Set-Based Extended Functions

Abstract

In this paper, inspired by the Zadeh approach to the fuzzy connectives in fuzzy set theory and by some applications, we introduce and study set-based extended functions on different universes. After presenting some results for set-based extended functions on a general universe, we focus our investigation on set-based extended functions on some particular universes, including lattices and (bounded) chains. A special attention is devoted to characterization of set-based extended aggregation functions on the unit interval [0, 1].

1 Introduction

Lotfi Zadeh proposed in his seminal paper [13] to use the minimum and maximum operators for modeling fuzzy intersection and fuzzy union, respectively. This paper focuses on such kinds of fusion procedures that share with Zadeh’s proposal a particular property, namely, that these fuzzy connectives can be seen as functions which, for any $n, m\in \mathbb {N}$ and any input vectors $\mathbf {x}= (x_1,\ldots , x_n)\in [0,1]^n$ and $\mathbf {z}= (z_1,\ldots ,z_m)\in [0,1]^m$ such that the sets $\{x_1,\ldots , x_n\}$ and $\{z_1,\ldots ,z_m\}$ coincide, provide for input vectors $\mathbf {x}$ and $\mathbf {z}$ the same output values, i.e.,

$$Min(\mathbf {x})= Min(\mathbf {z}) \ \ \mathrm{and}\ \ Max(\mathbf {x})= Max(\mathbf {z}).$$

In statistics, for a sample $(x_1,\ldots , x_n)$ several kinds of mean values have been introduced. For example, the arithmetic mean $AM(\mathbf {x})= \frac{1}{n}\sum _{i=1}^n x_i$ is the minimizer of the sum of squares $\sum _{i=1}^n (x_i - a)^2$ (Least Squares Method). Minimizing the maximal deviation, i.e., looking for the minimizer of $\max \{|x_i -a|\mid i= 1,\ldots ,n\}$ leads to the resulting mean M given by

$$M(\mathbf {x})= \frac{\min \{x_1,\ldots ,x_n\} + \max \{x_1,\ldots ,x_n\}}{2}.$$

Observe that repeating or rearrangement of observations does not have any influence on the output of M, i.e., for example, taking a sample

$$\mathbf {z}= (x_1,x_1, x_1, x_2,x_2,x_3,\ldots , x_n),$$

we obtain $M(\mathbf {z})= M(\mathbf {x})$.

Inspired by the mentioned observations, and taking into account that in most fusion problems the number of values to be fused cannot be fixed a priori, in this paper we will work with extended functions $F:\bigcup \limits _{n\in \mathbb {N}} X^n \rightarrow X$, $X\ne \emptyset$, satisfying, in addition, the above discussed property. They will be called set-based extended functions on X (for the definition see below). Evidently, each such set-based extended function depends on the set $\{y_1,\ldots ,y_k\}$ of values related to the input vector $(x_1,\ldots , x_n)$, where $\{x_1,\ldots , x_n\}=\{y_1,\ldots ,y_k\}$ and $\mathrm{card}(\{y_1,\ldots ,y_k\})=k$. Hence, neither the repetition of arguments to be fused nor their rearrangement have any influence on the output result.

We will proceed as follows. First, we propose the concept of set-based extended functions defined for arbitrary but finitely many inputs from some non-empty universe X, with outputs also from X. In the beginning, we examine properties of set-based extended functions acting on a general universe X. The obtained results are contained in Sect. 2. The next section is devoted to the investigation of set-based extended functions on a (bounded) lattice X. In Sect. 4, X is considered to be a (bounded) chain. This section also contains a characterization of set-based extended aggregation functions on $X=[0,1]$. Finally, some concluding remarks are added.

2 Set-Based Extended Functions on a General Universe

Suppose that we classify some products and their samples as good or bad only, i.e., we deal with the universe $X=\{g,b\}$. A function $F:\bigcup \limits _{n\in \mathbb {N}} X^n \rightarrow X$ assigns to a sample $\mathbf {x}=(x_1,\ldots ,x_n)\in X^n$ either the value good—if all the inputs $x_1,\ldots , x_n$ are good, or the value bad—in all other cases. The output value $F(\mathbf {x})$ depends on the set $\{x_1,\ldots ,x_n\}$ only, namely,

$$ F(x_1,\ldots ,x_n)= \left\{ \begin{array}{ll} b &{} \ \mathrm{if} \ b\in \{x_1,\ldots ,x_n\},\\ g &{} \ \mathrm{otherwise}.\end{array}\right. $$

Moreover, if we add any other inputs $y_1,\ldots , y_k$, but such that each of them has already appeared in the original sample, i.e., $y_1,\ldots , y_k\in \{x_1,\ldots ,x_n\}$, then

$$F(x_1,\ldots ,x_n, y_1,\ldots , y_k)= F(x_1,\ldots ,x_n).$$

In what follows, we formalize the above described situation, and define the notion of set-based extended function on a general universe X. We start by recalling the notion of extended function on X.

Definition 2.1

Let $X\ne \emptyset$. Any function $F:\bigcup \limits _{n\in \mathbb {N}} X^n \rightarrow X$ will be called an extended function on X.

Extended functions have open arity, i.e., they can work for any finite number of arguments.

Definition 2.2

Let $X\ne \emptyset$. A function $F:\bigcup \limits _{n\in \mathbb {N}} X^n \rightarrow X$ is called a set-based extended function on X if $F(\mathbf {y})= F(\mathbf {x})$ for any $n, k\in \mathbb {N}$ and all $\mathbf {x}= (x_1,\ldots ,x_n)\in X^n$, $\mathbf {y}= (y_1,\ldots , y_k)\in X^k$, such that $\{x_1,\ldots ,x_n\}= \{y_1,\ldots , y_k\}$.

Example 2.1

Consider a set X with cardinality $\mathrm{card}(X) >2$. Let E be a proper subset of X, and $a, b\in X$, $a\ne b$. Define $F_{E,a,b}:\bigcup \limits _{n\in \mathbb {N}} X^n \rightarrow X$ by

$$F_{E,a,b}(x_1,\ldots ,x_n) = \left\{ \begin{array}{ll} a &{} \ \mathrm{if} \ E\cap \{x_1,\ldots ,x_n\}\ne \emptyset ,\\ b &{} \ \mathrm{otherwise}.\end{array}\right. $$

Then $F_{E,a,b}$ is a set-based extended function on X. Note that $F_{E,a,b}$ is associative if and only if $a\in E$, where the associativity of a function $F:\bigcup \limits _{n\in \mathbb {N}} X^n \rightarrow X$ means that

$$F(\mathbf {x},\mathbf {y}) = F(F(\mathbf {x}), F(\mathbf {y}))$$

for all $\mathbf {x}, \mathbf {y} \in \bigcup \limits _{n\in \mathbb {N}} X^n$.

Example 2.1 is an example of a particular case of the construction of set-based extended functions described in the following proposition.

Proposition 2.1

Let $X\ne \emptyset$. Let $\mathcal {P}=\{E_1,\ldots ,E_k\}$ be a partition of X and $a_1,\ldots ,a_k\in X$. Define $F:\bigcup \limits _{n\in \mathbb {N}} X^n \rightarrow X$ by

$$\begin{aligned} F(\mathbf {x})= a_i, \ \ \mathrm{where} \ i=\min \{j\in \{1,\ldots ,k\}\,\mid \{x_1,\ldots ,x_n\} \cap E_j\ne \emptyset \}.\end{aligned}$$

(1)

Then F is a set-based extended function on X.

Example 2.2

Let $p\in \mathbb {N}$ and $X=\{1,\ldots ,p\}$. Then

• if we consider the partition $\mathcal {P}=\{E_i\}_{i=1}^p$, where $E_i=\{i\}$, and $a_i=i$, then (1) defines the function $Min:\bigcup \limits _{n\in \mathbb {N}} X^n \rightarrow X$ given by $Min(x_1,\ldots ,x_n)= \min \{x_1,\ldots ,x_n\}$;

• if $\mathcal {P}=\{E_i\}_{i=1}^p$, where $E_i= \{p-i+1\}$ and $a_i= p-i+1$, then (1) yields the function Max, $Max(x_1,\ldots ,x_n)= \max \{x_1,\ldots ,x_n\}$.

Lemma 2.1

Let $X\ne \emptyset$ and $\mathcal {H}(X)=\{\emptyset \ne E\subseteq X\,\mid E \ \mathrm{is\ finite} \}.$ Then each set-based extended function F on X corresponds in a one-to-one correspondence to a set function $G:\mathcal {H}(X)\rightarrow X$ given, for each $E= \{x_1,\ldots ,x_n\}$ in $\mathcal {H}(X)$, by

$$G(E)= F(x_1,\ldots ,x_n).$$

Clearly, $\mathcal {H}(X)$ is the power set of X except the empty set whenever X is finite.

Note that properties of the set function $G:\mathcal {H}(X)\rightarrow X$ can be transformed into new kinds of properties of the related set-based extended function F on X, as is shown in the following example.

Example 2.3

Consider $X=\mathbb {N}$ and define $G:\mathcal {H}(\mathbb {N})\rightarrow \mathbb {N}$ by $G(E)= \sum \limits _{i\in E} i$.

Obviously, G is monotone non-decreasing, because for all $E_1,E_2$ in $\mathcal {H}(\mathbb {N})$, $G(E_1)\le G(E_2)$ whenever $E_1\subseteq E_2$. G is also additive, i.e.,

$$G(E_1\cup E_2) = G(E_1) + G(E_2)\ \ \mathrm{whenever}\ \ E_1 \cap E_2 =\emptyset .$$

The set-based extended function $F:\bigcup \limits _{n\in \mathbb {N}} \mathbb {N}^n \rightarrow \mathbb {N}$ corresponding to G, is given by

$$ F(x_1,\ldots ,x_n)= \sum \limits _{i\in \mathbb {N}} i\cdot \min \,\left\{ 1, \sum \limits _{j=1}^n \mathbf {1}_{\{i\}}(x_j) \right\} ,$$

and is neither monotone non-decreasing nor additive in the standard case, because, given any $n\in \mathbb {N}$, the relation $\mathbf {x}\le \mathbf {y}$ does not imply $F(\mathbf {x})\le F(\mathbf {y})$ for all $\mathbf {x}, \mathbf {y}\in \mathbb {N}^n$, and similarly, the additivity property F( $\mathbf {x}+ \mathbf {y})= F(\mathbf {x}) +F(\mathbf {y})$ does not hold for all $\mathbf {x}, \mathbf {y}\in \mathbb {N}^n$.

However, F is monotone non-decreasing with respect to the partial order $\preceq$ on $\bigcup \limits _{n\in \mathbb {N}} \mathbb {N}^n$, defined as follows: for any $n,k\in \mathbb {N}$ and all $\mathbf {x}\in \mathbb {N}^n$, $\mathbf {y}\in \mathbb {N}^k$,

$$ \mathbf {x}\preceq \mathbf {y} \ \mathrm{whenever} \ n\le k \ \mathrm{and}\ x_i=y_i \ \mathrm{for\ all} \ i\le n.$$

Indeed, then for all $\mathbf {x}, \mathbf {y}\in \bigcup \limits _{n\in \mathbb {N}} \mathbb {N}^n$, if $\mathbf {x}\preceq \mathbf {y}$ then $F(\mathbf {x})\le F(\mathbf {y})$.

Similarly, F is concatenation additive, i.e., if $\{x_1,\ldots ,x_n\} \cap \{y_1,\ldots ,y_k\} = \emptyset$, then $F(\mathbf {x},\mathbf {y})= F(\mathbf {x}) +F(\mathbf {y})$.

We still give another example illustrating Lemma 2.1.

Example 2.4

Consider $X= \{0,1\}$. Then a function $F:\bigcup \limits _{n\in \mathbb {N}} \{0,1\}^n \rightarrow \{0,1\}$ is an extended Boolean function. The cardinality of X is $\mathrm{card}(X)=2$, $\mathcal {H}(X)=\{\{0\}, \{1\}, \{0,1\}\}$, i.e., $\mathrm{card}(\mathcal {H}(X))=3$, thus there are exactly $2^3=8$ set functions $G_i:\mathcal {H}(X)\rightarrow \{0,1\}$, $i =1,\ldots ,8$. Consequently, there are 8 set-based extended Boolean functions $F_i$, where $F_i$ corresponds to $G_i$ by Lemma 2.1. The results are summarized in Table 1.

Table 1. Set-based extended Boolean functions

Proposition 2.2

Fix $X=\{1,2,\ldots ,k\}$. Consider a permutation $\sigma :X\rightarrow X$ and a total order $\preceq _{\sigma }$ on X determined by $\sigma$, given by

$$x\preceq _{\sigma } y \ \ \mathrm{if\ and\ only\ if}\ \ \sigma ^{-1}(x)\le \sigma ^{-1}(y).$$

Let $G_{\sigma }:\mathcal {H}(X)\rightarrow X$, $G_{\sigma }(E)=\min _{\preceq _{\sigma }} \{x \mid x\in E\}.$ Then the set-based extended function $F_{\sigma }:\bigcup \limits _{n\in \mathbb {N}} X^n\rightarrow X$, $F_{\sigma }(\mathbf {x})= G_{\sigma }(\{x_1,\ldots ,x_n\})$, is symmetric, associative, and with neutral element $e=\sigma (n)$, but in general, $F_{\sigma }$ need not be monotone.

Recall that $e\in X$ is a neutral element of an extended function F on X, if for all $n\in \mathbb {N}$, and all $\mathbf {x}\in X^n$, with $e=x_i$ for some $i\in \{1,\ldots ,n\}$, we have

$$F(x_1,\ldots , x_{i-1},e,x_{i+1},\ldots , x_n)= F(x_1,\ldots , x_{i-1},x_{i+1},\ldots , x_n).$$

Obviously, in Proposition 2.2, there are k! set-based extended functions $F_{\sigma }$.

Remark 2.1

In Proposition 2.2, if for each $x,y\in X$,

$$x< y< e \ \Rightarrow \ \sigma ^{-1}(x)< \sigma ^{-1}(y)\ \ and \ \ x> y> e \ \Rightarrow \ \sigma ^{-1}(x) < \sigma ^{-1}(y),$$

then $F_{\sigma }$ is an idempotent uninorm (and only in that case). There are $2^{k-1}$ idempotent uninorms on X.

Note that the previous result for idempotent uninorms was also proved by Zemánková in [12].

We now summarize some properties related to general set-based functions.

Proposition 2.3

Let $X\ne \emptyset$. Set-based extended functions on X have the following properties.

• (i) Each set-based extended function on X is symmetric.

• (ii) For any function $V:X^k\rightarrow X$ and any set-based extended functions $F_1,\ldots , F_k$ on X, also the composite $F= V(F_1,\ldots , F_k):\bigcup \limits _{n\in \mathbb {N}} X^n \rightarrow X$ is a set-based extended function on X.

• (iii) For any function $V:X\rightarrow X$ and a any set-based extended function F on X, also the composites $V(F), F(V) :\bigcup \limits _{n\in \mathbb {N}} X^n \rightarrow X$, given by

  $$V(F)(\mathbf {x})=V(F(\mathbf {x}))\ \ \mathrm{and} \ \ F(V)(\mathbf {x})= F(V(x_1), \ldots , V(x_n)),$$

  respectively, are set-based extended functions.

Proposition 2.4

Let $X_i\ne \emptyset$, $i= 1,\ldots , k$, and let X be the Cartesian product of $X_i$, $X=X_1 \times \cdots \times X_k$. For any set-based extended functions $F_i$ on $X_i$, $i= 1,\ldots ,k$, the function $F:\bigcup \limits _{n\in \mathbb {N}} X^n \rightarrow X$, defined by

$$F((x_1^{(1)},\ldots ,x_k^{(1)}),\ldots , (x_1^{(n)},\ldots ,x_k^{(n)}))= (F_1(x_1^{(1)},\ldots ,x_1^{(n)}), \ldots , F_k(x_k^{(1)},\ldots ,x_k^{(n)})),$$

is a set-based extended function on X.

The following theorem shows that some algebraic properties of a function $F:\bigcup \limits _{n\in \mathbb {N}} X^n \rightarrow X$ already ensure that F is a set-based extended function on X.

Theorem 2.1

Let $X\ne \emptyset$. Let $F:\bigcup \limits _{n\in \mathbb {N}} X^n \rightarrow X$ be symmetric, idempotent and associative. Then F is a set-based extended function on X.

Proof:

Let F satisfy the given assumptions. For any $n\in \mathbb {N}$ and each $\mathbf {x}=(x_1,\ldots ,x_n)\in X^n$ with $\mathrm{card}(\{x_1,\ldots ,x_n\}) =k$, let $\{x_1,\ldots ,x_n\}= \{y_1,\ldots ,y_k\}$. Then there is a partition $\{I_1,\ldots ,I_k\}$ of $\{x_1,\ldots ,x_n\}$ given by

$$I_i= \{j\in \{1,\ldots ,n\}\,\mid x_j= y_i\}.$$

Then, writing $I_i= \{{j_i}_1,\ldots ,{j_i}_{m_i}\}$, where $m_i= \mathrm{card}(I_i)$, we have

$$\begin{aligned}F(\mathbf {x})= & {} F(x_{{j_1}_1},\ldots , x_{{j_1}_{m_1}}, x_{{j_2}_{1}},\ldots ,x_{{j_2}_{m_2}},\ldots , x_{{j_k}_{1}},\ldots ,x_{{j_k}_{m_k}})\\= & {} F(F(x_{{j_1}_1},\ldots , x_{{j_1}_{m_1}}),F(x_{{j_2}_{1}},\ldots ,x_{{j_2}_{m_2}}),\ldots , F(x_{{j_k}_{1}},\ldots ,x_{{j_k}_{m_k}}))\\= & {} F(y_1,\ldots ,y_k), \end{aligned}$$

where the first equality follows from the symmetry of F, the second one from its associativity, and the third one follows from the idempotency of F. Obviously, for all $\mathbf {x}, \mathbf {z}\in \bigcup \limits _{n\in \mathbb {N}} X^n$, such that $\{x_1,\ldots ,x_n\}=\{y_1,\ldots ,y_k\}= \{z_1,\ldots ,z_m\}$, we have $F(\mathbf {x})= F(\mathbf {z})$, and hence F is a set-based extended function on X.    $\square$

Note that neither idempotency nor associativity are necessary properties for being F a set-based extended function, see Example 2.1 and Proposition 2.1.

3 Set-Based Extended Functions on Lattices

In this section we consider X to be a carrier of a lattice $(X,\le )$. For any fixed $a\in X$, we define a function $F_a:\bigcup \limits _{n\in \mathbb {N}} X^n \rightarrow X$ by

$$F_a(\mathbf {x}) =\left\{ \begin{array}{lll} \bigvee \limits _i x_i &{} \ \mathrm{if }\ \bigvee \limits _i x_i < a,\\ \bigwedge \limits _i x_i &{} \ \mathrm{if }\ \bigwedge \limits _i x_i > a,\\ a &{} \ \mathrm{otherwise.}\end{array}\right. $$

Obviously, $F_a$ is symmetric and idempotent, and its associativity can also be verified. By Theorem 2.1, $F_a$ is a set-based extended function on X. Moreover, $F_a$ is monotone non-decreasing, and thus it is an extended aggregation function on X, see [7] (because of the idempotency of $F_a$ we need not consider X to be a bounded lattice). Observe that if X is bounded, with top and bottom elements $\mathbf {1}_X$ and $\mathbf {0}_X$, respectively, then $F_{\mathbf {1}_X}= \vee$ is the standard join on X, and $F_{\mathbf {0}_X} =\wedge$ is the standard meet on X. By Theorem 2.1, any idempotent uninorm F on a bounded (distributive) lattice X [8], is a set-based extended function on X. Similarly, idempotent nullnorms on bounded lattices, see [9], are set-based extended functions.

Proposition 3.1

Let $(X,\le )$ be an ordinal sum of lattices $(X_i,\le _i)_{i\in I}$, and let for any $i\in I$, $F_i :\bigcup \limits _{n\in \mathbb {N}} X_i^n \rightarrow X_i$ be a set-based extended function on $X_i$. Define $F:\bigcup \limits _{n\in \mathbb {N}} X^n \rightarrow X$ by

$$F(x_1,\ldots ,x_n)= F_i(y_1,\ldots ,y_k),$$

where

$$\begin{aligned} i= & {} \min \{j\in I\,\mid \, \{x_1,\ldots ,x_n\} \cap X_j \ne \emptyset \},\\ k= & {} \mathrm{card}(\{j\in \{1,\ldots ,n\}\, \mid x_j\in X_i\}),\\ \{y_1,\ldots , y_k\}= & {} \{x_j\,\mid x_j\in X_i\}. \end{aligned}$$

Then F is a set-based extended function on X. Moreover, F is monotone non-decreasing if and only if all $F_i$, $i\in I$, are of that property, and it is idempotent if and only if all $F_i$, $i\in I$, are idempotent.

More information on ordinal sum of lattices can be found, e.g., in [3].

4 Set-Based Extended Aggregation Functions on Chains

In this section we consider X to be a (bounded) chain. A total order on X has an important impact on characterization of monotone set-based extended functions on X.

Proposition 4.1

Let X be a chain. Then $F :\bigcup \limits _{n\in \mathbb {N}} X^n \rightarrow X$ is a monotone non-decreasing (non-increasing) set-based extended function if and only if for each $\mathbf {x} \in \bigcup \limits _{n\in \mathbb {N}} X^n$ we have

$$\begin{aligned} F(\mathbf {x})= D(Min(\mathbf {x}), Max(\mathbf {x})), \end{aligned}$$

(2)

for some monotone non-decreasing (non-increasing) function $D:X^2\rightarrow X$.

Proof:

It is not difficult to see that representation of F in the form (2) is sufficient for being F a monotone non-decreasing (non-increasing) set-based extended function on X. We only prove a necessary condition.

Let F be a monotone non-decreasing set-based extended function on a chain X. As F is symmetric, with no loss of generality, we can only consider elements $\mathbf {x}\in \bigcup \limits _{n\in \mathbb {N}} X^n$ such that $x_1\le \cdots \le x_n$. Then $x_1=Min(\mathbf {x})$, $x_n= Max(\mathbf {x})$ and we can write

$$\begin{aligned} F(x_1,x_n)= & {} F(x_1,\ldots ,x_1, x_n)\le F(x_1,x_2,\ldots , x_{n-1},x_n)\le F(x_1,x_n,\ldots ,x_n)\nonumber \\= & {} F(x_1,x_n), \end{aligned}$$

(3)

which yields $F(\mathbf {x})= F(Min(\mathbf {x}), Max(\mathbf {x}))$. Putting $D= F{|_{X^2}}$, we obtain the required representation in the form (2). The monotonicity of D follows from the monotonicity of F. To get the result for a monotone non-increasing F, it is enough to reverse the inequalities in (3).    $\square$

Now we provide a characterization of set-based extended aggregation functions acting on a bounded chain X, in particular on $X=[0,1]$. In what follows, we only recall the notion of extended aggregation function on [0, 1], for more details on (extended) aggregation functions and their properties we recommend, e.g., [4, 7, 10], see also [1, 2].

Definition 4.1

A function $A:\bigcup \limits _{n\in \mathbb {N}} [0,1]^n \rightarrow [0,1]$ is an extended aggregation function on [0, 1] if A is monotone non-decreasing and satisfies the boundary conditions, i.e.,

• (i) for all elements $\mathbf {0}=(0,\ldots ,0), \mathbf {1}=(1,\ldots ,1)\in \bigcup \limits _{n\in \mathbb {N}} [0,1]^n$, $A(\mathbf {0})=0$ and $A(\mathbf {1})=1$;

• (ii) for all $\mathbf {x}, \mathbf {y}\in \bigcup \limits _{n\in \mathbb {N}} [0,1]^n$ we have $A(\mathbf {x})\le A(\mathbf {y})$ whenever $\mathbf {x}\le \mathbf {y}$.

Note that for $\mathbf {x}, \mathbf {y}\in \bigcup \limits _{n\in \mathbb {N}} [0,1]^n$ we have $\mathbf {x}\le \mathbf {y}$ if and only if $\mathbf {x}$ and $\mathbf {y}$ are n-tuples of the same arity n satisfying $x_i\le y_i$ for each $i=1,\ldots ,n$.

We will also work with n-ary aggregation functions on [0, 1], i.e., functions

$$A_{(n)}:[0,1]^n\rightarrow [0,1]$$

which satisfy boundary conditions (i) and monotonicity conditions (ii) from Definition 4.1 for a considered fixed $n\in \mathbb {N}$. Clearly, given an extended aggregation function A on [0, 1], the function $A_{(n)}= A_{|_{[0,1]^n}}$ is an n-ary aggregation function.

Definition 4.2

A function $A:\bigcup \limits _{n\in \mathbb {N}} [0,1]^n\rightarrow [0,1]$ is a set-based extended aggregation function if A is an extended aggregation function on [0, 1] satisfying the set-based property, i.e., for all $n,k\in \mathbb {N}$, and all $\mathbf {x}= (x_1,\ldots ,x_n)\in [0,1]^n$ and $\mathbf {y}= (y_1,\ldots , y_k)\in [0,1]^k$, $A(\mathbf {y})= A(\mathbf {x})$ whenever $\{x_1,\ldots ,x_n\}= \{y_1,\ldots , y_k\}$.

It can be shown that set-based extended aggregation functions on [0, 1] can be completely characterized as follows.

Theorem 4.1

Let $A:\bigcup \limits _{n\in \mathbb {N}} [0,1]^n \rightarrow [0,1]$ be an extended aggregation function on [0, 1]. A is a set-based extended aggregation function on [0, 1] if and only if for all $\mathbf {x}\in \bigcup \limits _{n\in \mathbb {N}} [0,1]^n$ we have

$$\begin{aligned} A(\mathbf {x}) = A(Min(\mathbf {x}), Max(\mathbf {x})). \end{aligned}$$

(4)

For more results on set-based extended aggregation functions on [0, 1], see [11].

By the previous theorem, set-based extended aggregation functions on [0, 1] are generated by binary aggregation functions; there is a one-to-one correspondence between the set of all set-based extended aggregation functions and the set of all symmetric binary aggregation functions. Observe that in the case of an associative symmetric binary aggregation function $A:[0,1]^2\rightarrow [0,1]$ there are two possible ways how to extend it into an extended aggregation function. On the one hand, based on formula (2), one can define the function $A_{\Box } :\bigcup \limits _{n\in \mathbb {N}} [0,1]^n \rightarrow [0,1]$ by

$$ A_{\Box }(\mathbf {x})= A(Min(\mathbf {x}), Max(\mathbf {x})),$$

and on the other hand, using the associativity of A, one can define the function $A_{\triangle } :\bigcup \limits _{n\in \mathbb {N}} [0,1]^n \rightarrow [0,1]$ by

$$A_{\triangle }(x_1)= x_1,\ A_{\triangle } (x_1,x_2)= A(x_1,x_2),$$

and for all $n\ge 3$,

$$ A_{\triangle }(x_1,\ldots ,x_n)= A(A_{\triangle }(x_1,\ldots ,x_{n-1}),x_n). $$

Due to Proposition 2.1, $A_{\Box }= A_{\triangle }$ if and only if a binary aggregation function A is idempotent, i.e., $A(x,x) =x$ for all $x\in [0,1]$. Note that this is, e.g., the case of idempotent uninorms [6, 12], and also the case of idempotent nullnorms [5] (compare $F_a$ introduced in Sect. 3). As a negative example, consider the standard product $A(x_1,x_2)=x_1 x_2$. Then $A_{\triangle }(x_1,\ldots ,x_n)= \prod \limits _{i=1}^n x_i$ is the standard product, which, if $n\ne 2$, differs from $A_{\Box }(\mathbf {x})= (Min(\mathbf {x}))\cdot Max(\mathbf {x}))$.

5 Concluding Remarks

In this paper, we have introduced and discussed set-based extended functions, which can be seen as a generalization of extended functions $F:\bigcup \limits _{n\in \mathbb {N}} X^n \rightarrow X$, which are symmetric, idempotent and associative. In the case when X is a lattice, the introduced set-based extended functions can be viewed as a particular generalization of joins, meets, idempotent uninorms and idempotent nullnorms. In the case of bounded chains, we have shown the existence of a one-to-one correspondence between set-based aggregation functions A and symmetric binary aggregation functions D given by

$$A(\mathbf {x})= D(Min(\mathbf {x}),Max(\mathbf {x})).$$

Based on the presented approach, in our future research we intend to solve how to relate aggregation of input values $x_1,\ldots ,x_n$ to aggregation of inputs $x_1,\ldots ,x_n, x_{n+1},\ldots , x_{n+k}$, where $x_{n+1},\ldots , x_{n+k}$ are some additionally obtained observations.