key: cord-0044899-ttt746t1
authors: Santos, Helida; Dimuro, Graçaliz P.; Asmus, Tiago C.; Lucca, Giancarlo; Borges, Eduardo N.; Bedregal, Benjamin; Sanz, José A.; Fernández, Javier; Bustince, Humberto
title: General Grouping Functions
date: 2020-05-15
journal: Information Processing and Management of Uncertainty in Knowledge-Based Systems
DOI: 10.1007/978-3-030-50143-3_38
sha: 18f5e1a5dc7bb421bfc905f839b7fa94ce128571
doc_id: 44899
cord_uid: ttt746t1

Some aggregation functions that are not necessarily associative, namely overlap and grouping functions, have called the attention of many researchers in the recent past. This is probably due to the fact that they are a richer class of operators whenever one compares with other classes of aggregation functions, such as t-norms and t-conorms, respectively. In the present work we introduce a more general proposal for disjunctive n-ary aggregation functions entitled general grouping functions, in order to be used in problems that admit n dimensional inputs in a more flexible manner, allowing their application in different contexts. We present some new interesting results, like the characterization of that operator and also provide different construction methods.

Overlap functions are a kind of aggregation functions [3] that are not required to be associative, and they were introduced by Bustince et al. in [4] to measure the degree of overlapping between two classes or objects. Grouping functions are the dual notion of overlap functions. They were introduced by Bustince et al. [5] in order to express the measure of the amount of evidence in favor of either of two alternatives when performing pairwise comparisons [1] in decision making based on fuzzy preference relations [6] . They have also been used as the disjunction operator in some important problems, such as image thresholding [17] and the construction of a class of implication functions for the generation of fuzzy subsethood and entropy measures [13] .

Overlap and grouping functions have been largely studied since they are richer than t-norms and t-conorms [18] , respectively. Regarding, for instance, some properties like idempotency, homogeneity, and, mainly, the self-closeness feature with respect to the convex sum and the aggregation by generalized composition of overlap/grouping functions [7, 8, 10, 12] . For example, there is just one idempotent t-conorm (namely, the maximum t-conorm) and two homogeneous t-conorms (namely, the maximum and the probabilistic sum of t-conorms). On the contrary, there are uncountable numbers of idempotent, as well as homogenous, grouping functions [2, 11] . For comparisons among properties of grouping functions and t-conorms, see [2, 5, 17] However, grouping functions are bivariate functions. Since they may be non associative, they can only be applied in bi-dimensional problems (that is, when just two classes or objects are considered). In order to solve this drawback, Gómez et al. [16] introduced the concept of n-dimensional grouping functions, with an application to fuzzy community detection.

Recently, De Miguel et al. [20] introduced general overlap functions, by relaxing some boundary conditions, in order to apply to an n-ary problem, namely, fuzzy rule based classification systems, more specifically, in the determination of the matching degree in the fuzzy reasoning method. Then, inspired on the paper by De Miguel et al. [20] , the objective of this present paper is to introduce the concept of general grouping functions, providing their characterization and different construction methods. The aim is to define the theoretical basis of a tool that can be used to express the measure of the amount of evidence in favor of one of multiple alternatives when performing n-ary comparisons in multi-criteria decision making based on n-ary fuzzy heterogeneous, incomplete preference relations [14, 19, 26] , which we let for future work.

The paper is organized as follows. Section 2 presents some preliminary concepts. In Sect. 3, we define general grouping functions, studying some properties. In Sect. 4, we study the characterization of general grouping functions, providing some construction methods. Section 5 is the Conclusion.

In this section, we highlight some relevant concepts used in this work. 

For all properties and related concepts on overlap functions and grouping functions, see [2, 5, 7, 9, 10, 21, [23] [24] [25] . 

Following the ideas given in [20] , we can also generalize the idea of general grouping functions as follows. 

Note that (GG2) is the same of saying that 0 is an anhilator of the general grouping function GG. Proposition 1. If G : [0, 1] n → [0, 1] is an n-dimensional grouping function, then G is also a general grouping function.

Proof. Straighforward.

From this proposition, we can conclude that the concept of general grouping functions is a generalization of n-dimensional grouping functions, which on its turn is a generalization of the concepts of 0-grouping functions and 1-grouping functions. 

function, but it is not a bidimensional grouping function, since GG(0.5, 0.5) = 1.

4. Take any grouping function G, and a continuous t-conorm S. Then, the generalization of the previous item is the binary general grouping function given by: GG(x, y) = G(x, y)S(x, y) 5. Other examples are:

The generalization of the third item of Example 1 can be seen as follows.

Proposition 2. Take any grouping function G, and any t-conorm S. Then, the binary general grouping function given by:

.

be a commutative and continuous aggregation function. Then the following statements hold: x i = 0. Then, since F is an aggregation function, it

(ii) Suppose that F is a conjunctive aggregation function and it is either a general grouping function or an n-dimensional grouping function. Then, by either (GG3)

, which is a contradiction with the conjunctive property. Thus, one concludes that F is neither a general grouping function nor an n-dimensional grouping function.

We say that an element a ∈ [0, 1] is a neutral element of GG if for each 

Proof. (⇒) Suppose that (i) the neutral element of GG is a = 0 and (ii) GG(x 1 , . . . , x n ) = 0. Then, by (i), one has that, for each x 1 ∈ [0, 1], it holds that x 1 = GG(x 1 , 0 . . . , 0). By (ii) and since GG is increasing, it follows that

Similarly, one shows that x 2 , . . . , x n = 0, that is

Since a is the neutral element of GG, one has that GG(0, a, . . . , a) = 0, which means that a = 0, by (GG2 ).

Observe that the result stated by Proposition 4 does not mean that when a general grouping function has a neutral element, then it is necessarily equal to 0. In fact, for each a ∈ (0, 1), the function GG

is a general grouping function with a as neutral element. Proof. Since GG is idempotent and increasing in each argument, then one has that for all #"

Following a similar procedure described in [20] for general overlap functions on lattices, it is possible to characterize general grouping functions. In order to do that, first we introduce some properties and notations. Let us denote by G n the set of all general grouping functions. Define the ordering relation ≤ G n ∈ G n × G n , for all GG 1 , GG 2 ∈ G n by:

The supremum and infimum of two arbitrary general grouping functions GG 1 , GG 2 ∈ G n are, respectively, the general grouping functions GG 1 ∨GG 2 , GG 1 ∧ GG 2 ∈ G n , defined, for all #" x = (x 1 , . . . , x n ) ∈ [0, 1] n by:

The following result is immediate:

Remark 2. Note that the supremum of the lattice (G n , ≤ G n ) is given, for all

On the other hand, the infimum of (G n , ≤ G n ) is given, for all #" x = (x 1 , . . . , x n ) ∈ [0, 1] n , by:

However, neither GG sup nor GG inf are general grouping functions, since they are not continuous. Thus, in the lattice (G n , ≤ G n ) there is no bottom neither top elements. Then, similarly to general overlap functions, the lattice (G n , ≤ G n ) is not complete.

In this section we provide a characterization and some constructions methods for general grouping functions. 

for some f, h : [0, 1] n → [0, 1] the following properties hold, for all #" x ∈ [0, 1] n :

(i) f and h are commutative;

(ii) f is increasing and h is decreasing. Example 2. Observe that Theorem 2 provides a method for constructing general grouping functions. For example, take the maximum powered by p, defined by:

with p > 0. So, if we take the function T max p α : [0, 1] n → [0, 1], called αtruncated maximum powered by p, given, for all #" x ∈ [0, 1] n and α ∈ (0, 1), by:

then it is clear that T max p α is not continuous. However, one can consider the function CT max p α, : [0, 1] n → [0, 1], called the continuous truncated maximum powered by p, for all #" x ∈ [0, 1] n , α ∈ [0, 1] and ∈ (0, α], which is defined by:

Observe that taking f = CT max p α, , then f satisfies conditions (i)-(iii) and (v) in Theorem 2. Now, take h( #" x ) = min 1≤i≤n {1 − x i }, which satisfies conditions (i)-(ii) and (iv)-(v) required in Theorem 2. Thus, this assures that

is a general grouping function.

Observe that the maximum powered by p is an n-dimensional grouping function [15] and that CT max p α, is a general grouping function. However, CT max p α, is not an n-dimensional grouping function, for α − > 0, since CT max p α, (α − , . . . , α − ) = 0. . . . , x) .

Proof. (⇒) If GG is idempotent, then by Theorem 2 it holds that:

It follows that:

(⇐) It is immediate.

Example 3. Take the function αβ-truncated maximum powered by p, T max p αβ : [0, 1] n → [0, 1], for all #" x ∈ [0, 1] n ; α, β ∈ (0, 1) and α < β, defined by:

It is clear that T max p αβ is not continuous. However, we can define its continuous version, CT max p αβ, δ : [0, 1] n → [0, 1], for all #" x ∈ [0, 1] n ; α ∈ [0, 1); β, , δ ∈ (0, 1]; α + , β − δ ∈ (0, 1) and α + < β − δ, as follows:

Observe that CT max p αβ, δ satisfies conditions (GG1)-(GG5) from Definition 9, and then it is a general grouping function. But, whenever α = 0 or β = 1, then CT max p αβ, δ is not an n-dimensional grouping function, once CT max p αβ, δ (α − , . . . , α − ) = 0, for α − > 0, because max p (α − , . . . , α − ) = α − < α.

The following Theorem generalizes Example 3 providing a construction method for general grouping functions from truncated n-dimensional grouping functions.

Consider α ∈ [0, 1); β, , δ ∈ (0, 1]; α + , β − δ ∈ (0, 1) and α < β, α + < β − δ. Let G be an n-dimensional grouping function, whose αβ-truncated version is defined, for all #" x = (x 1 , . . . , x n ) ∈ [0, 1] n , by:

Then, the continuous version of T G αβ , for all #" x = (x 1 , . . . , x n ) ∈ [0, 1] n , is given by:

and it is a general grouping function. Besides that, whenever α = 0 and β = 1, then CT G αβ, δ is an n-dimensional grouping function.

The following proposition shows a construction method of general grouping functions that generalizes Example 1(4). x i = 0, then by (G2), it holds that G( #" x ) = 0, and, thus, GG GF ( #" x ) = G( #" x )F ( #" x ) = 0. For (GG3), whenever there exists i ∈ {1, . . . , n} such that x i = 1, then, by (G3), one has that G( #" x ) = 1, and, by the property of F , it holds that F ( #" x ) = 1. It follows that:

The following result is immediate. 

Note that G n is closed with respect to some aggregation functions, as stated by the following results, which provide construction methods of general grouping functions. 

Then, M GG 1 ,GG 2 ∈ G n if and only if M is a continuous aggregation function.

Proof. It follows that: (⇒) Suppose that M GG 1 ,GG 2 ∈ G n . Then it is immediate that M is continuous and increasing (A2). Now consider #" x = (x 1 , . . . , x n ) ∈ [0, 1] n and suppose that n i=1

x i = 0. Then, by (GG2), one has that: Proof. Since GG 2 , A 1 , . . . , A n are commutative, increasing and continuous functions, then GG 1 satisfies conditions (GG1), (GG4) and (GG5). So, it remains to prove:

Equivalently, one obtains A 2 ( #" x ), . . . , A n ( #" x ) = 0. Thus, since GG 2 is a general grouping function, one has that GG 1 ( #" x ) = GG 2 (A 1 ( #" x ) , . . . , A n ( #" x )) = GG 2 (0, . . . , 0) = 0. (GG3) Suppose that, for some #" x = (x 1 , . . . , x n ) ∈ [0, 1] n , there exists i ∈ {1, . . . , n} such that x i = 1. So, since A 1 is disjunctive then A 1 ( #" x ) ≥ max( #" x ) = 1, that is A 1 ( #" x ) = 1. Since GG 2 is a general grouping function, it follows that GG 1 ( #" x ) = GG 2 (A 1 ( #" x ) , . . . , A n ( #" x )) = GG 2 (1, A 2 ( #" x ), . . . , A n ( #" x )) = 1.

Next proposition uses the n-duality property. 

In this paper, we first introduced the concept of general grouping functions and studied some of their properties. Then we provided a characterization of general grouping functions and some construction methods. The theoretical developments presented here allow for a more flexible approach when dealing with decision making problems with multiple alternatives. Immediate future work is concerned with the development of an application in multi-criteria decision making based on n-ary fuzzy heterogeneous, incomplete preference relations.

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