Aggregation on lattices isomorphic to the lattice of closed subintervals of the real unit interval

doi:10.1016/j.fss.2022.02.013

Fuzzy Sets and Systems

Volume 441, 5 August 2022, Pages 262-278

https://doi.org/10.1016/j.fss.2022.02.013 Get rights and content

Abstract

In numerous generalizations of the original theory of fuzzy sets proposed by Zadeh, the considered membership degrees are often taken from lattices isomorphic to the lattice $L_{I}$ of closed subintervals of the unit interval $[0, 1]$ . This is, for example, the case of intuitionistic values, Pythagorean values or q-rung orthopair values. The mentioned isomorphisms allow to transfer the results obtained for the lattice $L_{I}$ directly to the other mentioned lattices. In particular, basic connectives in Zadeh's fuzzy set theory, i.e., special functions on the lattice $[0, 1]$ , can be extended to the interval-valued connectives, i.e., to special functions on the lattice $L_{I}$ , and then to the connectives on the lattices $L^{⁎}$ of intuitionistic values, P of Pythagorean values, and also on the lattice $L_{τ_{q}}$ of q-rung orthopair values. We give several examples of such connectives, in particular, of those which are related to strict t-norms. For all these connectives we show their link to an additive generator f of the considered strict t-norm T. Based on our approach, many results discussed in numerous papers can be treated in a unique framework, and the same is valid for possible newly proposed types of connectives based on strict t-norms. Due to this approach, a lot of tedious proofs of the properties of the proposed connectives could be avoided, which gives researchers more space for presented applications.

Introduction

Fuzzy sets characterized by membership functions with values from the real unit interval $[0, 1]$ were introduced in 1965 by Zadeh [40]. In this seminal paper, Zadeh introduced basic fuzzy connectives – the minimum and maximum for modeling, respectively, the intersection and union of fuzzy sets. There were also mentioned some other functions as, for example, the product, the arithmetic mean or the bounded sum (only in a partial form). Later, in the framework of fuzzy set theory, many other binary operations $F : {[0, 1]}^{2} \to [0, 1]$ were introduced and studied. We only recall triangular norms and conorms [22], fuzzy implications [6], and several types of averaging aggregation functions (also for higher arities), see, e.g., [7], [9], [17], [33]. Two years after Zadeh, Goguen proposed the concept of L-fuzzy sets [16], L being a bounded poset. This generalized view has opened the ways for using some particular lattices, mostly the lattices somehow related to the unit interval $[0, 1]$ . In 1974-1976, the lattice $L_{I} = {[a, b] | 0 \leq a \leq b \leq 1}$ of closed subintervals of $[0, 1]$ was independently introduced to fuzzy set theory by Grattan-Guinness [18], Jahn [20], and also by Zadeh [41]. For more details on the earlier developments in interval fuzzy set theory we refer to [15]. Later, in the framework of generalized fuzzy set theory, several other lattices (algebraically isomorphic to $L_{I}$ ) were studied and applied. As a most prominent example we recall intuitionistic fuzzy sets proposed in 1986 by Atanassov [3], related to the lattice $L^{⁎} = {(a, b) | a, b \in [0, 1], a + b \leq 1}$ , see also [5]. Among other lattices isomorphic to $L_{I}$ (equipped with the partial interval order defined by $[a, b] \leq_{L_{I}} [c, d]$ if and only if $a \leq c$ and $b \leq d$ ) we recall the Pythagorean membership grades lattice $P = {(a, b) \in {[0, 1]}^{2} | a^{2} + b^{2} \leq 1}$ , see [37], [38], and the lattice $L_{τ_{q}}$ of q-rung-orthopair membership grades, $L_{τ_{q}} = {(a, b) \in {[0, 1]}^{2} | a^{q} + b^{q} \leq 1}$ , $q \in] 0, \infty [$ , see [39], and also [2], [29], [31].

Fuzzy sets with membership grades from different lattices have mostly different semantics and thus their study is meaningful. On the other hand, due to possible algebraic relations expressed by some isomorphism, there is no need to reinvent “new” connectives separately in the framework of each bounded lattice L and related L-fuzzy sets, see [23]. For example, such link between interval fuzzy sets (based on the lattice $L_{I}$ ) and intuitionistic fuzzy sets (based on the lattice $L^{⁎}$ ) was observed and discussed by Deschrijver and Kerre in [11]. The aim of this paper is to extend their results to all above mentioned lattices $L_{I}$ , $L^{⁎}$ , P, and $L_{τ_{q}}$ with $q \in] 0, \infty [$ , and to help all interested researchers working with these lattices to avoid tedious proofs of some special results, in particular the results concerning connectives in the framework of L-fuzzy sets, when L is isomorphic to $L_{I}$ .

The rest of the paper is organized as follows. In the next section, several preliminary notions will be given. Section 3 is devoted to some connectives on $L_{I}$ , mostly generated by some additive generator $f : [0, 1] \to [0, \infty]$ , and in Section 4, we introduce the connectives on the lattice $L^{⁎}$ derived from the connectives discussed in Section 3. Section 5 is devoted to the connectives on general lattices isomorphic to $L_{I}$ , including the lattices P and $L_{τ_{q}}$ . In the end, several concluding remarks are added.

Section snippets

Preliminaries

We start with the basic bounded lattice $([0, 1], \leq)$ , ≤ being the standard total order of reals, considered by Zadeh in [40]. Obviously, its top element equals 1 and the bottom element is equal to 0. Any decreasing involutive function $N : [0, 1] \to [0, 1]$ is a continuous bijection, and it is called a strong negation. Due to Trillas [34], each strong negation N is related to an automorphism $φ : [0, 1] \to [0, 1]$ via $N (x) = φ^{- 1} (1 - φ (x))$ , i.e., N is isomophic to the strict negation $N_{Z}$ , $N_{Z} (x) = 1 - x$ , proposed by Zadeh [40]

Interval-valued connectives

In this section, we recall a method how to generalize the connectives discussed in Section 2 for the case of interval-valued connectives, i.e., if the lattice $L_{I}$ is considered, where $L_{I} = {[a, b] | 0 \leq a \leq b \leq 1} .$ We will present the so-called representable approach [14], though there are also other, more general approaches. The reason for making use of this approach is that almost all applications work with representable connectives.

Let us first recall that we consider the standard ordering of intervals

Connectives on the lattice $L^{⁎}$

In 1986, Atanassov [3] introduced intuitionistic fuzzy sets (AIFSs for short) with membership grades from the lattice $L^{⁎} = {(a, b) \in {[0, 1]}^{2} | a + b \leq 1}$ equipped with the partial order $\leq_{L^{⁎}}$ given by $(a, b) \leq_{L^{⁎}} (c, d)$ if and only if $a \leq c$ and $b \geq d$ . Clearly, $L^{⁎}$ is a bounded distributive lattice with the top element $1_{L^{⁎}} = (1, 0)$ and the bottom element $0_{L^{⁎}} = (0, 1)$ . An important information for us is that this lattice is isomorphic to the lattice $(L_{I}, \leq_{I})$ discussed in Section 3. The corresponding isomorphism $φ_{L^{⁎}} : L^{⁎} \to L_{I}$ is

Connectives on lattices related to $[0, 1]$ and isomorphic to $L_{I}$

Consider an automorphism $τ : [0, 1] \to [0, 1]$ and define a lattice $(L_{τ}, \leq_{L_{τ}})$ , where $L_{τ} = {(a, b) \in {[0, 1]}^{2} | τ (a) + τ (b) \leq 1},$ with the order $(a, b) \leq_{L_{τ}} (c, d)$ whenever $a \leq c$ and $b \geq d$ . Each of such lattices is isomorphic to the lattice $L^{⁎}$ , and the mapping $τ^{⁎} : L_{τ} \to L^{⁎}$ , given by $τ^{⁎} ((a, b)) = (τ (a), τ (b))$ , is one of possible isomorphisms between $L_{τ}$ and $L^{⁎}$ . Note that each lattice $L_{τ}$ is then isomorphic to the interval lattice $L_{I}$ , too.

Remark 5.1

An alternative view on $L_{τ}$ lattices follows from the fact that $τ (a) + τ (b) \leq 1$ if and only if $a \leq τ^{- 1} (1 - τ (b))$

Concluding remarks

Based on the standard fuzzy set theory connectives, such as negations, triangular norms and conorms, weighting functions, and weighted means, we have recalled introducing the corresponding interval-valued connectives on the lattice $L_{I}$ of closed subintervals of $[0, 1]$ using the so-called representable approach [14]. Most of these connectives are based on strict t-norms, i.e., they can be derived by means of a single decreasing bijective function $f : [0, 1] \to [0, \infty]$ , an additive generator of the

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.

Acknowledgement

R. Mesiar and A. Kolesárova kindly acknowledge the support of the project APVV-18-0052. A. Kolesárová was also partially supported by the grant VEGA 1/0267/21 and R. Mesiar by the grant Palacký University, Olomouc IGAPrF2021.

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Aggregation on lattices isomorphic to the lattice of closed subintervals of the real unit interval

Abstract

Introduction

Section snippets

Preliminaries

Interval-valued connectives

Connectives on the lattice L⁎

Connectives on lattices related to [0,1] and isomorphic to LI

Concluding remarks

Declaration of Competing Interest

Acknowledgement

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Inf. Sci.

Fuzzy Sets Syst.

J. Math. Anal. Appl.

Inf. Control

Inf. Sci.

Lectures on Functional Equations and Their Applications

Another view on q-rung orthopair fuzzy sets

Int. J. Intell. Syst.

Intuitionistic Fuzzy Sets: Theory and Applications

Fuzzy Implications

Aggregation Functions: A Guide for Practitioners

A Practical Guide to Averaging Functions

On the relationship between some extensions of fuzzy set theory

Fuzzy Sets Syst.

On the representation of intuitionistic fuzzy t-norms and t-conorms

IEEE Trans. Fuzzy Syst.

A generalization of operators on intuitionistic fuzzy sets using triangular norms and conorms

Notes IFS

Aggregation operators in interval-valued fuzzy and Atanassov's intuitionistic fuzzy set theory

Interval-valued fuzzy sets, possibility theory and imprecise probability

Aggregation Functions

Fuzzy membership mapped onto intervals and many-valued quantities

Z. Math. Log. Grundl. Math.

Connectives on the lattice $L^{⁎}$

Connectives on lattices related to $[0, 1]$ and isomorphic to $L_{I}$