Elsevier

Fuzzy Sets and Systems

Volume 291, 15 May 2016, Pages 82-97
Fuzzy Sets and Systems

Distributivity equation in the class of 2-uninorms

https://doi.org/10.1016/j.fss.2015.02.014Get rights and content

Abstract

This paper is mainly devoted to solving the functional equation of distributivity between aggregation operators with 2-neutral element. Our investigations are motivated by the couple of distributive logical connectives and their generalizations used in fuzzy set theory e.g., triangular norms, conorms, uninorms, nullnorms and implications. One of the recent generalizations covering both uninorms and nullnorms are 2-uninorms, which form a class of commutative, associative and increasing operators on the unit interval with an absorbing element separating two subintervals having their own neutral elements. In this work the distributivity of two binary operators from the class of 2-uninorms is considered. In particular, all possible solutions of the distributivity equation for the three defined subclasses of these operators depending on the position of its zero and neutral elements are characterized.

Introduction

The functional equations involving aggregation operators (e.g., [4], [5], [8], [26], [27]) play an important role in theories of fuzzy sets and fuzzy logic. A new direction of investigations is concerned with distributivity equation and inequalities for uninorms and nullnorms [3], [9], [10], [13], [14], [16], [17], [20], [27], [29], [30], [31], [32]. Uninorms, introduced by Yager and Rybalov [35], and studied by Fodor et al. [18], are special aggregation operators that have proven to be useful in many fields like fuzzy logic, expert systems, neural networks, utility theory and fuzzy system modeling (see [15], [19], [22], [24], [34]). Uninorms are interesting because their structure is a special combination of t-norms and t-conorms having a neutral element lying somewhere in the unit interval. The first notions of nullnorms and t-operators appeared respectively in [5], [25]. Both of these operators, which also generalize concepts of t-norms and t-conorms, where an absorbing (or zero) element is from the whole unit interval, have been studied further (e.g., [11], [28]). Moreover, in [5] and [27] it is proved that nullnorms and t-operators have the same structure so they are equivalent.

Our consideration was motivated by the logical connectives called 2-uninorms, which generalize both nullnorms and uninorms. Such generalization, further extended to the n-uninorms, was introduced by P. Akella in [2]. A 2-uninorm belongs to the class of increasing, associative and commutative binary operators on the unit interval with an absorbing element separating two subintervals having their own neutral elements.

In the case of nullnorms these neutral elements are respectively, 0 and 1 while for 2-uninorms they lie anywhere in the subintervals. Hence, in the structure of 2-uninorm we have two operators isomorphic with some uninorms, where in the case of nullnorms these operators are isomorphic with respectively, some t-conorm and t-norm. This makes the class of 2-uninorms significant generalization of both nullnorms and uninorms, which also includes certain other generalizations known from the literature e.g., S-uninorms, T-uninorms and Bi-uninorms [28].

This paper is organized as follows. In Section 2, we considered the algebraic structures of uninorms, nullnorms and 2-uninorms. We also reminded there the characterization of three subclasses of 2-uninorms. Then, the functional equation of distributivity is recalled (Section 3). In Sections 4 and 5, most of the possible solutions of distributivity equation for described subclasses of 2-uninorms are characterized. In particular, we investigate the distributivity between the couple of 2-uninorms as well as the distributivity between 2-uninorm and continuous triangular norm/conorm. It turns out that the full characterization depends on a number of cases what makes this problem interesting. In the last Section 6 we summarized the considerations and explained some of the encountered difficulties.

Section snippets

The class Uk(e,f) of 2-uninorms

We start with basic definitions and facts.

Definition 2.1

(See [35].) Let e[0,1]. An operation U:[0,1]2[0,1] is called a uninorm if it is associative, commutative, increasing with respect to both variables and fulfillingx[0,1]U(x,e)=x.(neutral elemente[0,1])

By Ue we denote the family of all uninorms with neutral element e[0,1].

Now we recall the general structure of a uninorm (for more details see [7], [12], [18]). We use the following notation De=[0,e)×(e,1](e,1]×[0,e) for e[0,1].

Theorem 2.2

(Cf. [18].) Let e[0,1]

Functional equation of distributivity

Now, we consider the distributivity equation of two binary operations. Let us remind some of the most important facts relating to this topic.

Definition 3.1

(Cf. [1], p. 318.) Let F,G:[0,1]2[0,1]. We say that operation F is distributive over G, ifF(x,G(y,z))=G(F(x,y),F(x,z))for allx,y,z[0,1].

Lemma 3.2

(See [29].) Let YX, F:X2X, G:Y2Y. If operation F with neutral element eY is distributive over operation G fulfilling G(e,e)=e, then G is idempotent in Y.

Lemma 3.3

(See [29].) Every increasing operation F:[0,1]2[0,1] is

Distributivity between operators from the class of 2-uninorms Uk(e,f) and continuous t-norms and t-conorms

At first our consideration will concern the distributivity of 2-uninorm F from all defined subclasses {Ck,Ck0,C10,C01,Ck1} over G{T,S}, where T, S denote the set of all triangular norms and triangular conorms, respectively.

Theorem 4.1

Let 0ekf1. A 2-uninorm FUk(e,f) is distributive over a continuous triangular norm G if and only if G=min.

Proof

Let FUk(e,f) be distributive over a continuous triangular norm G. To prove the necessary condition at first we show that G(e,e)=e and G(f,f)=f.

  • Since G is t-norm we

Distributivity of FUk1(e1,f1) over GUk2(e2,f2)

Now we consider the distributivity between operators FUk1(e1,f1) and GUk2(e2,f2) distinguishing both the order of their zero elements as well as their specific structures. We investigate the four following cases in which essential is the form of operator F and the proper order for the respective neutral elements that distributivity could occur.

Theorem 5.1

Let k1,k2[0,1] and k2k1. A 2-uninorm FC(e1,f1)k1 is distributive over a 2-uninorm GUk2(e2,f2) if and only if e1e2k2k1f2f1, G is idempotent,

Concluding remarks

Inspired by uninorms and nullnorms, we examined the functional equation of distributivity in the class of 2-uninorms. We gave the characterization of Eq. (7) including 3 subclasses of 2-uninorms, introduced in the paper of Akella from 2007, where he used U1,U2{Uemin,Uemax}. Surely, it is clear that it is possible to create much more other subclasses, in particular, using U1 and/or U2 as representable uninorms. Then the solutions of distributivity equation will be in another type. Nevertheless,

Acknowledgements

Authors wish to thank the editors and the anonymous referees whose valuable comments helped to improve final version of the paper.

This work was partially supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge No. RPPK.01.03.00-18-001/10.

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    It is well known that null-uninorms (uni-nullnorms) are very special 2-uninorms and the structures of 2-uninorms. However, other papers addressed this theme, demonstrating different approaches concerning the structural characterization of 2-uninorms like in Zong et al. [22], or in Drygaś and Rak where it is evaluated the solving of the functional equation of distributivity between aggregation operators [23]. Other papers in the literature highlighted the theme based on some different architectures of the 2-uninorms (see more in [24–27]).

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