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Fuzzy Sets and Systems 226 (2013) 99–114 www.elsevier.com/locate/fss

A construction method of semicopulas from fuzzy negations

I. Aguiló∗, J. Suñer, J. Torrens

Department of Mathematics and Computer Science, University of the Balearic Islands. Ctra. de Valldemossa, Km.7.5, 07122 Palma de Mallorca, Spain

Received 8 November 2012; received in revised form 23 January 2013; accepted 6 February 2013

Available online 13 February 2013

Abstract

In this paper a method of defining commutative semicopulas from fuzzy negations is introduced. Some properties are investigated that lead to understand these semicopulas as non-associative generalizations of the Łukasiewicz t-norm. In particular, it is proved that some well known examples of copulas and t-norms can be obtained by this method. Moreover, any commutative semicopula constructed by this method can be always obtained from a negation N which is symmetric with respect to the diagonal. Then, those symmetric fuzzy negations N for which the corresponding semicopula is a copula are characterized. Also, several examples of symmetric negations N are given such that the corresponding semicopula is a t-norm. © 2013 Elsevier B.V. All rights reserved.

Keywords: Fuzzy connectives and aggregation operators; Semicopula; Quasi-copula; Copula; t-Norm; Fuzzy negation; Id-symmetrical 1. Introduction

The process of merging all collected information into a concrete representative value is inherent to the human thinking and has currently become absolutely necessary because of the great quantity of data that is usually handled in many situations. From a mathematical point of view, this process of merging data is carried out by the so-called aggregation functions that have been extensively developed in last decades and many applications have been pointed out along this time. These applications include many subjects not only from mathematics and computer science but also from many applied fields like economics and social sciences. This great quantity of applications is one of the main reasons for the increasing interest in the topic of aggregation functions and this interest is endorsed by the publication of some monographs dedicated entirely to aggregation functions [3,4,14,27].

Aggregation functions have been also studied from the theoretical point of view and many authors have devoted their research to this topic. Usually, aggregation functions are divided into four classes: conjunctive, those that lie under the minimum, disjunctive, those that lie over the maximum, means or compensatory, those that lie in between, and mixed, those that are not in any of the above cases. It can be pointed out that these classes are not disjoint. In particular, uninorms [12] are an important example of mixed aggregations, although the idempotent ones [6,26] are in fact compensatory. Among conjunctive aggregation functions, there are some important classes like t-norms, useful ∗ Corresponding author. Tel.: +34 971172900; fax: +34 971173003.

E-mail addresses: isabel.aguilo@uib.es (I. Aguiló), jaume.sunyer@uib.es (J. Suñer), jts224@uib.es (J. Torrens). 0165-0114/$ - see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2013.02.003 100 I. Aguiló et al. / Fuzzy Sets and Systems 226 (2013) 99 –114 in many fields but specially in fuzzy set theory, copulas and quasi-copulas with applications mainly in statistics [25], and semicopulas [10]. From the theoretical point of view, one of the main questions has been the problem of constructing new aggregation functions (see the first chapter in [4] and references therein), specially in the case of copulas (see [7–9,11,13]).

One approach was given in [24] where a new class of binary aggregation functions is introduced from strong negations. 1 Based on this construction, in this paper we want to generalize it by constructing new binary aggregation functions from fuzzy negations in general, not necessarily strong nor even continuous. It will be proved that aggregation functions constructed by this method are always commutative semicopulas and it will be characterized when the constructed functions are in fact, copulas or quasi-copulas. Moreover, the general properties of the constructed semicopulas are quite similar to those of nilpotent t-norms and so they can be interpreted as a non-associative generalization of t-norms having a non-trivial zero-region. In fact, several examples are given that lead to a t-norm and so it is also investigated when the constructed semicopula is a t-norm. From these results it is derived that the construction method provided in this paper is an interesting way to obtain semicopulas in general, having additional properties in some special cases. 2. Preliminaries

In this section we give some basic results that will be used along the paper.

Definition 1. A function N : [0, 1] → [0, 1] is said to be a fuzzy negation if it is decreasing with N (0) = 1 and

N (1) = 0. A fuzzy negation N is said to be • Strict when it is strictly decreasing and continuous. • Strong when it is an involution, i.e., N (N (x)) = x for all x ∈ [0, 1].

It is well known that any strong negation is strict but not vice versa. Among the most used fuzzy negations we find the classical negation Nc(x) = 1 − x , and the weakest and the strongest fuzzy negations respectively given by

Nwt(x) = { 0 if x > 0, 1 if x = 0, Nst(x) = { 1 if x < 1, 0 if x = 1.

Clearly the first is strong whereas the other two are not continuous.

Definition 2. An n-ary aggregation function is a function F : [0, 1]n → [0, 1] which is increasing in each variable and such that F(0, . . . , 0) = 0 and F(1, . . . , 1) = 1. When n = 2 it is said that F is a binary aggregation function.

From now on, we deal with binary aggregation functions.

Definition 3. A binary aggregation function F is said to be a • Semicopula when F has neutral element 1, i.e., when F(x, 1) = F(1, x) = x for all x ∈ [0, 1]. • Quasi-copula when F is 1-Lipschitz, i.e., when