by $\sigma \left(1\right)=2$, $\sigma \left(2\right)=3$ and $\sigma \left(3\right)=1$. Then for any $\mathbf{x}=(x_1,x_2,x_3)\in {[0,1]}^3$ we have

$$F\left({\mathbf{x}}_{\sigma }^K\right)=F(1-x_2,1-x_3,x_1)=(1-x_2)x_3{x}_1=F\left(\mathbf{x}\right).$$

(ii) Another example of an n-K-symmetric aggregation function is given in Example 1 (iii), see (1).

Remark 3.

(i) Obviously, the n-K-symmetry of n-K-aggregation functions for $K\in \{\varnothing ,\{1,\dots ,n\left\}\right\}$ coincides with the standard symmetry.

(ii) Note that for any fixed $K\subseteq \{1,\dots ,n\}$ and any K-preserving permutation σ on $\{1,\dots ,n\}$, i.e., a permutation satisfying the property $\{i\in \{1,\dots ,n\}\mid \sigma (i)\in K\}=K$, we have ${\mathbf{x}}_{\sigma }^K={\mathbf{x}}_{\sigma }$ for each $\mathbf{x}\in {[0,1]}^n$. This fact allows us to introduce a weaker form of n-K-symmetry when the equality $F\left(\mathbf{x}\right)=F\left({\mathbf{x}}_{\sigma }\right)$ is required for all $\mathbf{x}\in {[0,1]}^n$ and any K-preserving permutation σ. Recall that if K is a proper subset of $\{1,\dots ,n\}$, this relaxed n-K-symmetry can be seen as the 2-symmetry property introduced and studied in [21]. Obviously, also in this case, the standard symmetry is recovered whenever $K\in \{\varnothing ,\{1,\dots ,n\left\}\right\}$.

Now, consider an n-K-aggregation function F that is generated by an aggregation function A, i.e., $F=A^{K^c}$. By definition, F has an n-K-neutral element e whenever A has neutral element e. Recall that an element $e\in [0,1]$ is a neutral element of an n-ary aggregation function A if and only if for any $i\in \{1,\dots ,n\}$, $A(x_1,\dots ,x_n)=x_i$ whenever $x_j=e$ for each $j\in \{1,\dots ,n\}\setminus \left\{i\right\}$. Now, we give the following characterization of n-K-neutral elements of hybrid n-K-aggregation functions.

Proposition 4.

A hybrid n-K-aggregation function $F:{[0,1]}^n\to [0,1]$ has an n-K-neutral element $e\in [0,1]$ if and only if

$F\left(\mathbf{x}\right)=x_i$ for each $i\in K$ and all $\mathbf{x}=(x_1,\dots ,x_n)\in {[0,1]}^n$ such that

$$x_j=\left\{\begin{array}{cc}e\hfill & ifj\in K\setminus \left\{i\right\},\hfill \\ 1-e\hfill & ifj\in K^c,\hfill \end{array}\right.$$

and

$F\left(\mathbf{x}\right)=1-x_i$ for each $i\in K^c$ and all $\mathbf{x}=(x_1,\dots ,x_n)\in {[0,1]}^n$ such that

$$x_j=\left\{\begin{array}{cc}e\hfill & ifj\in K,\hfill \\ 1-e\hfill & ifj\in K^c\setminus \left\{i\right\}.\hfill \end{array}\right.$$

Let us add that if an n-K-aggregation function F is decreasing (i.e., if $K=\varnothing$) then it has an n-K-neutral element e whenever $F\left(\mathbf{x}\right)=1-x_i$ for each $\mathbf{x}\in {[0,1]}^n$ such that $x_j=1-e$ for each $j\ne i$.

Example 3.

Let $n=3$. Consider again the function $F:{[0,1]}^3\to [0,1]$,

$$F(x_1,x_2,x_3)=x_1(1-x_2)x_3.$$

As mentioned in Example 2 (i), F is a 3-K-aggregation function with $K=\{1,3\}$ related to the standard ternary product Π, $F={\Pi }^{\left\{2\right\}}$. The neutral element of Π is $e=1$, and it is a 3-$\{1,3\}$-neutral element of F, because $F(x_1,0,1)=x_1$, $F(1,0,x_3)=x_3$ and $F(1,x_2,1)=1-x_2$.

In what follows, we will investigate the notions of n-K-idempotency and n-K-annihilator.

Recall that an n-ary aggregation function A is said to be idempotent if for any element $\mathbf{c}=(c,\dots ,c)\in {[0,1]}^n$ we have $A\left(\mathbf{c}\right)=c$. If such an aggregation function generates an n-K-aggregation function F then the idempotency of A results in the n-K-idempotency of F which can be directly characterized as follows.

Proposition 5.

An n-K-aggregation function $F:{[0,1]}^n\to [0,1]$ is n-K-idempotent if and only if for each $c\in [0,1]$ we have $F\left({\mathbf{c}}^K\right)=c$, where ${\mathbf{c}}^K$ denotes an n-tuple with components ${\left({\mathbf{c}}^K\right)}_i$, $i=1,\dots ,n$, given by

$${\left({\mathbf{c}}^K\right)}_i=\left\{\begin{array}{cc}c\hfill & \mathrm{if}i\in K,\hfill \\ 1-c\hfill & \mathrm{if}i\in K^c.\hfill \end{array}\right.$$

As an example, we mention an n-K-aggregation function G given by (1) which is n-K-idempotent. The following example shows a 2-K-idempotent 2-K-aggregation function.

Example 4.

Let $n=2$. Consider the function $F:{[0,1]}^2\to [0,1]$,

$$F(x_1,x_2)=min(x_1,1-x_2).$$

Then, F is a 2-K-aggregation function for $K=\left\{1\right\}$. If $K=\left\{1\right\}$, then for any $c\in [0,1]$ we have $F\left({\mathbf{c}}^K\right)=F(c,1-c)=min(c,1-(1-c))=c$, i.e., F is 2-K-idempotent. Note that F is generated by the aggregation function $A=Min$, i.e., $F=Mi{n}^{\left\{2\right\}}$.

By Definition 3, an n-K-aggregation function $F:{[0,1]}^n\to [0,1]$ has an n-K-annihilator if and only if the aggregation function A generating F has an annihilator, i.e., if there exists an element $a\in [0,1]$ such for each $\mathbf{x}\in {[0,1]}^n$, $A\left(\mathbf{x}\right)=a$ whenever $a\in \{x_1,\dots ,x_n\}$. Using this property and the relation between F and A given in Theorem 1, we get the following characterization of an n-K-annihilator.

Proposition 6.

An n-K-aggregation function $F:{[0,1]}^n\to [0,1]$ has an n-K-annihilator a if and only $F\left(\mathbf{x}\right)=a$ for any $\mathbf{x}\in {[0,1]}^n$ such that either $x_i=a$ for some $i\in K$ or $x_j=1-a$ for some $j\in K^c$.

Example 5.

Let us continue in Examples 2 (i) and 3. The function $F:{[0,1]}^3\to [0,1]$, $F(x_1,x_2,x_3)$$=x_1(1-x_2)x_3,$ is a ternary $\{1,3\}$-aggregation function related to the standard ternary product Π, $F={\Pi }^{\left\{2\right\}}$. The annihilator $a=0$ of Π is the 3-$\{1,3\}$-annihilator of F. Indeed, we have $F(x_1,x_2,x_3)=0$ whenever $0\in \{x_1,x_3\}$ or $x_2=1$.

Remark 4.

Consider an n-K-aggregation function F and its dual $F^d$. Then we have:

-

F is n-K-symmetric if and only if $F^d$ is n-K-symmetric;

-

F is n-K-idempotent if and only if $F^d$ is n-K-idempotent;

-

F has an n-K-neutral element e if and only if $F^d$ has an n-K-neutral element $1-e$;

-

F has an n-K-annihilator a if and only if $F^d$ has an n-K-annihilator $1-a$.

4. Binary $\mathit{K}$-Aggregation Functions

In this section, we fix $n=2$ and we will discuss 2-K-aggregation functions. There are four subsets of the basic set $\{1,2\}$, i.e., one can consider a set $K\in \{\varnothing ,\{1\},\{2\},\{1,2\left\}\right\}$. The set $K=\varnothing$ corresponds to decreasing fusion functions and the set $K=\{1,2\}$ corresponds to increasing fusion functions. For $K=\left\{1\right\}$ and $K=\left\{2\right\}$ we obtain two classes of hybrid monotone binary fusion functions; binary $\left\{1\right\}$-fusion functions that are increasing in the first variable and decreasing in the second one, and binary $\left\{2\right\}$-fusion functions that are decreasing in the first variable and increasing in the second one. A similar classification applies to 2-K-aggregation functions. We mention some of them below.

As already observed, binary $\{1,2\}$-aggregation functions are just standard binary aggregation functions, whose distinguished subclasses are, for example, conjunctive aggregation functions (i.e., monotone extensions of the Boolean conjunction), their duals—disjunctive aggregation functions (i.e., monotone extensions of the Boolean disjunction), or averaging aggregation functions (characterized by the inequalities $Min\le A\le Max$), etc. For an exhaustive overview of binary aggregation functions we refer to Refs. [1,3,9,16].

From Proposition 2 (ii), it follows that a binary fusion function F is a binary ∅-aggregation function if and only if $F=1-A$, where A is a binary aggregation function. So, for example, for any conjunctive binary aggregation function C, the function $F=1-C$ is a monotone extension of the Boolean Sheffer stroke [22]. It is a typical example of a binary ∅-aggregation function.

A similar relation is between binary $\left\{1\right\}$- and binary $\left\{2\right\}$-aggregation functions. A fusion function $F:{[0,1]}^2\to [0,1]$ is a binary $\left\{1\right\}$-aggregation function if and only if $1-F$ is a binary $\left\{2\right\}$-aggregation function. Consider a binary fusion function F which is a $\left\{2\right\}$-aggregation function. Then, due to the boundary conditions, we have $F(1,0)=0$ and $F(0,1)=1$. If we add the constraints $F(0,0)=1$ and $F(1,1)=1$, then F is a fuzzy implication function [7], i.e., a hybrid monotone extension of the Boolean implication. Note that each fuzzy implication function $F:{[0,1]}^2\to [0,1]$ has $\left\{2\right\}$-annihilator $a=1$. Moreover, it has $\left\{2\right\}$-neutral element $e=0$ if and only if it satisfies the neutrality principle $F(1,y)=y$, $y\in [0,1]$, and its related negation $N_F\left(x\right)=F(x,0)$, $x\in [0,1]$, is the standard Zadeh negation, i.e., $N_F\left(x\right)=1-x$. For any fuzzy implication function its dual $F^d$ is also a $\left\{2\right\}$-aggregation function, constrained by the conditions $F^d(0,0)=F^d(1,1)=0$. Note that $F^d$ is a hybrid monotone extension of the Boolean coimplication, and it is called a fuzzy coimplication, see Ref. [23].

A typical property studied for binary aggregation functions is their associativity. Recall that a binary aggregation function $A:{[0,1]}^2\to [0,1]$ is associative if and only if $A(A(x_1,x_2),x_3)=A(x_1,A(x_2,x_3))$ for all $x_1,x_2,x_3\in [0,1]$. For a set $K\subseteq \{1,2\}$, let F be a 2-K-aggregation function generated by an associative aggregation function A. Using the relation between F and A, see Theorem 1, and the associativity of A, we can characterize the 2-K-associativity of F by the equality

$$F^{K^c}(F^{K^c}(x_1,x_2),x_3)=F^{K^c}(x_1,F^{K^c}(x_2,x_3)).$$

(4)

required for all $x_1,x_2,x_3\in [0,1]$.

Example 6.

Consider the binary product $\Pi :{[0,1]}^2\to [0,1]$, $\Pi (x_1,x_2)=x_1{x}_2$, and $K=\left\{1\right\}$. The related binary $\left\{1\right\}$-aggregation function $F:{[0,1]}^2\to [0,1]$ is given by $F(x_1,x_2)=x_1(1-x_2)$. As Π is associative, F is $\left\{1\right\}$-associative. Indeed, $K^c=\left\{2\right\}$ and thus

$$F^{K^c}(x_1,x_2)=F(x_1,1-x_2)=x_1(1-(1-x_2))=x_1{x}_2,$$

and further,

$$\begin{array}{ccc}\hfill F^{K^c}(F^{K^c}(x_1,x_2),x_3)& =& F^{K^c}(x_1{x}_2,x_3)=F(x_1{x}_2,1-x_3)\hfill \\ & =& x_1{x}_2(1-(1-x_3))=x_1{x}_2{x}_3.\hfill \end{array}$$

On the other hand,

$$\begin{array}{ccc}\hfill F^{K^c}(x_1,F^{K^c}(x_2,x_3))& =& F^{K^c}(x_1,x_2{x}_3)=F(x_1,1-x_2{x}_3)\hfill \\ & =& x_1(1-(1-x_2{x}_3))=x_1{x}_2{x}_3,\hfill \end{array}$$

which proves the equality required in (4).

Example 7.

As a typical example of a binary $\left\{2\right\}$-associative $\left\{2\right\}$-aggregation function F we mention an $(S,N)$-implication function F, see Ref. [7], where S is a t-conorm and N is the Zadeh negation, i.e., such an F is given by $F(x_1,x_2)=S(1-x_1,x_2)$. Consider the probabilistic sum $S_P$ given by $S_P(x_1,x_2)=x_1+x_2-x_1{x}_2$. Then the related binary $\left\{2\right\}$-associative $\left\{2\right\}$-aggregation function F is just the Reichenbach fuzzy implication function $I_R$, see Section 1.

Finally, let us note that all the discussed n-K-properties of n-K-aggregation functions are $d_K$-dual to the corresponding properties of aggregation functions (as mentioned in Remark 1). Therefore, the presented definitions and the proofs of the mentioned properties are rather simple and transparent. We have included them into the paper for the convenience of the readers and to have an alternative independent view on the presented n-K-properties.

5. Concluding Remarks

It is known that any n-ary aggregation function $A:{[0,1]}^n\to [0,1]$ can be seen as a lattice order homomorphism between the bounded lattices $({[0,1]}^n,{\le }_n)$ and $\left(\right[0,1],\le )$. Note that $({[0,1]}^n,{\le }_n)$ is a bounded lattice equipped by the standard order ${\le }_n$ of real n-tuples, which is a linear order if $n=1$ and a partial order if $n>1$. A similar representation can be considered for the introduced n-K-aggregation functions. For a subset $K\subseteq \{1,\dots ,n\}$, we can introduce a (partial) order ${\le }_K$ on ${[0,1]}^n$ as follows:

$$\mathbf{x}{\le }_K\mathbf{y}\iff ((\forall i\in K,x_i\le y_i)\wedge (\forall i\in K^c,x_i\ge y_i)).$$

Then $({[0,1]}^n,{\le }_K)$ is a bounded lattice with the top element ${\mathbf{1}}_K$ and the bottom element ${\mathbf{1}}_{K^c}$. Then, any n-K-aggregation function $F:{[0,1]}^n\to [0,1]$ can be seen as a lattice order homomorphism between $({[0,1]}^n,{\le }_K)$ and $\left(\right[0,1],\le )$ (and any n-K-fusion function as a lattice homomorphism). Our approach can be seen as a generalization of several branches of fuzzy set theory. For example, for $n=2$, the ordering of fuzzy intuitionistic values introduced by Atanassov [24] or q-rung orthopair values discussed by Yager in [25] is just the ordering ${\le }_{\left\{1\right\}}$ on a related subdomain of ${[0,1]}^2$.

As already mentioned, the concept of n-K-aggregation functions was inspired, among others, by fuzzy implications, i.e., binary $\left\{2\right\}$-aggregation functions with additional constraints $F(0,0)=F(1,1)=1$, and by utility functions when some positive and some negative criteria evaluated on the scale $[0,1]$ are considered. In addition, to the best of our knowledge, no conceptual theory of mixed aggregation functions has been elaborated so far. Our approach fills this gap by translating and shifting it into a classical aggregation theory, where the methods and concepts have been broadly investigated. The basic idea of this contribution is a generalization of aggregation functions, replacing the mandatory requirement of coordinate-wise increasingness by the coordinate-wise monotonicity. Our approach is closely related to the concept of directional monotonicity [12] generalizing monotone fusion functions. While aggregation functions are monotone increasing in each variable, which, expressed by the notion of directional monotonicity, means that they are directionally increasing in all directions ${\overrightarrow{e}}_i=\left(e_{i,j}\right)$ with $i,j\in \{1,\dots ,n\}$, $e_{i,j}=1$ if $i=j$ and $e_{i,j}=0$ otherwise, n-K-aggregation functions are in all these directions directionally monotone. To stress the difference, observe that while n-K-aggregation functions can be seen as lattice order homomorphisms, in general, this is not the case of pre-aggregation functions [16]. Note that fuzzy implications are binary $\left\{2\right\}$-aggregation functions, but they are not pre-aggregation functions. Since our paper contributes to the basics of the aggregation theory, we expect near applications of the introduced concept.

As an example, one can expect possible applications of our newly introduced mathematical tools in optimal control in large-scale multiagent systems, see, e.g., Refs. [26,27]. Similarly, our approach has the potential to contribute to the specification of dynamic aggregation functions in optimal control, dynamical programming, and multiple attribute dynamical decision making, see, e.g., Ref. [28].