Some Errors on Hesitant Fuzzy Set Theory

1. Introduction

Fuzzy sets [1,2,3,4], interval-valued fuzzy sets [5,6], intuitionistic fuzzy sets [7], and hesitant fuzzy sets [8,9] are commonly used tools for dealing with uncertain information. Hesitant fuzzy sets with a membership degree represented by a discrete array are better suited for capturing the indecisiveness in real world compared to fuzzy sets, interval-valued fuzzy sets, and intuitionistic fuzzy sets. Hesitant fuzzy sets find extensive applications in various domains, including decision-making [10,11], attribute reduction [12], classification [13], linguistic perceptual [14], and forecasting [15], among others.

The hesitant fuzzy element is a discrete array, characterized by diversity and complexity. These attributes have led some scholars, in their research on hesitant fuzzy sets, to formulate imprecise or flawed propositions due to insufficient consideration. This article reviews some literatures on hesitant fuzzy sets, discusses some propositions of hesitant fuzzy topology spaces, hesitant fuzzy approximate spaces and hesitant fuzzy algebras, and provides counterexamples.

The construction of this paper is organized as follows. Section 2 introduces foundations of hesitant fuzzy sets. Section 3 discusses absorption law and distributive law of hesitant fuzzy sets. Section 4 discusses some propositions of hesitant fuzzy topology spaces. Section 5 discusses some propositions of hesitant fuzzy approximate spaces. Section 6 discusses some propositions of hesitant fuzzy algebras. Section 7 provides a conclusion.

2. Foundations of Hesitant Fuzzy Sets

This section reviews some foundations about hesitant fuzzy sets.

2.1. Reviews of Fuzzy Sets and Hesitant Fuzzy Sets

Definition 2.1.

[1] A fuzzy set F on U is a mapping $F:U\to [0,1]$.

Definition 2.2.

[1] $F_1$ and $F_2$ are two fuzzy sets on U, $F_1$ is a subset of $F_2$ if $F_1\left(x\right)⩽F_2\left(x\right)$ for all $x\in U$, denoted as $F_1\subset F_2$.

Definition 2.3.

[16] A hesitant fuzzy element is a non-empty, finite subset of $[0,1]$.

Definition 2.4.

[8] A hesitant fuzzy set on U is defined as a function that when applied to U returns a subset of $[0,1]$.

In the following, $HF\left(U\right)$ denotes the set of all hesitant fuzzy sets defined over U.

Definition 2.5.

[8] For each $x\in U$, and a hesitant fuzzy set H, the lower bound and upper bound of $H\left(x\right)$ are defined

lower bound $H^{-}\left(x\right)=infH\left(x\right)$,

upper bound $H^{+}\left(x\right)=supH\left(x\right)$.

Definition 2.6.

[8] Given two hesitant fuzzy sets represented by their membership functions $H_1$ and $H_2$, their union and intersection are defined

union $(H_1\cup H_2)\left(x\right)=\{h\in H_1\left(x\right)\cup H_2\left(x\right):h⩾sup(H_1^{-}\left(x\right),H_2^{-}\left(x\right))\}$,

intersection $(H_1\cap H_2)\left(x\right)=\{h\in H_1\left(x\right)\cup H_2\left(x\right):h⩽inf(H_1^{+}\left(x\right),H_2^{+}\left(x\right))\}$.

Example 2.7.

Let $U=\left\{x\right\}$, $H_1\left(x\right)=\{0.3,0.3,0.6\}$ and $H_2\left(x\right)=\{0.5,0.8\}$.

$H_1^{-}\left(x\right)=0.3$, $H_2^{-}\left(x\right)=0.5$, $max\{H_1^{-}\left(x\right),H_2^{-}\left(x\right)\}=0.5$. $(H_1\cup H_2)\left(x\right)=\{0.5,0.6,0.8\}$.

$H_1^{+}\left(x\right)=0.6$, $H_2^{+}\left(x\right)=0.8$, $min\{H_1^{+}\left(x\right),H_2^{+}\left(x\right)\}=0.6$. $(H_1\cap H_2)\left(x\right)=\{0.3,0.3,0.5,0.6\}$.

Definition 2.8.

[8] For $x\in U$ and $H\in HF\left(U\right)$, the complement of H is denoted as $H^c$, where

$H^c\left(x\right)={\cup }_{\gamma \in H\left(x\right)}\{1-\gamma \}$.

2.2. Inclusion Definitions of Hesitant Fuzzy Sets

For two hesitant fuzzy sets $H_1$ and $H_2$, the references [17,18,19,20,21] introduced that $H_1$ is a hesitant fuzzy subset of $H_2$ if $H_1\left(x\right)\subset H_2\left(x\right)$ for all $x\in U$. However, there is no clear description for $H_1\left(x\right)\subset H_2\left(x\right)$ in references [17,18,19,20,21]. For example, $H_1\left(x\right)=\{0.1,0.9\}$ and $H_2\left(x\right)=\{0.6,0.8\}$, how to determine $H_1\left(x\right)\subset H_2\left(x\right)$ or $H_2\left(x\right)\subset H_1\left(x\right)$ based on the values of $\{0.1,0.9\}$ and $\{0.6,0.8\}$?

The hesitant fuzzy element contains multiple values, a single inclusion relationship is not sufficient to fully describe the various relationships between hesitant fuzzy sets. There should be multiple inclusion relationships among various hesitant fuzzy sets. Lu and Xu et al [9] presented several kinds of inclusion relationships among various hesitant fuzzy sets and provided an example to describe the various relationships (shown in Example 2.9).

Example 2.9.

[9] Let $U=\{x_1,x_2,x_3,x_4,x_5,x_6\}$ be a set of decision-making schemes and H be an expert team that consists of three experts. $H\left(U\right)={\displaystyle \frac{\{0.9,0.2\}}{x_1}}+{\displaystyle \frac{\{0.6,0.6,0.5\}}{x_2}}+{\displaystyle \frac{\{0.7,0.5,0.5\}}{x_3}}+{\displaystyle \frac{\{0.8,0.6,0.5\}}{x_4}}+{\displaystyle \frac{\{0.9,0.3,0.1\}}{x_5}}+{\displaystyle \frac{\{0.9,0.8,0.7\}}{x_6}}$ are the estimated values of schemes provided by experts, in which the estimated values for $x_1$ are 0.9 and 0.2 that are obtained through the evaluations made by experts. One of the three experts fails to evaluate the scheme $x_1$.

(1) Here, $x_1$ has an estimated value of 0.9, which is greater than or equal to all the estimated values for $x_2$, it is possible that the scheme $x_1$ is better than the scheme $x_2$, denoted as $H\left(x_2\right){\subset }_{p}H\left(x_1\right)$.

(2) Here, $0.55=mean\left[H\left(x_1\right)\right]<mean\left[H\left(x_2\right)\right]=0.567$, where $mean[·]$ is the mean value operator. In comparing the mean values of estimated values, we find that the scheme $x_2$ is better than the scheme $x_1$, denoted as $H\left(x_1\right){\subset }_{m}H\left(x_2\right)$.

(3) The best estimated value of $x_3$ is 0.7, which is greater than or equal to the best estimated value of $x_2$. On the other hand, the worst estimated value of $x_3$ is 0.5, which is greater than or equal to the worst estimated value of $x_2$. To compare the respective best and worst cases of schemes $x_2$ and $x_3$, it is acceptable that the scheme $x_3$ is better than the scheme $x_2$, denoted as $H\left(x_2\right){\subset }_{a}H\left(x_3\right)$.

(4) To compare the estimated values for schemes $x_3$ and $x_4$ one by one ($0.7⩽0.8$; $0.5⩽0.6$; and $0.5⩽0.5$), it is strongly credible that the scheme $x_4$ is better than the scheme $x_3$, denoted as $H\left(x_3\right){\subset }_{s}H\left(x_4\right)$.

(5) We can obtain $H\left(x_1\right){\subset }_{s}H\left(x_5\right)$ after truncating the tail estimated value of $x_5$, i.e., deleting the estimated value 0.1 of $x_5$. This case is denoted as $H\left(x_1\right){\subset }_{st}H\left(x_5\right)$ and is recorded briefly as $H\left(x_1\right){\subset }_{t}H\left(x_5\right)$.

(6) The worst estimated value of $x_6$ is greater than or equal to the best estimated value of $x_3$. Thus, it is necessary that the scheme $x_6$ is better than the scheme $x_3$, which is denoted as $H\left(x_3\right){\subset }_{n}H\left(x_6\right)$.

Definition 2.10.

[9] Let $H_1$ and $H_2$ be two hesitant fuzzy sets on U. Several kinds of inclusion relationships of two hesitant fuzzy sets are defined as follows,

(1) If $H_1^{+}\left(x\right)⩽H_2^{+}\left(x\right)$, then $H_1\left(x\right){\subset }_p{H}_2\left(x\right)$. If $H_1\left(x\right){\subset }_p{H}_2\left(x\right)$ for all $x\in U$, then $H_1{\subset }_p{H}_2$. If $H_1{\subset }_p{H}_2$ and $H_2{\subset }_p{H}_1$, then $H_1{=}_p{H}_2$.

(2) If $H_1^{+}\left(x\right)⩽H_2^{+}\left(x\right)$ and $H_1^{-}\left(x\right)⩽H_2^{-}\left(x\right)$, then $H_1\left(x\right){\subset }_a{H}_2\left(x\right)$. If $H_1\left(x\right){\subset }_a{H}_2\left(x\right)$ for all $x\in U$, then $H_1{\subset }_a{H}_2$. If $H_1{\subset }_a{H}_2$ and $H_2{\subset }_a{H}_1$, then $H_1{=}_a{H}_2$.

(3) If $mean\left[H_1\left(x\right)\right]⩽mean\left[H_2\left(x\right)\right]$, then $H_1\left(x\right){\subset }_m{H}_2\left(x\right)$. If $H_1\left(x\right){\subset }_m{H}_2\left(x\right)$ for all $x\in U$, then $H_1{\subset }_m{H}_2$. If $H_1{\subset }_m{H}_2$ and $H_2{\subset }_m{H}_1$, then $H_1{=}_m{H}_2$.

(4) Let $H_1\left(x\right)=V=\{v_1,v_2,\dots ,v_k\}$ and $H_2\left(x\right)=W=\{w_1,w_2,\dots ,w_l\}$ be two descending sequences. If $k⩾l$ and $w_i⩾v_i$ for $1⩽i⩽l$, then $H_1\left(x\right){\subset }_s{H}_2\left(x\right)$. If $H_1\left(x\right){\subset }_s{H}_2\left(x\right)$ for all $x\in U$, then $H_1{\subset }_s{H}_2$. If $H_1{\subset }_s{H}_2$ and $H_2{\subset }_s{H}_1$, then $H_1{=}_s{H}_2$.

(5) Let $H_1\left(x\right)=V=\{v_1,v_2,\dots ,v_k\}$ and $H_2\left(x\right)=W=\{w_1,w_2,\dots ,w_l\}$ be two descending sequences. If $k<l$ and $w_i⩾v_i$ for $1⩽i⩽k$, then $H_1\left(x\right){\subset }_t{H}_2\left(x\right)$. If $H_1\left(x\right){\subset }_t{H}_2\left(x\right)$ for all $x\in U$, then $H_1{\subset }_t{H}_2$. It is obvious that $H_1{\subset }_t{H}_2$ and $H_2{\subset }_t{H}_1$ cannot hold simultaneously.

(6) If $H_1^{+}\left(x\right)⩽H_2^{-}\left(x\right)$, then $H_1\left(x\right){\subset }_n{H}_2\left(x\right)$. If $H_1\left(x\right){\subset }_n{H}_2\left(x\right)$ for all $x\in U$, then $H_1{\subset }_n{H}_2$. If $H_1{\subset }_n{H}_2$ and $H_2{\subset }_n{H}_1$, then $H_1{=}_n{H}_2$.

3. Absorption Law and Distributive Law of Hesitant Fuzzy Sets

The hesitant fuzzy sets do not satisfy the absorption law and distributive law. We have provided two counterexamples, shown in Example 3.1 and 3.2.

3.1. Absorption Law

The following example demonstrates that the hesitant fuzzy sets do not satisfy the absorption law.

Example 3.1.

Let $U=\left\{x\right\}$, $A={\displaystyle \frac{0.3,0.6}{x}}$ and $B={\displaystyle \frac{0.55,0.7}{x}}$.

$A\cup B={\displaystyle \frac{0.55,0.6,0.7}{x}}$, $(A\cup B)\cap A={\displaystyle \frac{0.3,0.55,0.6,0.6}{x}}$. $0.55\in \left(\right(A\cup B)\cap A)\left(x\right)$, however, $0.55\notin A\left(x\right)$. Hence, $(A\cup B)\cap A\ne A$.

$A\cap B={\displaystyle \frac{0.3,0.55,0.6}{x}}$, $(A\cap B)\cup A={\displaystyle \frac{0.3,0.3,0.55,0.6,0.6}{x}}$. $0.55\in \left(\right(A\cap B)\cup A)\left(x\right)$, however, $0.55\notin A\left(x\right)$. Hence, $(A\cap B)\cup A\ne A$.

3.2. Distributive Law

The Theorem 2.8 in [22] and the Theorem 1 in [23] show that hesitant fuzzy sets satisfy the distributive laws about intersection and union operations. We raise doubts about this proposition. A counterexample is shown as below.

Example 3.2.

Let $U=\left\{x\right\}$, $A={\displaystyle \frac{\{0.1,0.35\}}{x}}$, $B={\displaystyle \frac{\{0.55,0.7\}}{x}}$, $C={\displaystyle \frac{\{0.3,0.6\}}{x}}$.

$\left(\right(A\cap B)\cup C)\left(x\right)=\{0.3,0.35,0.6\}$, $\left(\right(A\cup C)\cap (B\cup C\left)\right)\left(x\right)=\{0.3,0.35,0.55,0.6,0.6\}$.

$0.55\notin \left(\right(A\cap B)\cup C)\left(x\right)$, however, $0.55\in \left(\right(A\cup C)\cap (B\cup C\left)\right)\left(x\right)$, then $(A\cap B)\cup C\ne (A\cup C)\cap (B\cup C)$.

$\left(\right(A\cup B)\cap C)\left(x\right)=\{0.3,0.55,0.6\}$, $\left(\right(A\cap C)\cup (B\cap C\left)\right)\left(x\right)=\{0.3,0.3,0.35,0.55,0.6\}$.

$0.35\notin \left(\right(A\cup B)\cap C)\left(x\right)$, however, $0.35\in \left(\right(A\cap C)\cup (B\cap C\left)\right)\left(x\right)$, then $(A\cup B)\cap C\ne (A\cap C)\cup (B\cap C)$.

4. Hesitant Fuzzy Topology Spaces

Topology [24] is an important foundation of mathematics and computer science, it is widely applied in various fields, such as precision feeding system [25], object detection and localization [26], structural analysis [27].

This section presents our viewpoints on the possible errors that may exist in the literatures on hesitant fuzzy topological spaces. If the researchers need to cite the propositions mentioned in this section in their subsequent work, they should verify them again.

4.1. Empty Hesitant Fuzzy Set and Full Hesitant Fuzzy Set

In current literatures of hesitant fuzzy sets, there are two different definitions for empty hesitant fuzzy set $H^0$ and full hesitant fuzzy set $H^1$ in hesitant fuzzy topology spaces, (i) $H^0\left(x\right)=\left\{0\right\}$ and $H^1\left(x\right)=\left\{1\right\}$ for all $x\in U$[8,21], (ii) $H^0\left(x\right)=\varnothing$ and $H^1\left(x\right)=[0,1]$ for all $x\in U$ [18,19,20,28]. The references [18,19,20,28] cited the reference [8] or [21] to obtain the definitions of empty hesitant fuzzy set and full hesitant fuzzy set. However, the references [18,19,20,28] generated a new different type of definitions to compare with the cited references [8,21].

First, let’s review the construction of classical sets [29] and fuzzy sets [1,30,31,32,33]. Let $U=\{x_1,x_2,x_3\}$ be the universe of discourse, two classical sets $A=\left\{x_1\right\}$ and $B=\{x_2,x_3\}$ can be described as the form of fuzzy sets, $A=\{{\displaystyle \frac{1}{x_1}},{\displaystyle \frac{0}{x_2}},{\displaystyle \frac{0}{x_3}}\}$ and $B=\{{\displaystyle \frac{0}{x_1}},{\displaystyle \frac{1}{x_2}},{\displaystyle \frac{1}{x_3}}\}$. $A=\{{\displaystyle \frac{1}{x_1}},{\displaystyle \frac{0}{x_2}},{\displaystyle \frac{0}{x_3}}\}$ indicates that the probability of $x_1$ belonging to set A is 100%, while the probabilities of $x_2$ and $x_3$ belonging to set A are both 0%. The range of probability values, which was originally $\{0,1\}$, is extended to $[0,1]$, resulting in a fuzzy set, such as $F=\{{\displaystyle \frac{0.3}{x_1}},{\displaystyle \frac{0.8}{x_2}},{\displaystyle \frac{0.5}{x_3}}\}$. To compared with fuzzy sets, hesitant fuzzy sets have an additional element of hesitation, such as $H=\{{\displaystyle \frac{0.3,0.5}{x_1}},{\displaystyle \frac{0.8,0.9}{x_2}},{\displaystyle \frac{0.5,0.6,0.8}{x_3}}\}$. When the values in $H\left(x\right)$ are the same values, the hesitant fuzzy sets degenerate into fuzzy sets, such as $H=\{{\displaystyle \frac{0,0}{x_1}},{\displaystyle \frac{1,1,1}{x_2}},{\displaystyle \frac{0.8,0.8}{x_3}}\}=\{{\displaystyle \frac{0}{x_1}},{\displaystyle \frac{1}{x_2}},{\displaystyle \frac{0.8}{x_3}}\}$.

$H^0\left(x\right)=\varnothing$ indicates that the probability domain for x belonging to $H^0$ is ∅, i.e., it is meaningless to discuss whether x belongs to $H^0$. $H^1\left(x\right)=[0,1]$ In a classical topological space, all elements of the domain belong to the full set. $H^1\left(x\right)=\left\{1\right\}$ for all $x\in U$ means that the probability that x belongs to $H^1$ is 100% for all $x\in U$. However, $H^1\left(x\right)=[0,1]$ means that the probability that x belongs to $H^1$ is a random number in $[0,1]$. The definitions in [8,21] are more reasonable than the other.

4.2. The Intersection and Union Operations of the Elements in Hesitant Fuzzy Topology

For a hesitant fuzzy topology $\tau$, (i) if $H,G\in \tau$, then $H\cap G\in \tau$; (ii) if $H_{\alpha }\in \tau$ for $\alpha \in \Gamma$, then ${\cup }_{\alpha \in \Gamma }{H}_{\alpha }\in \tau$. However, it is very difficult to obtain a hesitant fuzzy topology to satisfy the conditions (i) and (ii).

For instance, the Example 1 in reference [18], $H_1\left(b\right)=\{0.3,0.6,0.9\}$, $H_2\left(b\right)=\{0.3,0.6,0.8\}$, $(H_1\cap H_2)\left(b\right)=\{0.3,0.3,0.6,0.6,0.8\}$. $(H_1\cap H_2)\left(b\right)\ne H^0\left(b\right)$, $(H_1\cap H_2)\left(b\right)\ne H_1\left(b\right)$, $(H_1\cap H_2)\left(b\right)\ne H_2\left(b\right)$, $(H_1\cap H_2)\left(b\right)\ne H_3\left(b\right)=\{0.3,0.6\}$, $(H_1\cap H_2)\left(b\right)\ne H_4\left(b\right)=\{0.3,0.6,0.8,0.9\}$, $(H_1\cap H_2)\left(b\right)\ne H^1\left(b\right)$, i.e., $H_1\cap H_2\notin \tau$, where $\tau =\{H^0,H_1,H_2,H_3,H_4,H^1\}$ [18].

The conditions (i) and (ii) above can be weakened to obtain hesitant fuzzy m-topology and a-topology as follows.

Definition 4.1.

Let U be a nonempty set, and let $\tau \subset HF\left(U\right)$, $\tau$ is called a hesitant fuzzy m-topology on U if it satisfies the following axioms:

(1) $H^0,H^1\in \tau$.

(2) For $H,G\in \tau$, there is $H^{\prime }\in \tau$ such that $H\cap G{=}_m{H}^{\prime }$.

(3) For $\alpha \in \Gamma$ and $H_{\alpha }\in \tau$, there is $H^{\prime \prime }\in \tau$ such that ${\cup }_{\alpha \in \Gamma }{H}_{\alpha }{=}_m{H}^{\prime \prime }$.

Definition 4.2.

Let U be a nonempty set, and let $\tau \subset HF\left(U\right)$, $\tau$ is called a hesitant fuzzy a-topology on U if it satisfies the following axioms:

(1) $H^0,H^1\in \tau$.

(2) For $H,G\in \tau$, there is $H^{\prime }\in \tau$ such that $H\cap G{=}_a{H}^{\prime }$.

(3) For $\alpha \in \Gamma$ and $H_{\alpha }\in \tau$, there is $H^{\prime \prime }\in \tau$ such that ${\cup }_{\alpha \in \Gamma }{H}_{\alpha }{=}_a{H}^{\prime \prime }$.

Example 4.3.

Let $U=\{x,y\}$, $H^0=\{{\displaystyle \frac{0}{x}},{\displaystyle \frac{0}{y}}\}$, $H_1=\{{\displaystyle \frac{0.1,0.2}{x}},{\displaystyle \frac{0.6,0.7}{y}}\}$, $H_2=\{{\displaystyle \frac{0.1,0.15,0.2}{x}},{\displaystyle \frac{0.6,0.65,0.7}{y}}\}$, $H_3=\{{\displaystyle \frac{0.3,0.4}{x}},{\displaystyle \frac{0.8,0.9}{y}}\}$, $H_4=\{{\displaystyle \frac{0.1,0.2,0.3,0.4}{x}},{\displaystyle \frac{0.6,0.7,0.8,0.9}{y}}\}$, and $H^1=\{{\displaystyle \frac{1}{x}},{\displaystyle \frac{1}{y}}\}$.

(1) Let $\tau =\{H^0,H_1,H_2,H^1\}$. $H_1\cap H_2=\{{\displaystyle \frac{0.1,0.1,0.15,0.2,0.2}{x}},{\displaystyle \frac{0.6,0.6,0.65,0.7,0.7}{y}}\}=H_1\cup H_2$, $H_1\cap H_2{=}_m{H}_1$ and $H_1\cup H_2{=}_m{H}_1$. Obvious, $H_1\cap H_2\notin \tau$. Then, $\tau$ is a hesitant fuzzy m-topology not a hesitant fuzzy topology.

(2) Let $\tau =\{H^0,H_1,H_3,H_4,H^1\}$. $H_1\cap H_4{=}_a{H}_1$, $H_1\cup H_4{=}_a{H}_4$, $H_3\cap H_4{=}_a{H}_4$, $H_3\cup H_4{=}_a{H}_3$. Obvious, $H_3\cup H_4\notin \tau$. Then, $\tau$ is a hesitant fuzzy a-topology not a hesitant fuzzy topology.

(3) Let $\tau =\{H^0,H_1,H_3,H^1\}$. $\tau$ is a hesitant fuzzy topology, hesitant fuzzy m-topology and hesitant fuzzy a-topology.

4.3. Inferences Based on Inclusion Relationships

The proof of the mathematical theory is logically rigorous and clear. The logical relationship of the definitions used in the proof process is also clear. Unclear or inexact logical relationships can lead to incorrect proofs and the construction of erroneous theorems that are then cited by later generations.

For example, the Definition 2 in [18] is not clear enough, “$h_1\subset h_2$ if $h_1\left(x\right)\subset h_2\left(x\right)$ for each $x\in X$”. It do not express the condition of $h_1\left(x\right)\subset h_2\left(x\right)$. $h_1\left(x\right)$ and $h_2\left(x\right)$ do not necessarily have to be two real numbers; they could be two discrete arrays, then the condition of $h_1\left(x\right)\subset h_2\left(x\right)$ needs a clear expression. The logical relationship of Definition 2 in [18] in not clear enough. However, the proof of Theorem 1 in [18] used the inclusion relationships “$B\subset B_1\tilde{\cap }{B}_2$”.

The similar situations where the definition is unclear also occurs in Definition 2.2 of reference [19], Definition 2 of reference [20], Definition 2.14 of reference [21], etc.

5. Hesitant Fuzzy Approximate Spaces

This section presents our viewpoints on the possible errors that may exist in the literatures on hesitant fuzzy approximate spaces. If the researchers need to cite the propositions mentioned in this section in their subsequent work, they should verify them again.

5.1. Discussions on $\underline{R}(A\cap B)$ and $\underline{R}\left(A\right)\cap \underline{R}\left(B\right)$

Are $\underline{R}(A\cap B)$ and $\underline{R}\left(A\right)\cap \underline{R}\left(B\right)$ equal? The references [23,34,35] give this proposition $\underline{R}(A\cap B)=\underline{R}\left(A\right)\cap \underline{R}\left(B\right)$. This review challenges this proposition and provides a counterexample as follows. Subsequent researchers should verify this proposition again before citing it.

Example 5.1.

Let $U=\{x_1,x_2\}$, $A={\displaystyle \frac{\{0.3,0.6\}}{x_1}}+{\displaystyle \frac{\{0.3,0.5\}}{x_2}}$ and $B={\displaystyle \frac{\{0.4,0.7\}}{x_1}}+{\displaystyle \frac{\{0.55,0.6\}}{x_2}}$. $R(U,U)$ is shown as follows,

$R(U,U)=\left[\begin{array}{cc}\left\{1\right\}& \{0.3,0.4,0.7\}\\ \{0.3,0.4,0.7\}& \left\{1\right\}\end{array}\right].$

$A\cap B={\displaystyle \frac{\{0.3,0.4,0.6\}}{x_1}}+{\displaystyle \frac{\{0.3,0.5\}}{x_2}}$.

$\underline{R}(A\cap B)\left(x_1\right)=\bigcap _{y\in U}[R^c(x_1,y)\cup (A\cap B)\left(y\right)]=[\left\{0\right\}\cup \{0.3,0.4,0.6\}]\cap [\{0.3,0.6,0.7\}\cup \{0.3,0.5\}]=\{0.3,0.4,0.6\}\cap \{0.3,0.3,0.5,0.6,0.7\}=\{0.3,0.3,0.3,0.4,0.5,0.6,0.6\}$.

$\underline{R}\left(A\right)\left(x_1\right)=\bigcap _{y\in U}[R^c(x_1,y)\cup A\left(y\right)]=[\left\{0\right\}\cup \{0.3,0.6\}]\cap [\{0.3,0.6,0.7\}\cup \{0.3,0.5\}]=\{0.3,0.6\}\cap \{0.3,0.3,0.5,0.6,0.7\}=\{0.3,0.3,0.3,0.5,0.6,0.6\}$.

$\underline{R}\left(B\right)\left(x_1\right)=\bigcap _{y\in U}[R^c(x_1,y)\cup B\left(y\right)]=[\left\{0\right\}\cup \{0.4,0.7\}]\cap [\{0.3,0.6,0.7\}\cup \{0.55,0.6\}]=\{0.4,0.7\}\cap \{0.55,0.6,0.6,0.7\}=\{0.4,0.55,0.6,0.6,0.7,0.7\}$.

$(\underline{R}\left(A\right)\cap \underline{R}\left(B\right))\left(x_1\right)=\{0.3,0.3,0.3,0.5,0.6,0.6\}\cap \{0.4,0.55,0.6,0.6,0.7,0.7\}=\{0.3,0.3,0.3,0.4,0.5,0.55,0.6,0.6,0.6,0.6\}$.

$0.55\in (\underline{R}\left(A\right)\cap \underline{R}\left(B\right))\left(x_1\right)$ and $0.55\notin \underline{R}(A\cap B)\left(x_1\right)$, then $(\underline{R}\left(A\right)\cap \underline{R}\left(B\right))\left(x_1\right)\ne \underline{R}(A\cap B)\left(x_1\right)$. Hence, $\underline{R}\left(A\right)\cap \underline{R}\left(B\right)\ne \underline{R}(A\cap B)$.

5.2. Discussions on $\overline{R}(A\cup B)$ and $\overline{R}\left(A\right)\cup \overline{R}\left(B\right)$

Are $\overline{R}(A\cup B)$ and $\overline{R}\left(A\right)\cup \overline{R}\left(B\right)$ equal? The references [23,34,35] give this proposition $\overline{R}(A\cup B)=\overline{R}\left(A\right)\cup \overline{R}\left(B\right)$. This review challenges this proposition and provides a counterexample as follows. Subsequent researchers should verify this proposition again before citing it.

Example 5.2.

Let $U=\{x_1,x_2\}$, $A={\displaystyle \frac{\{0.3,0.6\}}{x_1}}+{\displaystyle \frac{\{0.3,0.45\}}{x_2}}$ and $B={\displaystyle \frac{\{0.4,0.7\}}{x_1}}+{\displaystyle \frac{\{0.5,0.6\}}{x_2}}$. $R(U,U)$ is shown as follows,

$R(U,U)=\left[\begin{array}{cc}\left\{1\right\}& \{0.3,0.4,0.7\}\\ \{0.3,0.4,0.7\}& \left\{1\right\}\end{array}\right].$

$A\cup B={\displaystyle \frac{\{0.4,0.6,0.7\}}{x_1}}+{\displaystyle \frac{\{0.5,0.6\}}{x_2}}$.

$\overline{R}(A\cup B)\left(x_1\right)=\bigcup _{y\in U}[R(x_1,y)\cap (A\cup B)\left(y\right)]=[\left\{1\right\}\cap \{0.4,0.6,0.7\}]\cup [\{0.3,0.4,0.7\}\cap \{0.5,0.6\}]=\{0.4,0.6,0.7\}\cup \{0.3,0.4,0.5,0.6\}=\{0.4,0.4,0.5,0.6,0.6,0.7\}$.

$\overline{R}\left(A\right)\left(x_1\right)=\bigcup _{y\in U}[R(x_1,y)\cap A\left(y\right)]=[\left\{1\right\}\cap \{0.3,0.6\}]\cup [\{0.3,0.4,0.7\}\cap \{0.3,0.45\}]=\{0.3,0.6\}\cup \{0.3,0.3,0.4,0.45\}=\{0.3,0.3,0.3,0.4,0.45,0.6\}$.

$\overline{R}\left(B\right)\left(x_1\right)=\bigcup _{y\in U}[R(x_1,y)\cap B\left(y\right)]=[\left\{1\right\}\cap \{0.4,0.7\}]\cup [\{0.3,0.4,0.7\}\cap \{0.5,0.6\}]=\{0.4,0.7\}\cup \{0.3,0.4,0.5,0.6\}=\{0.4,0.4,0.5,0.6,0.7\}$.

$(\overline{R}\left(A\right)\cup \overline{R}\left(B\right))\left(x_1\right)=\{0.3,0.3,0.3,0.4,0.45,0.6\}\cup \{0.4,0.4,0.5,0.6,0.7\}=\{0.4,0.4,0.4,0.45,0.5,0.6,0.6,0.7\}$.

$0.45\in (\overline{R}\left(A\right)\cup \overline{R}\left(B\right))\left(x_1\right)$ and $0.45\notin \overline{R}(A\cup B)\left(x_1\right)$, then $(\overline{R}\left(A\right)\cup \overline{R}\left(B\right))\left(x_1\right)\ne \overline{R}(A\cup B)\left(x_1\right)$. Hence, $\overline{R}\left(A\right)\cup \overline{R}\left(B\right)\ne \overline{R}(A\cup B)$.

6. Hesitant Fuzzy Algebras

This section presents our viewpoints on the possible errors that may exist in the literatures on hesitant fuzzy algebras. If the researchers need to cite the propositions mentioned in this section in their subsequent work, they should verify them again.

6.1. Inferences Based on Inclusion Relationships

The proof of the mathematical theory is logically rigorous and clear. The logical relationship of the definitions used in the proof process is also clear. The references [36,37,38,39,40,41,42,43,44,45,46,47] on hesitant fuzzy algebras have used the inclusion relationships of hesitant fuzzy sets, but there is no clear definition of the inclusion relationships of hesitant fuzzy sets.

6.2. Inferences Based on Union Operation of Hesitant Fuzzy Sets

The formula (3.3) in [36] describes that $G\left(a\right)\subset G\left(x\right)$ and $G\left(b\right)\subset G\left(y\right)$ then $G\left(a\right)\cup G\left(b\right)\neg \subset G\left(x\right)\cup G\left(y\right)$. This proposition holds for ${\subset }_a$, but does not hold for ${\subset }_m$. For example, let $G\left(a\right)=\{0.1,0.9\}$, $G\left(x\right)=\{0.5,0.6\}$, $G\left(b\right)=\{0.5,0.6\}$ and $G\left(y\right)=\{0.5,0.7\}$. $G\left(a\right)\cup G\left(b\right)=\{0.5,0.6,0.9\}=0.666$, $G\left(y\right)=\{0.5,0.5,0.6,0.7\}=0.575$. $G\left(a\right){\subset }_{m}G\left(x\right)$ and $G\left(b\right){\subset }_{m}G\left(y\right)$, however, $G\left(a\right)\cup G\left(b\right)\neg {\subset }_{m}G\left(x\right)\cup G\left(y\right)$. The reference [36] does not provide the definition of the inclusion relationships of hesitant fuzzy sets.

The proof of the Proposition 3.9 in [36] uses an inconclusive conclusion that $G\left(a\right)\subset G\left(x\right)$ and $G\left(b\right)\subset G\left(y\right)$ then $G\left(a\right)\cup G\left(b\right)\subset G\left(x\right)\cup G\left(y\right)$. Even though $G\left(x\right)\subset G\left(x\right)$ and $G(0*y)\subset G\left(0\right)\cup G\left(y\right)$ are true, but $G\left(x\right)\cup G(0*y)\subset G\left(x\right)\cup G\left(0\right)\cup G\left(y\right)$ may not be true. For example, let $G\left(x\right)=\{0.1,0.9\}$, $G(0*y)=\{0.4,0.5\}$ and $G\left(0\right)\cup G\left(y\right)=\{0.1,0.8\}$, then $G\left(x\right)\cup G(0*y)=\{0.4,0.5,0.9\}$ and $G\left(x\right)\cup G\left(0\right)\cup G\left(y\right)=\{0.1,0.1,0.8,0.9\}$. $G\left(x\right){\subset }_{m}G\left(x\right)$ and $G(0*y){\subset }_{m}G\left(0\right)\cup G\left(y\right)$, however $G\left(x\right)\cup G(0*y)\neg {\subset }_{m}G\left(x\right)\cup G\left(0\right)\cup G\left(y\right)$. Hence, the propositions about ∪-hesitant fuzzy subalgebra G of a BCI-algebra X may need to be re-verified.

6.3. Inferences Based on Intersection Operation of Hesitant Fuzzy Sets

The proof of Proposition 3.3 in the reference [40] describes that $C_1\cap C_2\subset C_1$ and $C_1\cap C_2\subset C_2$. $C_1\cap C_2{\subset }_a{C}_1$ and $C_1\cap C_2{\subset }_a{C}_2$ hold simultaneously. However, $C_1\cap C_2{\subset }_m{C}_1$ and $C_1\cap C_2{\subset }_m{C}_2$ do not necessarily both hold at the same time. For example, let $U=\left\{x\right\}$, $C_1\left(x\right)=\{0.1,0.7\}$ and $C_2\left(x\right)=\{0.6,0.8\}$, then $(C_1\cap C_2)\left(x\right)=\{0.1,0.6,0.7\}$ and $(C_1\cap C_2)\left(x\right)\neg {\subset }_m{C}_1\left(x\right)$. Then, the conclusion of the Proposition 3.3 in [40] that $C_1\cap C_2$ is a chained hesitant fuzzy $\mathcal{R}$-module needs to be re-verified.

6.4. Inferences Based on Distributive Law of Hesitant Fuzzy Sets

The Example 3.2 shows that the hesitant fuzzy sets do not satisfy the distributive laws for intersection and union operations.

The Theorem 2.8 in reference [22] needs to be re-verified. The proof of the Theorem 4.2 (in [22]) uses the result of the Theorem 2.8 (in [22]), the Corollary 4.3 (in [22]) is obtained by the result of Theorem 4.2 (in [22]). Hence, the result in Corollary 4.3 (in [22]), i.e., $(\mathcal{G}(U,E),\widetilde{⋒},\tilde{\cup })$ is a distributive quasilattice, needs to be re-verified.

7. Conclusion

In the mathematical system, no errors can occur; otherwise, it will lead to the collapse of the mathematical knowledge system. This review raises some questions regarding some propositions about hesitant fuzzy sets and their mathematical structures. For instance, propositions about the absorption rate and distributive law of hesitant fuzzy sets, the universal set and empty set of hesitant fuzzy topological spaces, propositions about upper and lower approximations of hesitant fuzzy approximation spaces, and propositions about the distributive law of hesitant fuzzy algebras.

This review only randomly selects some propositions from the literature that may be incorrect for discussion. Regarding the correctness of other literature, scholars should be cautious when citing them.