Distributivity between extended nullnorms and uninorms on fuzzy truth values

doi:10.1016/j.ijar.2020.06.006

International Journal of Approximate Reasoning

Volume 125, October 2020, Pages 1-13

https://doi.org/10.1016/j.ijar.2020.06.006 Get rights and content

Abstract

This paper mainly investigates the distributive laws between extended nullnorms and uninorms on fuzzy truth values under the condition that the nullnorm is conditionally distributive over the uninorm. It presents the distributive laws between the extended nullnorm and t-conorm, and the left and right distributive laws between the extended generalized nullnorm and uninorm, where a generalized nullnorm is an operator from the class of aggregation operators with absorbing element that generalizes a nullnorm.

Introduction

The concept of a type-2 fuzzy set was introduced by Zadeh in 1975 [26] as an extension of type-1 fuzzy sets, and it has been heavily investigated both as a mathematical object and for use in applications [21], [26]. The algebra of truth values for fuzzy sets of type-2 consists of all mappings from the unit interval into itself and their operations which are convolutions of operations on the unit interval [21]. The algebra theory was studied extensively by Harding, C. and E. Walker [9], and C. and E. Walker [21], [22], [23]. Theory of aggregation of real numbers plays an important role in many different theoretical and practical fields, e.g., decision making theory, fuzzy set theory, integration theory, etc. Aggregation operators for real numbers are extended to ones for type-2 fuzzy sets. For example, Gera and Dombi [8] proposed computationally simple, pointwise formulas for extended t-norms and t-conorms on fuzzy truth values; Takáč [19] investigated extended aggregation operations on the algebra of convex normal fuzzy truth values with their left and right parts; Torres-Blanc, Cubillo, and Hernández [20] applied the Zadeh's extension principle to extend the aggregation operations of type-1 to the case of type-2 fuzzy sets. In particular, the distributive laws between those convolution operations on fuzzy truth values become an interesting and natural research area. The significance of these topics follows not only from the theoretical point of view in the logical algebraic structure for fuzzy sets of type-2, but also from their applications in efficient inferencing in approximate reasoning, especially in fuzzy control systems, so that they are discussed in many articles. For instance, Harding, C. and E. Walker [9] and C. and E. Walker [21], [23] discussed the distributive laws between extended minimums and maximums, and extended maximums and minimums, respectively, the distributive laws between extended t-norms and maximums, and the distributive laws between extended t-conorms and minimums. Hu and Kwong [11] also presented the distributive laws between extended t-norms and maximums, and the distributive laws between extended t-conorms and minimums. Xie [24] extended type-1 proper nullnorms and proper uninorms to fuzzy truth values and studied the distributive laws between the extended uninorms and minimums, and the distributive laws between the extended uninorms and maximums. Recently, Liu and Wang [15] discussed distributivity between extended t-norms and t-conorms on fuzzy truth values under the condition that the t-norm is conditionally distributive over the t-conorm or the t-conorm is conditionally distributive over the t-norm. It is well known that uninorms [25] and nullnorms [1] are aggregation operations with neutral elements and absorbing elements on $[0, 1]$ , respectively. They are generalizations of t-norms and t-conorms as well. However, the distributive laws between the extended nullnorms and uninorms on fuzzy truth values are not discussed till now, so that this paper will investigate these problems based on the results of conditionally distributivity of nullnorms over the uninorms in [5], [6], [13].

This paper is organized as follows. In Section 2 we recall some necessary definitions and previous results. In Section 3 we investigate the distributive laws between extended nullnorms and uninorms on fuzzy truth values under the condition that the nullnorm is conditionally distributive over the uninorm. In Section 4 we study distributivity of extended continuous operators with absorbing element and extended uninorms. A conclusion is given in Section 5.

Section snippets

Previous results

In this section, we recall some basic concepts and terminologies used throughout the paper.

Definition 2.1

[12]

A t-norm (resp. t-conorm) is a binary operation $T : {[0, 1]}^{2} \to [0, 1]$ (resp. $S : {[0, 1]}^{2} \to [0, 1]$ ) that is commutative, associative, non-decreasing in each variable, and has a neutral element 1 (resp. 0).

Definition 2.2

[12]

(i)
A t-norm T is said to be strict, if T is continuous and strictly monotone.
(ii)
A t-norm T is said to be nilpotent, if T is continuous and if each $x \in (0, 1)$ is a nilpotent element of T.

The basic continuous t-norms are

Distributive laws between the extended nullnorms and uninorms

In this section, the distributive laws between the extended nullnorms and uninorms on fuzzy truth values are discussed.

Theorem 3.1

Let $F : {[0, 1]}^{2} \to [0, 1]$ be a continuous non-decreasing operator. If $f \in F$ is convex, then the following statements hold for all $g, h \in F$ .

(i)
$f ⊙_{F} (g ⊓ h) = (f ⊙_{F} g) ⊓ (f ⊙_{F} h)$ ;
(ii)
$f ⊙_{F} (g ⊔ h) = (f ⊙_{F} g) ⊔ (f ⊙_{F} h)$ .

Proof

We only provide the proof of statement (i), the statement of (ii) being analogous.

According to formulas (2.1) and (2.3), for all $z \in [0, 1]$ , we have $(f ⊙_{F} (g ⊓ h)) (z) = \underset{F (y, x) = z}{⋁} (f (y) \land (g ⊓ h) (x)) = \underset{F (y, x) = z}{⋁} (f (y) \land (\underset{u \land v = x}{⋁} g$

Distributivity of extended continuous operators and uninorms

In order to investigate the distributive laws between the extended continuous operators and uninorms, we first need the following three theorems.

Theorem 4.1

[5]

A continuous operator $F \in Z_{k}$ and a continuous t-conorm S satisfy (CDl) if and only if exactly one of the following cases is fulfilled:

(i) $S = S_{M}$ ,

(ii) there is an $a \in [k, 1)$ such that $S, F$ are given by $S (x, y) = {\begin{matrix} a + (1 - a) S_{L} (\frac{x - a}{1 - a}, \frac{y - a}{1 - a}), & if (x, y) \in {[a, 1]}^{2}, \\ \max (x, y), & otherwise \end{matrix}$ and $F = {\begin{matrix} A, & on {[0, k]}^{2}, \\ B, & on {[k, 1]}^{2}, \\ k, & otherwise, \end{matrix}$ where $A : {[0, k]}^{2} \to [0, k]$ is a continuous increasing

Conclusions

The main contributions are the distributive laws between the extended nullnorm and uninorm on fuzzy truth values under the condition that the nullnorm is conditionally distributive over the uninorm, and the left and right distributive laws between the extended generalization nullnorms and uninorms. It is worth to be pointed out that the results in this paper are developed from the ones in [5], [13], and they generalize the corresponding results in [9], [21], [24].

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors thank the referees for their valuable comments and suggestions.

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^☆: This work is supported by National Natural Science Foundation of China (No. 11171242).

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Distributivity between extended nullnorms and uninorms on fuzzy truth values☆

Abstract

Introduction

Section snippets

Previous results

[12]

[12]

Distributive laws between the extended nullnorms and uninorms

Distributivity of extended continuous operators and uninorms

[5]

Conclusions

Declaration of Competing Interest

Acknowledgements

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Inf. Sci.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Inf. Sci.

Inf. Sci.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Inf. Sci.

Fuzzy Sets Syst.