Keywords

1 Introduction

Although the use of some particular aggregation functions, such as for instance the arithmetic mean, dates from long ago, the systematic study of mathematical functions that could be used in merging several input data into a representative output value is quite more recent. The boom of the study of aggregation functions can be probably dated in the 1980s and from then they have been extensively studied (see [4, 5, 7, 12]). The great quantity of application fields where these operators play a key role has been the main reason for which the theoretical study of aggregation functions have experienced this important growth in last decades.

The usual definition of n-ary aggregation function (increasing functions \(f:[0,1]^n \rightarrow [0,1]\) with \(f(0,\ldots ,0)=0\) and \(f(1,\ldots ,1)=1\)) is rather general. This fact leads usually to require additional properties to the aggregation functions depending on each concrete study in the literature. In this sense, the following classification of n-ary aggregation functions is generally accepted (see [5]): the class of conjunctive aggregation functions (those that take values below the minimum), disjunctive aggregation functions (those that take values over the maximum), averaging aggregation functions (those that take values between the minimum and the maximum) and mixed aggregation functions (the remaining ones, that is, those with different types of behaviour on different parts of the domain). Many examples in each one of these classes are well-known. For instance,

  • Conjunctive (respectively, disjunctive) aggregation functions include t-norms, copulas and quasi-copulas (respectively, t-conorms, co-copulas and duals of quasi-copulas). These kinds of aggregations have been used mainly as logical operators with consequent applications in fuzzy logic and approximate reasoning, but also in image processing, probability and statistics, and economy, among others (see for instance [1, 13,14,15, 23]). In these topics the relation between aggregation functions and fuzzy implication functions becomes of great interest as it can be seen in many references (see [2, 3, 8, 11, 18, 24, 25]).

  • Averaging aggregation functions include all weighted means, OWA’s, Choquet and Sugeno integrals and many generalizations. They are mainly used in all processes related to aggregation of information, decision making, consensus, optimization, image analysis, and so on (see [4, 5, 7, 12, 26]).

  • Finally, mixed aggregation functions include uninorms and nullnorms among others, and they have been proved to be useful in many of the applications already mentioned for the other types of aggregations like fuzzy logic and approximate reasoning, image processing, decision making and so on (see for instance [6, 10, 16, 27] and the recent survey on uninorms [17]).

Among the theoretical aspects on aggregation functions, one of special interest is the study of their expressions in order to define aggregation functions whose expression is as simple as possible in order to make easier their implementation and computation in applications. In this direction, a first step could be to study aggregation functions whose expression is given by polynomial or rational functions of different degrees. This was already done in some particular cases. See for instance [1], where all the rational Archimedean continuous t-norms are characterized leading to the well-known Hamacher class (in particular, the only polynomial t-norm is the product t-norm \(T_P(x,y)=xy\)). In [9], all the rational uninorms were characterized as those whose expression is given by

$$U_e(x,y)={\left\{ \begin{array}{ll} \frac{(1-e)xy}{(1-e)xy+e(1-x)(1-y)} &{} \text {if }(x,y)\in [0,1]^2\setminus \{(0,1),(1,0)\},\\ 0 \ (\text {or } 1) &{} \text {otherwise.} \end{array}\right. }$$

In this case, there do not exist any polynomial uninorm since they are never continuous. A similar study for polynomial and rational fuzzy implication functions was done in the successive papers [19,20,21], where many different examples are shown.

Recently, an initial study of polynomial aggregation functions in general of degrees one and two was presented in [22]. Whereas weighted arithmetic means are the only polynomial aggregation functions of degree one, many different families appear in the case of degree 2. Such study was firstly done for binary aggregation functions and after that, some of these results were extended to n-ary aggregations (see [22]). Following this line of research, in this paper we want to deal with the case of binary aggregation functions given by polynomial expressions only in some partial domains of the unit square. The idea is that many aggregation functions have a non trivial 0 region (recall for instance the Łukasiewicz or the nilpotent minimum t-norm) or dually a non trivial 1 region (recall the dual t-conorms of the previous mentioned t-norms).

Thus, this paper is organized as follows. After some preliminaries in Sect. 2, we will focus in Sect. 3 firstly on binary aggregation functions with a non trivial 0 region limited by the classical negation \(N_c\) and given by a polynomial of degrees 1 or 2 over \(N_c\). In this study we will analyse some of their common additional properties like commutativity, associativity, idempotency and neutral element. Then in Sect. 4, using the duality, all these results can be adequately applied to the study of binary aggregation functions with a non trivial 1 region limited by the classical negation \(N_c\) and given by a polynomial of degrees 1 or 2 under \(N_c\). We end the paper with Sect. 5 devoted to conclusions and future work.

2 Preliminaries

Let us recall some concepts and results that will be used throughout this paper. First, we give the definition of an aggregation function.

Definition 1

([5, 7]). An n-ary aggregation function is a function of \(n>1\) arguments that maps the (n-dimensional) unit cube onto the unit interval \(f:[0,1]^n\rightarrow [0,1]\), with the properties

  1. (i)

    \(f(\underbrace{0,0,\ldots ,0}_{n-\text {times}})=0\) and \(f(\underbrace{1,1,\ldots ,1}_{n-\text {times}})=1\).

  2. (ii)

    \(\mathbf {x}\le \mathbf {y}\) implies \(f(\mathbf x )\le f(\mathbf y )\) for all \(\mathbf x \), \(\mathbf y \in [0,1]^n\).

In particular, a 2-ary aggregation function will be called a binary aggregation function.

As we have already mentioned in the introduction, many families of aggregation functions have been introduced in the literature. One of such families which will play an important role in this paper is the family of weighted arithmetic means.

Definition 2

([5, 7]). The weighted arithmetic mean is the n-ary function given by

$$M_w(\mathbf x )=w_1x_1+w_2x_2+\ldots +w_nx_n=\displaystyle \sum _{i=1}^n{w_ix_i}$$

where \(\mathbf w =(w_1,\ldots ,w_n)\) is the so-called weighting vector satisfying \(w_i\in [0,1]\) for all \(1\le i \le n\) and \(\displaystyle \sum _{i=1}^n{w_i}=1\).

Some additional properties of binary aggregation functions which will be used in this work are:

  • The idempotency,

  • The symmetry,

    for all \((x,y)\in [0,1]^2\).

  • The associativity,

  • It is said that \(a\in (0,1)\) is a zero divisor when

    for some \(y>0\).

  • It is said that \(a\in (0,1)\) is a one divisor when

    for some \(y<1\).

  • The left neutral element property with a fixed \(e\in [0,1]\),

  • The right neutral element property with a fixed \(e\in [0,1]\),

  • The neutral element property with a fixed \(e\in [0,1]\),

  • The left absorbing element property with a fixed \(a\in [0,1]\),

  • The right absorbing element property with a fixed \(a\in [0,1]\),

  • The absorbing element property with a fixed \(a\in [0,1]\),

Finally, we will call conjunctors to those binary aggregation functions with absorbing element 0 and disjunctors to those aggregation functions with absorbing element 1. Note that conjunctive (disjunctive) aggregation functions are trivially conjunctors (disjunctors) but not vice versa.

3 Polynomial Binary Aggregation Functions with a Non Trivial 0 Region

In this section, we want to deal with binary aggregation functions given by polynomial expressions only in some partial domains of the unit square. As we have already commented, the idea is that in many cases the aggregation function has a non trivial 0 region as for instance the Łukasiewicz or the nilpotent minimum t-norms. Thus, we want to focus first on aggregation functions with a non trivial 0 region limited by the classical negation \(N_c\) and given by a polynomial of degree one or two over \(N_c\). Note that this ensures in particular the existence of zero divisors. Formally, we will study functions of the form

$$\begin{aligned} f(x,y)= {\left\{ \begin{array}{ll}0 &{} \text {if } y\le 1-x,\\ P(x,y) &{} \text {if } y > 1-x,\end{array}\right. } \end{aligned}$$
(1)

where P(x, y) is a polynomial of degree 1 or 2.

Remark 1

Of course, any other region could be considered as the zero-region of the aggregation function. For instance, it could be delimited by any other polynomial negation of the form \(N(x)=1-x^n\). However, in this paper we will focus on the classical negation \(N_c\) which in particular will allow us to retrieve the Łukasiewicz t-norm as one of the searched aggregation functions.

First of all note that there exist a lot of aggregation functions of the form (1) as the following examples show.

Example 1

All the following cases are aggregation functions of the form (1):

  1. (i)

    It is clear that if we take as P(x, y) any binary polynomial aggregation function of degree one or two from those characterized in [22] we trivially obtain aggregation functions of the desired form. However, we will see along the paper that there are many other examples different from these ones, as for instance the next example.

  2. (ii)

    The Łukasiewicz t-norm \(T_\mathbf{L}\) is also an aggregation function of the form (1), just taking as P(x, y) the polynomial \(P(x,y)=x+y-1\).

  3. (iii)

    Note that, although we will deal only with polynomials of degree one or two, it can be deduced from the first item that there are examples with P(x, y) of any degree. Take for instance

    $$f(x,y)={\left\{ \begin{array}{ll}0 &{} \text {if } y\le 1-x,\\ \frac{x^n+y^n}{2} &{} \text {if } y > 1-x.\end{array}\right. } $$

Let us divide the following results in two different sections, one devoted to the case when P(x, y) is a polynomial of degree smaller than or equal to one, and the other devoted to the case when P(x, y) is a polynomial of degree two.

3.1 The Case of Degree 0 or 1

We investigate in this section which functions of the form (1) are in fact aggregation functions where P(x, y) is a polynomial of degree 0 or 1. That is, we will deal with functions of the form

$$\begin{aligned} f(x,y)= {\left\{ \begin{array}{ll}0 &{} \text {if } y\le 1-x,\\ ax + by +c &{} \text {if } y > 1-x,\end{array}\right. } \end{aligned}$$
(2)

where a, b, c are real numbers. In this case it is easy to characterize all possibilities as it is done in the next result.

Theorem 1

Let f be a binary function of the form (2). The following statements are equivalent:

  1. (i)

    f is a binary aggregation function.

  2. (ii)

    There exist \(a,b \in [0,1]\) such that \(f=f_{a,b}\) where \(f_{a,b}:[0,1]^2 \rightarrow [0,1]\) is given by

    $$\begin{aligned} f_{a,b}(x,y)= {\left\{ \begin{array}{ll}0 &{} \text {if } y\le 1-x,\\ a x+ b y + 1-a-b&{} \text {if } y > 1-x.\end{array}\right. } \end{aligned}$$
    (3)

The theorem above includes the following particular cases.

Example 2

  1. (i)

    The case \(a=b=0\) gives the only aggregation function of the form (2) with P(x, y) a polynomial of degree zero. That is, the function:

    $$f_{0,0}(x,y)= {\left\{ \begin{array}{ll}0 &{} \text {if } y\le 1-x,\\ 1 &{} \text {if } y > 1-x.\end{array}\right. }$$
  2. (ii)

    The first and second projections with a 0 region limited by \(N_c\) are also obtained in the cases \(a=1, b=0\), and \(a=0, b=1\), respectively.

  3. (iii)

    The Łukasiewicz t-norm is obtained taking \(a=b=1\).

  4. (iv)

    Taking \(b=1-a\) we obtain the family of weighted arithmetic means with a 0 region limited by \(N_c\), that is,

    $$\begin{aligned} f_{a,1-a}(x,y)= {\left\{ \begin{array}{ll}0 &{} \text {if } y\le 1-x,\\ ax+(1-a)y &{} \text {if } y > 1-x.\end{array}\right. } \end{aligned}$$
    (4)

Note that the last item in the previous example corresponds with those given in Example 1-(i) when P(x, y) is of degree 1 (see Theorem 11 in [22]). Consequently, even in the case of degree one, we obtain many other examples than the trivial ones given in Example 1-(i).

From Theorem 1, it is easy to characterize all binary aggregation functions of the form (3) fulfilling some additional properties. We collect some of them in the following proposition.

Proposition 1

Let \(f=f_{a,b}\) be a binary aggregation function of the form (3). The following statements are true:

  1. (i)

    \(f_{a,b}\) is continuous if and only if \(a=b=1\), that is, if and only if \(f_{a,b}=f_{1,1}\) is the Łukasiewicz t-norm.

  2. (ii)

    \(f_{a,b}\) satisfies (SYM) if and only if \(a=b\). That is, when \(f_{a,b}=f_{a,a}\) is given by

    $$f_{a,a}(x,y)={\left\{ \begin{array}{ll}0 &{} \text {if } y\le 1-x,\\ a(x+y)+1-2a &{} \text {if } y > 1-x.\end{array}\right. }$$
  3. (iii)

    \(f_{a,b}\) satisfies (ID) in its positive region, if and only if \(b=1-a\). That is, when \(f_{a,b}=f_{a,1-a}\) is given by Eq. (4).

  4. (iv)

    \(f_{a,b}\) satisfies (L-NE(e)) if and only if the neutral element is \(e=1\) and \(b=1\). That is, when \(f_{a,b}=f_{a,1}\) is given by

    $$f_{a,1}(x,y)={\left\{ \begin{array}{ll}0 &{} \text {if } y\le 1-x,\\ ax+y-a &{} \text {if } y > 1-x.\end{array}\right. }$$
  5. (v)

    \(f_{a,b}\) satisfies (R-NE(e)) if and only if and only if the neutral element is \(e=1\) and \(a=1\). That is, when \(f_{a,b}=f_{1,b}\) is given by

    $$f_{1,b}(x,y)={\left\{ \begin{array}{ll}0 &{} \text {if } y\le 1-x,\\ x+by-b &{} \text {if } y > 1-x.\end{array}\right. }$$
  6. (vi)

    \(f_{a,b}\) satisfies (NE(e)) if and only if the neutral element is \(e=1\) and \(f_{a,b}=f_{1,1}\) is the Łukasiewicz t-norm.

  7. (vii)

    \(f_{a,b}\) satisfies (ASS) if and only if \(f_{a,b}=f_{1,1}\) is the Łukasiewicz t-norm.

3.2 The Case of Degree 2

In this section, those functions of the form (1) where P(x, y) is a polynomial of degree two are studied. In particular, we are interested in characterizing which functions of the form

$$\begin{aligned} f(x,y)= {\left\{ \begin{array}{ll}0 &{} \text {if } y\le 1-x\\ ax^2 + by^2+cxy +dx+ey+f &{} \text {if } y > 1-x,\end{array}\right. } \end{aligned}$$
(5)

where a, b, c, d, e, f are real numbers such that \(a^2+b^2+c^2>0\), are aggregation functions. This case is more complex than the previous one and it leads to a huge number of possibilities for the values of a, b, c, d, e, f. Due to this great quantity of cases and the lack of space, we will present in the next example just some families of binary aggregation functions of the form (5) where P(x, y) is a polynomial of degree two.

Example 3

  1. (i)

    The family

    $$f(x,y)=\left\{ \begin{array}{ll} 0 &{}\hbox {if } y\le 1-x,\\ by^2+bxy+dx+ey+1-2b-d-e&{}\hbox {if }y>1-x,\end{array}\right. $$

    where \(-1<b<0\), \(-b\le d \le 1-d\) and \(-3b\le e\le 1-2b\) is a family of binary aggregation functions with a 0-region limited by \(N_c\) and given by a polynomial of degree 2 over \(N_c\).

  2. (ii)

    The family

    $$f(x,y)=\left\{ \begin{array}{ll} 0 &{}\hbox {if } y\le 1-x,\\ cxy+2\sqrt{-c}x-cy+1-2\sqrt{-c}&{}\hbox {if }y>1-x,\end{array}\right. $$

    where \(-4\le c\le -1\) is also a family of these aggregation functions which depends only on one parameter.

By requiring the fulfilment of some additional properties, we can reduce the number of possibilities for the values of a, b, c, d, e, f making feasible to present the complete families of aggregation functions of the form (5) satisfying these properties. Let us start studying the continuity and the existence of a neutral element independently.

Proposition 2

Let f be a binary function of the form (5). The following statements are equivalent:

  1. (i)

    f is a continuous aggregation function.

  2. (ii)

    f is given by \(f(x,y)=\)

    $$\left\{ \begin{array}{ll} 0&{}\hbox {if } y\le 1-x,\\ ax^2+by^2+(a+b)xy+(1-2a-b)x+(1-a-2b)y+a+b-1&{}\hbox {if } y>1-x, \end{array}\right. $$

    where \(-1\le a,b\le 1\) and \(a^2+b^2>0\).

Proposition 3

Let f be a binary function of the form (5). The following statements are equivalent:

  1. (i)

    f is an aggregation function satisfying (NE(e)).

  2. (ii)

    \(e=1\) and f is given by

    $$f(x,y)=\left\{ \begin{array}{ll} 0&{}\hbox {if } y\le 1-x,\\ cxy+(1-c)x+(1-c)y+c-1&{}\hbox {if } y>1-x, \end{array}\right. $$

    where \(0<c\le 1\).

Remark 2

Propositions 2 and 3 again clearly show that the study of these aggregation functions does not reduce to truncate by 0 the family of polynomial functions, i.e., there are more possibilities than the aggregation functions presented in Example 1-(i). Indeed, there are no continuous aggregation functions within the family presented in Example 1-(i). Moreover, since the unique polynomial aggregation function of degree 2 having 1 as neutral element is \(f(x,y)=xy\), the product truncated by 0 must be obtained in Proposition 3 (as it is, just take \(c=1\)). However, for any value \(0<c<1\), other aggregation functions are obtained.

Other additional properties can be also fully characterized but there still remain too many possibilities for the parameter values. Thus, let us consider pairs of additional properties in order to reduce the number of possible values. We will begin with the symmetry property jointly with other additional properties.

Proposition 4

Let f be a binary function of the form (5). The following statements are equivalent:

  1. (i)

    f is a continuous aggregation function satisfying (SYM).

  2. (ii)

    f is given by

    $$f(x,y)=\left\{ \begin{array}{ll} 0&{}\hbox {if } y\le 1-x,\\ a(x^2+y^2)+2axy+(1-3a)(x+y)+2a-1&{}\hbox {if } y>1-x, \end{array}\right. $$

    where \(a\in [-1,0[ \ \cup \ ]0,1]\).

From Proposition 3, the following result is straightforward.

Proposition 5

Let f be a binary function of the form (5). If f satisfies (NE(e)) then f satisfies also (SYM).

Let us study now the idempotency in the positive region jointly with other additional properties.

Proposition 6

Let f be an aggregation function of the form (5). If f is continuous or satisfies (NE(e)), then f is not idempotent in its positive region.

Contrarily, there exist solutions when we join (ID) in the positive region with (SYM).

Proposition 7

Let f be a binary function of the form (5). The following statements are equivalent:

  1. (i)

    f is an aggregation function satisfying (SYM) and (ID) in its positive region.

  2. (ii)

    f is given by

    $$ f(x,y)=\left\{ \begin{array}{ll} 0&{}\hbox {if } y\le 1-x,\\ a(x^2+y^2)-2axy+\frac{1}{2}(x+y)&{}\hbox {if } y>1-x, \end{array}\right. $$

    where \(a\in \left[ -\frac{1}{2},0\left[ \ \cup \ \right] 0,\frac{1}{2}\right] \).

From Corollary 17 in [22], it can be checked that the aggregation functions of the form (5) satisfying (SYM) and (ID) in its positive region are exactly those polynomial aggregation functions of degree 2 truncated by 0.

Besides the results presented in this section, more combinations of additional properties can be analysed but they are not included due to the lack of space.

4 Polynomial Binary Aggregation Functions with a Non Trivial 1-region

Analogously to the study carried out in Sect. 3 for polynomial binary aggregation functions with a non trivial 0-region, a similar study can be done for polynomial binary aggregation functions with a non trivial 1-region. This family of aggregation functions includes the well-known Łukasiewicz or the nilpotent maximum t-conorms. This construction ensures the existence of one divisors and taking again the classical negation \(N_c\) as limit of the 1-region, the expression of these operators can be explicitly given as

$$\begin{aligned} f(x,y)= {\left\{ \begin{array}{ll}P(x,y) &{} \text {if } y<1-x,\\ 1 &{} \text {if } y \ge 1-x,\end{array}\right. } \end{aligned}$$
(6)

where P(x, y) is a polynomial of degree 1 or 2.

This family of aggregation functions is huge and there exist a lot of aggregation functions of the form (6) as the following examples show.

Example 4

All the following cases are aggregation functions of the form (6):

  1. (i)

    A construction method of aggregation functions of the form (6) is straightforward from any binary polynomial aggregation function of degree one or two from those characterized in [22]. It is based on assigning a polynomial aggregation function as P(x, y) and impose the value 1 all over \(N_c\). This construction method does not provide the whole family of aggregation functions of the form (6) since for instance, it does not provide any continuous aggregation function.

  2. (ii)

    The Łukasiewicz t-conorm \(S_\mathbf{L}\) is an aggregation function of the form (6), just taking as P(x, y) the polynomial \(P(x,y)=x+y\). Note that this operator is continuous.

  3. (iii)

    The following family of aggregation functions of the form (6) proves the existence of operators of this kind for any degree n

    $$ f(x,y)={\left\{ \begin{array}{ll}1-\frac{(1-x)^n+(1-y)^n}{2} &{} \text {if } y<1-x,\\ 1 &{} \text {if } y \ge 1-x.\end{array}\right. } $$

As the reader can perceive from the previous examples is that all of them are the dual operators (with respect to \(N_c\)) of the corresponding ones in Sect. 4. Indeed, both forms are connected through the duality as the following result states.

Proposition 8

f(x, y) is a binary function of the form (1) if and only if \(1-f(1-x,1-y)\) is a binary function of the form (6).

This result allows us to rewrite easily all the results presented in Sect. 4 to results involving aggregation functions of the form (6). For the sake of clarity, we show for instance the characterization of all aggregation functions of the form (6) with degree less or equal to 1.

Theorem 2

Let f be a binary function of the form (6) with P(x, y) a polynomial of degree less or equal to 1. The following statements are equivalent:

  1. (i)

    f is a binary aggregation function.

  2. (ii)

    There exist \(a,b \in [0,1]\) such that \(f=f_{a,b}\) where \(f_{a,b}:[0,1]^2 \rightarrow [0,1]\) is given by

    $$\begin{aligned} f_{a,b}(x,y)= {\left\{ \begin{array}{ll}a x+ b y &{} \text {if } y<1-x,\\ 1&{} \text {if } y\ge 1-x.\end{array}\right. } \end{aligned}$$
    (7)

5 Conclusions and Future Work

This paper continues the line of research initiated in [22] about the construction of aggregation functions whose expressions are as simple as possible. Specifically, aggregation functions which have a 0 (or 1)-region delimited by \(N_c\) and polynomial function as expression in the other sub-domain have been deeply analysed. Indeed, we have characterized all binary aggregation functions with 0 (or 1)-region delimited by \(N_c\) and defined by a polynomial of degree less than or equal to 1 in the other sub-domain. Moreover, other additional desirable properties such as the idempotency in the non-constant region, symmetry, continuity or the existence of a neutral element among others, have been studied for the members of this family of aggregation functions. Furthermore, a similar study for polynomials of degree 2 has been also performed. In this case, due to the huge number of possibilities, some particular families have been presented as well as some additional properties, often in pairs, have been studied. Finally, using the duality, every result on this class of aggregation functions with the 0-region can be translated to the class of aggregation functions with a 1-region.

As future work, we want to complete the study of the additional properties when an underlying polynomial of degree 2 is considered. Also, some general results for polynomials of degree n could also be proved similarly to as it was done in [22]. Finally, another fuzzy negation to delimitate the constant region could be also considered.