Generalized fuzzy rough sets based on fuzzy coverings

Authors: Guangming Langa, Qingguo Lia∗ and Lankun Guob

∗Corresponding author<EMAIL_ADDRESS>
aCollege of Mathematics and Econometrics, Hunan University, Changsha, Hunan
410082, P.R. China

bCollege of Information Science and Engineering, Hunan University, Changsha,
Hunan 410082, P.R. China

Abstract. This paper further studies the fuzzy rough sets based on fuzzy
coverings. We first present the notions of the lower and upper approximation
operators based on fuzzy coverings and derive their basic properties. To
facilitate the computation of fuzzy coverings for fuzzy covering rough sets,
the concepts of fuzzy subcoverings, the reducible and intersectional elements,
the union and intersection operations are provided and their properties are
discussed in detail. Afterwards, we introduce the concepts of consistent
functions and fuzzy covering mappings and provide a basic theoretical
foundation for the communication between fuzzy covering information systems.
In addition, the notion of homomorphisms is proposed to reveal the
relationship between fuzzy covering information systems. We show how large-
scale fuzzy covering information systems and dynamic fuzzy covering
information systems can be converted into small-scale ones by means of
homomorphisms. Finally, an illustrative example is employed to show that the
attribute reduction can be simplified significantly by our proposed approach.

#  Generalized fuzzy rough sets based on fuzzy coverings

Guangming Langa Qingguo Lia Lankun Guob
a College of Mathematics and Econometrics, Hunan University
Changsha, Hunan 410082, P.R. China
b College of Information Science and Engineering, Hunan University
Changsha, Hunan 410082, P.R. China Corresponding author. Tel./fax: +86 731
8822855, liqingguoli@yahoo.com.cn(Q. G. Li)
E-mail address: langguangming1984@126.com(G. M. Lang), lankun.guo@gmail.com(L.
K. Guo).

> Abstract. This paper further studies the fuzzy rough sets based on fuzzy
> coverings. We first present the notions of the lower and upper approximation
> operators based on fuzzy coverings and derive their basic properties. To
> facilitate the computation of fuzzy coverings for fuzzy covering rough sets,
> the concepts of fuzzy subcoverings, the reducible and intersectional
> elements, the union and intersection operations are provided and their
> properties are discussed in detail. Afterwards, we introduce the concepts of
> consistent functions and fuzzy covering mappings and provide a basic
> theoretical foundation for the communication between fuzzy covering
> information systems. In addition, the notion of homomorphisms is proposed to
> reveal the relationship between fuzzy covering information systems. We show
> how large-scale fuzzy covering information systems and dynamic fuzzy
> covering information systems can be converted into small-scale ones by means
> of homomorphisms. Finally, an illustrative example is employed to show that
> the attribute reduction can be simplified significantly by our proposed
> approach.
>
> Keywords: Rough set; Fuzzy covering; Information system; Homomorphism;
> Attribute reduction
>

##  1 Introduction

Rough set theory, originally constructed on the basis of an equivalence
relation, was proposed by Pawlak[17] for solving inexact or uncertain
problems. But the condition of the equivalence relation is so restrictive that
the applications of rough sets are limited in many practical problems. To deal
with more complex data sets, many researchers have derived a large number of
generalized models by replacing the equivalence relation with a few
mathematical concepts such as fuzzy relations[5, 6, 19, 20, 22, 14, 1] and
coverings[3, 4, 7, 8, 15, 18, 21, 23, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39]
of the universe of discourse.

Recently, the theory of fuzzy rough sets has become a rapidly developing
research area and got a lot of attention. For example, Dubois et al.[5, 6]
initially provided the rough fuzzy sets and the fuzzy rough sets. Then
Radzikowska et al.[19, 20] defined the fuzzy rough sets (respectively, the
L-fuzzy rough sets) based on fuzzy similarity relations (respectively,
residuated lattices). Afterwards, many researchers[15, 33, 4, 7] investigated
fuzzy rough sets based on fuzzy coverings. In practice, we need to compute the
approximations of fuzzy sets in fuzzy covering approximation spaces. But the
classical approximation operators based on coverings are incapable of
computing the approximations of fuzzy sets in the fuzzy covering approximation
space. It motivates us to extend approximation operators of covering
approximation spaces for fuzzy covering approximation spaces. In addition,
there are a large number of fuzzy coverings for the universal set in general.
To facilitate the computation of fuzzy coverings for fuzzy covering rough
sets, it is interesting to investigate the relationship among the elements of
a fuzzy covering and operations on fuzzy coverings.

Meanwhile, many researches[10, 12, 11, 16, 24, 25, 26, 27, 28, 9, 40, 41, 42]
have been conducted on homomorphisms between information systems with the aim
of attribute reductions. For instance, Grzymala-Busse[10, 12, 11] initially
introduced the concept of information system homomorphisms and investigated
its basic properties. Then Li et al.[16] discussed invariant characters of
information systems under some homomorphisms. Afterwards, Wang et al.[24]
found that a complex massive covering information system could be compressed
into a relatively small-scale one under the condition of a homomorphism, and
their attribute reductions are equivalent to each other. Actually, we often
deal with attribute reductions of large-scale fuzzy covering information
systems in practical situations, and the work of Wang et al. mentioned above
inspires that the attribute reduction of large-scale fuzzy covering
information systems may be conducted by means of homomorphisms. But so far we
have not seen any work on homomorphisms between fuzzy covering information
systems. Additionally, the fuzzy covering information system varies with time
due to the dynamic characteristics of data collection, and the non-incremental
approach to compressing the dynamic fuzzy covering information system is often
very costly or even intractable. For this issue, we attempt to apply an
incremental updating scheme to maintain the compression dynamically and avoid
unnecessary computations by utilizing the compression of the original system.

The purpose of this paper is to investigate further fuzzy coverings based
rough sets. First, we present the notions of the lower and upper approximation
operators based on fuzzy coverings by extending Zhu’s model[37], and examine
their basic properties. Particularly, we find that the upper approximation
based on neighborhoods can not be represented without using the neighborhoods
as the classical covering approximation space[37] in the fuzzy approximation
space. Second, we propose the concepts of fuzzy subcoverings, reducible and
intersectional elements, union and intersection operations and investigate
their basic properties in detail. Third, the theoretical foundation is
established for the communication between fuzzy covering information systems.
Concretely, we construct a consistent function by combining the fuzzy
covering, proposed by Deng et al. [4], with the approach in [24], and explore
its main properties known from the consistent function for the classical
covering approximation space in [24]. We also provide the concepts of fuzzy
covering mappings and study their basic properties in detail. Fourth, the
notion of homomorphisms between fuzzy covering information systems is
introduced for attribute reductions. We find that a large-scale fuzzy covering
information system can be compressed into a relatively small-scale one, and
attribute reductions of the original system and image system are equivalent to
each other under the condition of a homomorphism. In addition, we give the
algorithm to construct attribute reducts and employ an example to illustrate
the efficiency of our approach for attribute reductions of fuzzy covering
information systems. We also discuss how to compress the dynamic fuzzy
covering information system.

The rest of this paper is organized as follows: Section 2 briefly reviews the
basic concepts related to the covering information systems and fuzzy covering
information systems. In Section 3, we put forward some concepts such as the
neighborhood operators, the approximation operators and reducible elements for
fuzzy covering approximation spaces, and investigate their basic properties.
Section 4 is devoted to introducing the concept of consistent functions which
provides the theoretical foundation for the communication between fuzzy
covering information systems. In Section 5, we present the notion of
homomorphisms between fuzzy covering information systems and discuss its basic
properties. We also investigate data compressions of fuzzy covering
information systems and dynamic fuzzy covering information systems. An example
is given to illustrate that how to conduct attribute reductions of the fuzzy
covering information system by means of homomorphisms. We conclude the paper
and set further research directions in Section 6.

##  2 Preliminaries

In this section, we briefly recall some basic concepts related to the covering
information system and fuzzy covering information system. Three examples are
given to illustrate two types of covering information systems.

###### Definition 2.1

[2] Let $U$ be a non-empty set $($the universe of discourse$)$. A non-empty
sub-family $\mathscr{C}\subseteq\mathscr{P}(U)$ is called a covering of $U$ if

$(1)$ every element in $\mathscr{C}$ is non-empty;

$(2)$ $\bigcup\\{C|C\in\mathscr{C}\\}=U$, where $\mathscr{P}(U)$ is the
powerset of $U$.

It is clear that the concept of a covering is an extension of the notion of a
partition. In what follows, $(U,\mathscr{C})$ is called a classical covering
approximation space.

To investigate further coverings based rough sets, Chen et al. proposed the
following concepts on coverings.

###### Definition 2.2

[3] Let $\mathscr{C}$=$\\{C_{1},C_{2},...,C_{N}\\}$ be a covering of $U$,
$C_{x}$=$\bigcap\\{C_{i}|x\in C_{i}\text{ and }C_{i}\in\mathscr{C}\\}$ for any
$x\in U$, and $Cov(\mathscr{C})$=$\\{C_{x}|x\in U\\}$. Then $Cov(\mathscr{C})$
is called the induced covering of $\mathscr{C}$.

Suppose $c$ is an attribute, the domain of $c$ is
$\\{c_{1},c_{2},...,c_{N}\\}$, $C_{i}$ means the set of objects in $U$ taking
a certain attribute value $c_{i}$, and $C_{x}=C_{i}\cap C_{j}$, it implies
that the possible value of $x$ regarding the attribute $c$ is $c_{i}$ or
$c_{j}$, and $C_{x}$ is the minimal set containing $x$ in $Cov(\mathscr{C})$.

###### Definition 2.3

[3] Let $\Delta$=$\\{\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}\\}$
be a family of coverings of $U$,
$\Delta_{x}$=$\bigcap\\{C_{ix}|C_{ix}\in\mathscr{C}_{i}$, $1\leq i\leq m\\}$
for any $x\in U$, and $Cov(\Delta)$=$\\{\Delta_{x}|x\in U\\}$. Then
$Cov(\Delta)$ is called the induced covering of $\Delta$.

That is to say, $\Delta_{x}$ is the intersection of all the elements including
$x$ of each $\mathscr{C}_{i}$, and it is the minimal set including $x$ in
$Cov(\Delta)$. Furthermore, $Cov(\Delta)$ is a partition if every covering in
$\Delta$ is a partition. In what follows, $(U,\Delta)$ is called a covering
information system. To illustrate how covering information systems are
constructed, we present two examples which have different application
backgrounds.

###### Example 2.4

Let $U=\\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}$ be eight houses,
$C=\\{price,color\\}$ the attribute set, the domains of $price$ and $color$
are $\\{high,middle,low\\}$ and $\\{good,bad\\}$, respectively. To evaluate
these houses, specialists $A$ and $B$ are employed and their evaluation
reports are shown as follows:

$\displaystyle high_{A}$ $\displaystyle=$
$\displaystyle\\{x_{1},x_{4},x_{5},x_{7}\\},middle_{A}=\\{x_{2},x_{8}\\},low_{A}=\\{x_{3},x_{6}\\};$
$\displaystyle high_{B}$ $\displaystyle=$
$\displaystyle\\{x_{1},x_{2},x_{4},x_{7},x_{8}\\},middle_{B}=\\{x_{5}\\},low_{B}=\\{x_{3},x_{6}\\};$
$\displaystyle good_{A}$ $\displaystyle=$
$\displaystyle\\{x_{1},x_{2},x_{3},x_{6}\\},bad_{A}=\\{x_{4},x_{5},x_{7},x_{8}\\};$
$\displaystyle good_{B}$ $\displaystyle=$
$\displaystyle\\{x_{1},x_{2},x_{3},x_{5}\\},bad_{B}=\\{x_{4},x_{6},x_{7},x_{8}\\},$

where $high_{A}$ denotes the houses belonging to high price by the specialist
$A$. The meanings of other symbols are similar. Since their evaluations are of
equal importance, we should consider all their advice. Consequently, we obtain
the following results:

$\displaystyle high_{A\vee B}$ $\displaystyle=$ $\displaystyle high_{A}\cup
high_{B}=\\{x_{1},x_{2},x_{4},x_{5},x_{7},x_{8}\\};$ $\displaystyle
middle_{A\vee B}$ $\displaystyle=$ $\displaystyle middle_{A}\cup
middle_{B}=\\{x_{2},x_{5},x_{8}\\};$ $\displaystyle low_{A\vee B}$
$\displaystyle=$ $\displaystyle low_{A}\cup low_{B}=\\{x_{3},x_{6}\\};$
$\displaystyle good_{A\vee B}$ $\displaystyle=$ $\displaystyle good_{A}\cup
good_{B}=\\{x_{1},x_{2},x_{3},x_{5},x_{6}\\};$ $\displaystyle bad_{A\vee B}$
$\displaystyle=$ $\displaystyle bad_{A}\cup
bad_{B}=\\{x_{4},x_{5},x_{6},x_{7},x_{8}\\}.$

Based on the above statement, we derive a covering information system
$(U,\Delta)$, where $\Delta=\\{\mathscr{C}_{price},$ $\mathscr{C}_{color}\\}$,
$\mathscr{C}_{price}=\\{high_{A\vee B},middle_{A\vee B},low_{A\vee B}\\}$ and
$\mathscr{C}_{color}=\\{good_{A\vee B},bad_{A\vee B}\\}$.

###### Example 2.5

Let Table 1 be an incomplete information system, where $\ast$ stands for the
lost value. According to the interpretation in [11], the lost value is
considered to be similar to any value in the domain of the corresponding
attribute. Consequently, we obtain three coverings of $U$ by the attribute set
as follows:
$\mathscr{C}_{structure}=\\{\\{x_{1},x_{2},x_{4},x_{6}\\},\\{x_{2},x_{3},x_{5},x_{6}\\}\\}$,
$\mathscr{C}_{color}=\\{\\{x_{1},x_{2},x_{5}\\},\\{x_{3},x_{4},x_{5},x_{6}\\}\\}$,
$\mathscr{C}_{price}=\\{\\{x_{1},x_{4},x_{5},x_{6}\\},\\{x_{2},x_{3},x_{4},x_{6}\\}\\}$.
Hence, $(U,\Delta)$ is a covering information system, where
$\Delta=\\{\mathscr{C}_{structure},\mathscr{C}_{color},$
$\mathscr{C}_{price}\\}$.

To conduct the communication between covering information systems, Wang et al.
provided the concept of consistent functions based on coverings.

###### Definition 2.6

[24] Let $f$ be a mapping from $U_{1}$ to $U_{2}$,
$\mathscr{C}$=$\\{C_{1},C_{2},...,C_{N}\\}$ a covering of $U_{1}$,
$C_{x}$=$\bigcap\\{C_{i}|x\in C_{i}\text{ and }C_{i}\in\mathscr{C}\\}$, and
$[x]_{f}=\\{y\in U_{1}|f(x)=f(y)\\}$. If $[x]_{f}\subseteq C_{x}$ for any
$x\in U_{1}$, then $f$ is called a consistent function with respect to
$\mathscr{C}$.

Based on Definition 2.6, Wang et al. constructed a homomorphism between a
complex massive covering information system and a relatively small-scale
covering information system. It has also been proved that their attribute
reductions are equivalent to each other under the condition of a homomorphism.
Hence, the notion of the consistent function provides the foundation for the
communication between covering information systems.

In order to deal with uncertainty and more complex problems, Zadeh[31]
proposed the theory of fuzzy sets by extending the classical set theory. Let
$U$ be a non-empty universe of discourse, a fuzzy set of $U$ is a mapping
$A:U\longrightarrow[0,1]$. We denote by $\mathscr{F}(U)$ the set of all fuzzy
sets of $U$. For any $A,B\in\mathscr{F}(U)$, we say that $A\subseteq B$ if
$A(x)\leq B(x)$ for any $x\in U$. The union of $A$ and $B$, denoted as $A\cup
B$, is defined by $(A\cup B)(x)=A(x)\vee B(x)$ for any $x\in U$, and the
intersection of $A$ and $B$, denoted as $A\cap B$, is defined by $(A\cap
B)(x)=A(x)\wedge B(x)$ for any $x\in U$. The complement of $A$, denoted as
$-A$, is defined by $(-A)(x)=1-A(x)$ for any $x\in U$. Furthermore, a fuzzy
relation on $U$ is a mapping $R:U\times U\longrightarrow[0,1]$. We denote by
$\mathscr{F}(U\times U)$ the set of all fuzzy relations on $U$.

In practical situations, there exist a lot of fuzzy information systems as a
generalization of crisp information systems, and the investigations of fuzzy
information systems have powerful prospects in applications. To conduct the
communication between fuzzy information systems, Wang et al. proposed a
consistent function with respect to a fuzzy relation.

###### Definition 2.7

[27] Let $U_{1}$ and $U_{2}$ be two universes, $f$ a mapping from $U_{1}$ to
$U_{2}$, $R\in\mathscr{F}(U_{1}\times U_{1})$, $[x]_{f}=\\{y\in
U_{1}|f(x)=f(y)\\}$, and $\\{[x]_{f}|x\in U_{1}\\}$ is a partition on $U_{1}$.
For any $x,y\in U_{1}$, if $R(u,v)=R(s,t)$ for any two pairs
$(u,v),(s,t)\in[x]_{f}\times[y]_{f}$, then $f$ is said to be consistent with
respect to $R$.

Based on the consistent function, Wang et al. constructed a homomorphism
between a large-scale fuzzy information system and a relatively small-scale
fuzzy information system. It has been proved that their attribute reductions
are equivalent to each other under the condition of a homomorphism. In this
sense, the notion of the consistent function provides an approach to studying
the communication between fuzzy information systems.

Recently, Deng et al.[4] proposed the concept of a fuzzy covering.

###### Definition 2.8

[4] A fuzzy covering of $U$ is a collection of fuzzy sets
$\mathscr{C}^{\ast}\subseteq\mathscr{F}(U)$ which satisfies

$(1)$ every fuzzy set $C^{\ast}\in\mathscr{C}^{\ast}$ is non-null, i.e.,
$C^{\ast}\neq\emptyset$;

$(2)$ $\forall x\in U,\bigvee_{C^{\ast}\in\mathscr{C}^{\ast}}C^{\ast}(x)>0$.

Unless stated otherwise, $U$ is a finite universe, and $\mathscr{C}^{\ast}$
consists of finite number of sets in this work. In what follows,
$(U,\mathscr{C}^{\ast})$ is called a fuzzy covering approximation space, and
$(U,\Delta^{\ast})$ is called a fuzzy covering information system, where
$\Delta^{\ast}=\\{\mathscr{C}_{i}^{\ast}|1\leq i\leq m\\}$. Throughout the
paper, we denote the set of all fuzzy coverings of $U$ as $C(U)$ for
simplicity.

In the following, we employ an example to illustrate the fuzzy covering
information system.

###### Example 2.9

Let $U=\\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}$ be eight houses,
$C=\\{price,color\\}$ the attribute set, the domains of $price$ and $color$
are $\\{high,middle,low\\}$ and $\\{good,bad\\}$, respectively. To evaluate
these houses, specialists $A$ and $B$ are employed and their evaluation
reports are shown as follows:

$\displaystyle high^{\ast}_{A}$ $\displaystyle=$
$\displaystyle\frac{1}{x_{1}}+\frac{0.7}{x_{2}}+\frac{0}{x_{3}}+\frac{1}{x_{4}}+\frac{1}{x_{5}}+\frac{0}{x_{6}}+\frac{1}{x_{7}}+\frac{0.65}{x_{8}};$
$\displaystyle middle^{\ast}_{A}$ $\displaystyle=$
$\displaystyle\frac{0.6}{x_{1}}+\frac{1}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.4}{x_{4}}+\frac{0.45}{x_{5}}+\frac{0.5}{x_{6}}+\frac{0.5}{x_{7}}+\frac{1}{x_{8}};$
$\displaystyle low^{\ast}_{A}$ $\displaystyle=$
$\displaystyle\frac{0}{x_{1}}+\frac{0.5}{x_{2}}+\frac{1}{x_{3}}+\frac{0}{x_{4}}+\frac{0.5}{x_{5}}+\frac{1}{x_{6}}+\frac{0}{x_{7}}+\frac{0.5}{x_{8}};$
$\displaystyle good^{\ast}_{A}$ $\displaystyle=$
$\displaystyle\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\frac{0.5}{x_{4}}+\frac{0.6}{x_{5}}+\frac{1}{x_{6}}+\frac{0}{x_{7}}+\frac{0}{x_{8}};$
$\displaystyle bad^{\ast}_{A}$ $\displaystyle=$
$\displaystyle\frac{0}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0}{x_{3}}+\frac{1}{x_{4}}+\frac{1}{x_{5}}+\frac{0.2}{x_{6}}+\frac{1}{x_{7}}+\frac{1}{x_{8}};$
$\displaystyle high_{B}$ $\displaystyle=$
$\displaystyle\frac{0.9}{x_{1}}+\frac{0.7}{x_{2}}+\frac{0}{x_{3}}+\frac{1}{x_{4}}+\frac{1}{x_{5}}+\frac{0}{x_{6}}+\frac{1}{x_{7}}+\frac{0.8}{x_{8}};$
$\displaystyle middle^{\ast}_{B}$ $\displaystyle=$
$\displaystyle\frac{0.6}{x_{1}}+\frac{1}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.4}{x_{4}}+\frac{0.45}{x_{5}}+\frac{0.7}{x_{6}}+\frac{0.5}{x_{7}}+\frac{1}{x_{8}};$
$\displaystyle low^{\ast}_{B}$ $\displaystyle=$
$\displaystyle\frac{0}{x_{1}}+\frac{0.5}{x_{2}}+\frac{1}{x_{3}}+\frac{0}{x_{4}}+\frac{0.5}{x_{5}}+\frac{0.9}{x_{6}}+\frac{0}{x_{7}}+\frac{0.5}{x_{8}};$
$\displaystyle good^{\ast}_{B}$ $\displaystyle=$
$\displaystyle\frac{0.8}{x_{1}}+\frac{1}{x_{2}}+\frac{0.9}{x_{3}}+\frac{0.5}{x_{4}}+\frac{0.6}{x_{5}}+\frac{1}{x_{6}}+\frac{0}{x_{7}}+\frac{0}{x_{8}};$
$\displaystyle bad^{\ast}_{B}$ $\displaystyle=$
$\displaystyle\frac{0}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.4}{x_{3}}+\frac{1}{x_{4}}+\frac{1}{x_{5}}+\frac{0.2}{x_{6}}+\frac{0.9}{x_{7}}+\frac{1}{x_{8}},$

where $high_{A}$ is the membership degree of each house belonging to the high
price by the specialist $A$. The meanings of the other symbols are similar.
Based on the above results, we obtain a fuzzy covering information system
$(U,\Delta^{\ast})$, where
$\Delta^{\ast}=\\{\mathscr{C}^{\ast}_{price},\mathscr{C}^{\ast}_{color}\\}$,
$\mathscr{C}^{\ast}_{price}=\\{C_{high},C_{middle},C_{low}\\}$ and
$\mathscr{C}^{\ast}_{color}=\\{C_{good},C_{bad}\\}$.

$\displaystyle C_{high}$ $\displaystyle=$ $\displaystyle high^{\ast}_{A}\cup
high^{\ast}_{B}=\frac{1}{x_{1}}+\frac{0.7}{x_{2}}+\frac{0}{x_{3}}+\frac{1}{x_{4}}+\frac{1}{x_{5}}+\frac{0}{x_{6}}+\frac{1}{x_{7}}+\frac{0.8}{x_{8}};$
$\displaystyle C_{middle}$ $\displaystyle=$ $\displaystyle
middle^{\ast}_{A}\cup
middle^{\ast}_{B}=\frac{0.6}{x_{1}}+\frac{1}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.4}{x_{4}}+\frac{0.45}{x_{5}}+\frac{0.7}{x_{6}}+\frac{0.5}{x_{7}}+\frac{1}{x_{8}};$
$\displaystyle C_{low}$ $\displaystyle=$ $\displaystyle low^{\ast}_{A}\cup
low^{\ast}_{B}=\frac{0}{x_{1}}+\frac{0.5}{x_{2}}+\frac{1}{x_{3}}+\frac{0}{x_{4}}+\frac{0.5}{x_{5}}+\frac{1}{x_{6}}+\frac{0}{x_{7}}+\frac{0.5}{x_{8}};$
$\displaystyle C_{good}$ $\displaystyle=$ $\displaystyle good^{\ast}_{A}\cup
good^{\ast}_{B}=\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\frac{0.5}{x_{4}}+\frac{0.6}{x_{5}}+\frac{1}{x_{6}}+\frac{0}{x_{7}}+\frac{0}{x_{8}};$
$\displaystyle C_{bad}$ $\displaystyle=$ $\displaystyle bad^{\ast}_{A}\cup
bad^{\ast}_{B}=\frac{0}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.4}{x_{3}}+\frac{1}{x_{4}}+\frac{1}{x_{5}}+\frac{0.2}{x_{6}}+\frac{1}{x_{7}}+\frac{1}{x_{8}}.$

It is obvious that we can construct a fuzzy covering of the universe with an
attribute. Since the fuzzy covering rough set theory is effective to handle
uncertain information, the investigation of this theory becomes an important
task in rough set theory.

##  3 The basic properties of the fuzzy covering approximation space

In this section, we introduce the concepts of neighborhoods, the lower and
upper approximation operators to facilitate the computation of fuzzy sets for
fuzzy covering approximation spaces. Then we propose the concepts of fuzzy
subcoverings, irreducible and reducible elements, non-intersectional and
intersectional elements of fuzzy coverings. Afterwards, the union and
intersection operations on two fuzzy coverings are provided. We also construct
two roughness measures and employ several examples to illustrate the proposed
notions.

###  3.1 The lower and upper approximation operations

Before introducing approximation operators, we present the concepts of
neighborhoods and induced fuzzy coverings based on fuzzy coverings.

###### Definition 3.1

Let $(U,\mathscr{C}^{\ast})$ be a fuzzy covering approximation space, and
$x\in U$. Then $C^{\ast}_{\mathscr{C}^{\ast}x}=\bigcap\\{C^{\ast}|C^{\ast}(x)$
$>0\text{ and }C^{\ast}\in\mathscr{C}^{\ast}\\}$ is called the neighborhood of
$x$ concerning $\mathscr{C}^{\ast}$.

We notice that $C_{x}^{\ast}$ is the intersection of all fuzzy subsets whose
membership degrees of $x\in U$ are not zeroes. Assume that
$C_{1}^{\ast},C_{2}^{\ast}\in\mathscr{C}^{\ast}$,
$C_{1}^{\ast}(x)>0,C_{2}^{\ast}(x)>0$, and $C^{\ast}_{x}=C^{\ast}_{1}\cap
C^{\ast}_{2}$ for $x\in U$, it implies that the membership degree of $x$ in
$C^{\ast}_{x}$ is $min\\{C_{1}^{\ast}(x),C_{2}^{\ast}(x)\\}$. In addition, we
observe that the classical neighborhood of a point $C_{x}=\bigcap\\{C|x\in
C\in\mathscr{C}\\}$ is the same as that in Definition 3.1 if the membership
degree for any $x\in U$ has its value only from the set $\\{0,1\\}$, where
$\mathscr{C}$ is a covering of $U$. For convenience, we denote $C^{\ast}$,
$\mathscr{C}^{\ast}$, $C^{\ast}_{\mathscr{C}_{i}x}$ and $C_{\mathscr{C}x}$ as
$C$, $\mathscr{C}$, $C_{ix}$ and $C_{x}$, respectively.

We present the properties of the neighborhood operator below.

###### Proposition 3.2

Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $x,y\in U$.
If $C_{x}(y)>0$, then $C_{y}\subseteq C_{x}.$

Proof. Assume that $\mathscr{A}=\\{C|C\in\mathscr{C},C(x)>0\\}$ and
$\mathscr{B}=\\{C^{\prime}|C^{\prime}\in\mathscr{C},C^{\prime}(y)>0\\}$. Since
$C_{x}(y)>0$, it follows that $C(y)>0$ for any $C\in\mathscr{A}$.
Consequently, $C\in\mathscr{B}$. It implies that
$\\{C|C\in\mathscr{C},C(x)>0\\}\subseteq\\{C^{\prime}|C^{\prime}\in\mathscr{C},C^{\prime}(y)>0\\}.$
Therefore, $C_{y}\subseteq C_{x}.$ $\Box$

###### Proposition 3.3

Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $x,y\in U$.
If $C_{x}(y)>0$ and $C_{y}(x)>0$, then $C_{y}=C_{x}.$

Proof. Straightforward from Proposition 3.2. $\Box$

Based on Definition 3.1, we present the concept of a fuzzy covering induced by
the original fuzzy covering.

###### Definition 3.4

Let $\mathscr{C}$=$\\{C_{1},C_{2},...,C_{N}\\}$ be a fuzzy covering of $U$,
$C_{x}$=$\bigcap\\{C_{i}|C_{i}(x)>0\text{ and }C_{i}\in\mathscr{C}\\}$ for any
$x\in U$, and $Cov(\mathscr{C})$=$\\{C_{x}|x\in U\\}$. Then $Cov(\mathscr{C})$
is called the induced fuzzy covering of $\mathscr{C}$.

It is clear that $C_{x}$ has the minimal membership degree of $x$ in
$Cov(\mathscr{C})$, and each element of $Cov(\mathscr{C})$ can not be
represented as the union of other elements of $Cov(\mathscr{C})$. In other
words, $C_{x}$ is the minimal set containing $x$ in $Cov(\mathscr{C})$.
Furthermore, $Cov(\mathscr{C})$ is a fuzzy covering of $U$, and it is easy to
prove that the concept presented in Definition 2.2 is a special case of
Definition 3.4 when the values of membership degree are taken from the set
$\\{0,1\\}$.

An example is employed to illustrate the induced fuzzy covering.

###### Example 3.5

Let $U_{1}=\\{x_{1},x_{2},x_{3},x_{4}\\}$, and
$\mathscr{C}_{1}=\\{C^{\prime}_{1},C^{\prime}_{2},C^{\prime}_{3}\\}$, where
$C^{\prime}_{1}=\frac{1}{x_{1}}+\frac{0.5}{x_{2}}+\frac{1}{x_{3}}+\frac{0.5}{x_{4}}$,
$C^{\prime}_{2}=\frac{0.5}{x_{1}}+\frac{0.6}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.6}{x_{4}}$,
and
$C^{\prime}_{3}=\frac{0}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0}{x_{3}}+\frac{0.5}{x_{4}}$.
By Definition 3.4, we obtain the induced fuzzy covering
$Cov(\mathscr{C}_{1})=\\{C_{x_{i}}|i=1,2,3,4\\}$, where
$C_{x_{1}}=C_{x_{3}}=\frac{0.5}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$,
and
$C_{x_{2}}=C_{x_{4}}=\frac{0}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0}{x_{3}}+\frac{0.5}{x_{4}}$.

For convenience, we denote $C^{\prime}_{i}$ as $C_{i}$ in the following
examples.

We also propose the notion of a fuzzy covering induced by a family of fuzzy
coverings.

###### Definition 3.6

Let $\Delta$=$\\{\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}\\}$ be a
family of fuzzy coverings of $U$, $\Delta_{x}$=$\bigcap\\{C_{ix}|C_{ix}\in
Cov(\mathscr{C}_{i})$, $1\leq i\leq m\\}$ for any $x\in U$, and
$Cov(\Delta)$=$\\{\Delta_{x}|x\in U\\}$. Then $Cov(\Delta)$ is called the
induced fuzzy covering of $\Delta$.

In other words, $\Delta_{x}$ is the intersection of all the elements whose
membership degrees of $x$ are not zeroes in each $\mathscr{C}_{i}$, and it is
the set whose membership degree of $x$ is the minimal in $Cov(\Delta)$.
Furthermore, given $x,y\in U$, if $\Delta_{x}(y)>0$, then
$\Delta_{y}\subseteq\Delta_{x}$. Consequently, $\Delta_{x}(y)>0$ and
$\Delta_{y}(x)>0$ imply that $\Delta_{x}=\Delta_{y}$. In addition,
$Cov(\Delta)$ is a fuzzy covering of $U$. Therefore, it is easy to verify that
the notion given in Definition 2.3 is a special case of Definition 3.6 when
the values of membership degree are taken from the set $\\{0,1\\}$.

Next, we give an example to illustrate Definition 3.6.

###### Example 3.7

Let $U_{1}=\\{x_{1},x_{2},x_{3},x_{4}\\}$,
$\Delta=\\{\mathscr{C}_{1},\mathscr{C}_{2},\mathscr{C}_{3}\\}$,
$\mathscr{C}_{1}=\\{C_{4},C_{5},C_{6}\\}$,
$\mathscr{C}_{2}=\\{C_{7},C_{8},C_{9}\\}$, and
$\mathscr{C}_{3}=\\{C_{10},C_{11},C_{12}\\}$, where
$C_{4}=\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$,
$C_{5}=\frac{0.5}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.6}{x_{3}}+\frac{0.6}{x_{4}}$,
$C_{6}=\frac{0}{x_{1}}+\frac{0}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$,
$C_{7}=\frac{0}{x_{1}}+\frac{0}{x_{2}}+\frac{1}{x_{3}}+\frac{1}{x_{4}}$
$C_{8}=\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{0.7}{x_{3}}+\frac{0.7}{x_{4}}$,
$C_{9}=\frac{0.6}{x_{1}}+\frac{0.6}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$,
$C_{10}=\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\frac{1}{x_{4}}$
$C_{11}=\frac{0.5}{x_{1}}+\frac{0.5}{x_{2}}+\frac{1}{x_{3}}+\frac{1}{x_{4}}$,
and
$C_{12}=\frac{0.8}{x_{1}}+\frac{0.8}{x_{2}}+\frac{0.7}{x_{3}}+\frac{0.7}{x_{4}}$.
By Definition 3.6, we obtain that
$Cov(\Delta)=\\{\Delta_{x_{i}}|i=1,2,3,4\\}$, where
$\Delta_{x_{1}}=\Delta_{x_{2}}=\frac{0.5}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$,
and
$\Delta_{x_{3}}=\Delta_{x_{4}}=\frac{0}{x_{1}}+\frac{0}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$.

In practice, the classical approximation operators based on coverings are not
fit for computing the approximations of fuzzy sets in the fuzzy covering
approximation space. To solve this issue, we propose the concepts of the lower
and upper approximation operators based on fuzzy coverings by extending
approximation operators in [37].

###### Definition 3.8

Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $X\subseteq
U$. Then the lower and upper approximations of X are defined as

$\displaystyle\underline{X}_{\mathscr{C}}$ $\displaystyle=$
$\displaystyle\bigcup\\{C|C\subseteq X\text{ and }C\in\mathscr{C}\\};$
$\displaystyle\overline{X}_{\mathscr{C}}$ $\displaystyle=$
$\displaystyle\left(\bigcup\\{C_{x}|X(x)>0\text{ and
}\underline{X}_{\mathscr{C}}(x)=0,x\in
U\\}\right)\cup\underline{X}_{\mathscr{C}}.$

The physical meaning of the lower and upper approximations of $X$ is that we
can approximate $X$ by $\underline{X}_{\mathscr{C}}$ and
$\overline{X}_{\mathscr{C}}$. Particularly, if
$\overline{X}_{\mathscr{C}}=\underline{X}_{\mathscr{C}}=X$, then $X$ can be
understood as a definable set. Otherwise, $X$ is undefinable. It is clear that
the lower and upper approximation operations are the same as those[37] in the
classical covering approximation space if $\mathscr{C}$ is a covering of $U$.
In this sense, the notions given in Definition 3.8 are generalizations of the
classical ones into the fuzzy setting. In the following, we investigate their
basic properties in detail.

###### Proposition 3.9

Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and
$X,Y\subseteq U$. Then

$(1)$ $\overline{\emptyset}_{\mathscr{C}}=\emptyset$,
$\underline{\emptyset}_{\mathscr{C}}=\emptyset$;

$(2)$ $\underline{U}_{\mathscr{C}}\subseteq U$,
$\overline{U}_{\mathscr{C}}\subseteq U$;

$(3)$ $\underline{X}_{\mathscr{C}}\subseteq\overline{X}_{\mathscr{C}}$,
$\underline{X}_{\mathscr{C}}\subseteq X$;

$(4)$
$\overline{X}_{\mathscr{C}}\cup\overline{Y}_{\mathscr{C}}\subseteq\overline{(X\cup
Y)}_{\mathscr{C}}$;

$(5)$ $X\subseteq
Y\Longrightarrow\underline{X}_{\mathscr{C}}\subseteq\underline{Y}_{\mathscr{C}},\overline{X}_{\mathscr{C}}\subseteq\overline{Y}_{\mathscr{C}}$;

$(6)$ $\forall C\in\mathscr{C}$, $\underline{C}=C,\overline{C}=C;$

$(7)$
$\underline{(\underline{X}_{\mathscr{C}})}_{\mathscr{C}}=\underline{X}_{\mathscr{C}}$,
$\overline{X}_{\mathscr{C}}=\overline{(\overline{X}_{\mathscr{C}})}_{\mathscr{C}}$;

$(8)$
$\overline{(\underline{X}_{\mathscr{C}})}_{\mathscr{C}}=\underline{X}_{\mathscr{C}}$,
$\underline{(\overline{X}_{\mathscr{C}})}_{\mathscr{C}}\subseteq\overline{X}_{\mathscr{C}}$.

Proof. Straightforward from Definition 3.8. $\Box$

###### Proposition 3.10

The following properties do not hold generally in the fuzzy covering
approximation space:

$(1)$ $\underline{(X\cap
Y)}_{\mathscr{C}}=\underline{X}_{\mathscr{C}}\cap\underline{Y}_{\mathscr{C}}$;

$(2)$ $\underline{(-X)}_{\mathscr{C}}=-(\overline{X}_{\mathscr{C}})$;

$(3)$ $\overline{(-X)}_{\mathscr{C}}=-(\underline{X}_{\mathscr{C}})$;

$(4)$
$\underline{(-\underline{X}_{\mathscr{C}})}_{\mathscr{C}}=-(\underline{X}_{\mathscr{C}})$;

$(5)$
$\overline{(-\overline{X}_{\mathscr{C}})}_{\mathscr{C}}=-(\overline{X}_{\mathscr{C}})$;

$(6)$ $\underline{U}_{\mathscr{C}}=U$, $\overline{U}_{\mathscr{C}}=U$;

$(7)$ $\overline{(X\cup
Y)}_{\mathscr{C}}\subseteq\overline{X}_{\mathscr{C}}\cup\overline{Y}_{\mathscr{C}}$;

$(8)$ $X\subseteq\overline{X}_{\mathscr{C}}$.

Example 2 in [35] can illustrate that Proposition 3.10(1-5) does not hold
generally in the fuzzy covering approximation space. Specially, we obtain that
$\underline{U}_{\mathscr{C}}=U$ and $X\subseteq\overline{X}_{\mathscr{C}}$ do
not necessarily hold for any $X\subseteq U$. Consequently, the lower and upper
approximation operations are not interior and closure operators, respectively,
in the fuzzy covering approximation space.

We employ an example to illustrate that Proposition 3.10(6-8) does not hold
generally.

###### Example 3.11

Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$, $\mathscr{C}=\\{C_{13},C_{14},C_{15},$
$C_{16}\\}$, where
$C_{13}=\frac{0.3}{x_{1}}+\frac{0}{x_{2}}+\frac{0}{x_{3}}+\frac{0}{x_{4}},$
$C_{14}=\frac{0}{x_{1}}+\frac{0}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}},$
$C_{15}=\frac{0.3}{x_{1}}+\frac{0}{x_{2}}+\frac{0}{x_{3}}+\frac{0.4}{x_{4}}$,
and
$C_{16}=\frac{0}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0}{x_{4}}.$
By Definition 3.8, it follows that
$\overline{U}_{\mathscr{C}}=\underline{U}_{\mathscr{C}}=\frac{0.3}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}\neq
U$. For
$X=\frac{0.4}{x_{1}}+\frac{0}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.5}{x_{4}}$ and
$Y=\frac{0}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0}{x_{4}}$, we
have that
$\overline{X}_{\mathscr{C}}=\frac{0.3}{x_{1}}+\frac{0}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.4}{x_{4}}$,
$\overline{Y}_{\mathscr{C}}=\frac{0}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0}{x_{4}}$
and $\overline{(X\cup
Y)}_{\mathscr{C}}=\frac{0.3}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$.
Consequently, $\overline{(X\cup
Y)}_{\mathscr{C}}\neq\overline{X}_{\mathscr{C}}\cup\overline{Y}_{\mathscr{C}}$
and $X\nsubseteq\overline{X}_{\mathscr{C}}$.

Some relationships among $\underline{X}_{\mathscr{C}}$,
$\overline{X}_{\mathscr{C}}$ and $X$ are explored in the following.

###### Proposition 3.12

Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $X\subseteq
U$.

$(1)$ If $\underline{X}_{\mathscr{C}}=X$, then
$\overline{X}_{\mathscr{C}}=\underline{X}_{\mathscr{C}}$;

$(2)$ If $\underline{X}_{\mathscr{C}}=X$, then $\overline{X}_{\mathscr{C}}=X$;

$(3)$ $\underline{X}_{\mathscr{C}}=X$ if and only if $X$ is a union of
elements in $\mathscr{C}$;

$(4)$ If $X$ is a union of elements in $\mathscr{C}$, then
$\overline{X}_{\mathscr{C}}=X$.

Next, an example is given to illustrate that the converses of Proposition
3.12(1), (2) and (4) do not hold generally.

###### Example 3.13

Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$, $\mathscr{C}=\\{C_{17},C_{18},C_{19},$
$C_{20}\\}$, where
$C_{17}=\frac{0.2}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.1}{x_{4}},$
$C_{18}=\frac{0.1}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.1}{x_{4}},$
$C_{19}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0.1}{x_{4}}$,
and
$C_{20}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.2}{x_{4}}.$
For
$X=\frac{0.5}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$,
it follows that
$\underline{X}_{\mathscr{C}}=\frac{0.2}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0.2}{x_{4}}=\overline{X}_{\mathscr{C}}$.
But $X$ is not a union of some subsets in the fuzzy covering $\mathscr{C}$.
For
$Y=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.1}{x_{4}}$,
according to Definition 3.1, it follows that
$C_{x_{1}}=C_{x_{2}}=C_{x_{3}}=C_{x_{4}}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.1}{x_{4}}$.
Then we have that $\overline{Y}=Y$, but $Y$ is not a union of some elements of
$\mathscr{C}$.

From Proposition 3.10, we see that
$\underline{X}_{\mathscr{C}}\cap\underline{Y}_{\mathscr{C}}=\underline{(X\cap
Y)}_{\mathscr{C}}$ does not hold generally for any $X,Y\subseteq U$ in the
fuzzy covering approximation space. But if
$\underline{X}_{\mathscr{C}}\cap\underline{Y}_{\mathscr{C}}=\underline{(X\cap
Y)}_{\mathscr{C}}$ for any $X,Y\subseteq U$, then we can obtain the following
results.

###### Proposition 3.14

If
$\underline{X}_{\mathscr{C}}\cap\underline{Y}_{\mathscr{C}}=\underline{(X\cap
Y)}_{\mathscr{C}}$ for any $X,Y\subseteq U$, then $C_{1}\cap C_{2}=\emptyset$
or $C_{1}\cap C_{2}$ is a union of elements of $\mathscr{C}$ for any
$C_{1},C_{2}\in\mathscr{C}$.

Proof. Taking any $C_{1},C_{2}\in\mathscr{C},$ it follows that
$\underline{C_{1}}_{\mathscr{C}}\cap\underline{C_{2}}_{\mathscr{C}}=\underline{(C_{1}\cap
C_{2})}_{\mathscr{C}}=C_{1}\cap C_{2}.$ By Proposition 3.12, we have that
$C_{1}\cap C_{2}=\emptyset$ or $C_{1}\cap C_{2}$ is a union of elements of
$\mathscr{C}$ for any $C_{1},C_{2}\in\mathscr{C}$. $\Box$

This proposition shows that the intersection of two elementary elements in a
fuzzy covering $\mathscr{C}$ can be represented as a union of elements of
$\mathscr{C}$ if
$\underline{X}_{\mathscr{C}}\cap\underline{Y}_{\mathscr{C}}=\underline{(X\cap
Y)}_{\mathscr{C}}$ for any $X,Y\subseteq U$.

It is clear that
$\underline{X}_{\mathscr{C}}\subseteq\underline{X}_{Cov(\mathscr{C})}$ and
$\overline{X}_{\mathscr{C}}=\overline{X}_{Cov(\mathscr{C})}$ for any
$X\subseteq U$ in the classical covering approximation space
$(U,\mathscr{C})$. But they do not necessarily hold in the fuzzy covering
approximation space. To illustrate this point, we employ the following
example.

###### Example 3.15

Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$, $\mathscr{C}=\\{C_{21},C_{22},$
$C_{23}\\}$, where
$C_{21}=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0}{x_{4}},$
$C_{22}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0}{x_{4}}$
and
$C_{23}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.5}{x_{4}}.$
According to Definition 3.1, we have that
$C_{x_{1}}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0}{x_{4}},$
$C_{x_{2}}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0}{x_{4}},$
$C_{x_{3}}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0}{x_{4}}$
and
$C_{x_{4}}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.5}{x_{4}}.$
Taking
$X=\frac{0.2}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.6}{x_{3}}+\frac{0}{x_{4}}$,
according to Definition 3.8, it follows that
$\underline{X}_{\mathscr{C}}=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0}{x_{4}}$
and
$\underline{X}_{Cov(\mathscr{C})}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0}{x_{4}}$.
Consequently,
$\underline{X}_{\mathscr{C}}\nsubseteq\underline{X}_{Cov(\mathscr{C})}.$
Similarly, we obtain that
$\overline{X}_{\mathscr{C}}\neq\overline{X}_{Cov(\mathscr{C})}.$

It is well known that the upper approximation can be represented with
neighborhoods in the classical covering approximation space. But we do not
have the same result in the fuzzy covering approximation space.

###### Theorem 3.16

Let $(U,\mathscr{C})$ be a fuzzy covering approximation space. Then
$\bigcup\\{C_{x}|X(x)>0\\}\subseteq\overline{X}_{\mathscr{C}}$ holds for
$X\subseteq U$.

Proof. Taking any $X\subseteq U$, according to Definition 3.8, we see that
$\overline{X}_{\mathscr{C}}=(\bigcup\\{C_{x}|X(x)>0\text{ and
}\underline{X}_{\mathscr{C}}(x)=0\\})\cup\underline{X}_{\mathscr{C}}$ and
$\bigcup\\{C_{x}|X(x)>0\\}=(\bigcup\\{C_{x}|X(x)>0\text{ and
}\underline{X}_{\mathscr{C}}(x)=0\\})\cup(\bigcup\\{C_{x}|\underline{X}_{\mathscr{C}}(x)>0\\})$.
It follows that there exist $C\in\mathscr{C}$ and $C\subseteq X$ for any $x$
satisfying $\underline{X}(x)>0$. Consequently, $C_{x}\subseteq C\subseteq X$.
Hence, $\bigcup\\{C_{x}|\underline{X}_{\mathscr{C}}(x)>0\\}\subseteq X$.
Therefore, $\bigcup\\{C_{x}|X(x)>0\\}\subseteq\overline{X}_{\mathscr{C}}$
holds for $X\subseteq U$. $\Box$

We see that $\overline{X}_{\mathscr{C}}\subseteq\bigcup\\{C_{x}|X(x)>0\\}$
does not necessarily hold for any $X\subseteq U$. So the upper approximation
can not be represented with neighborhoods only in the fuzzy covering
approximation space. To show this point, we give an example below.

###### Example 3.17

Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$, and
$\mathscr{C}=\\{C_{17},C_{18},C_{19},$ $C_{20}\\}$. For
$X=\frac{0.2}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0.2}{x_{4}}$,
it follows that
$\underline{X}_{\mathscr{C}}=\frac{0.2}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0.2}{x_{4}}=\overline{X}_{\mathscr{C}}$.
Furthermore,
$\bigcup\\{C_{x}|X(x)>0\\}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.1}{x_{4}}$.
Obviously, $\overline{X}_{\mathscr{C}}\nsubseteq\bigcup\\{C_{x}|X(x)>0\\}$.

We now investigate the relationship between the lower and upper approximation
operators.

###### Theorem 3.18

Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. If
$\underline{X}_{\mathscr{C}_{1}}=\underline{X}_{\mathscr{C}_{2}}$ for any
$X\subseteq U$, then
$\overline{X}_{\mathscr{C}_{1}}=\overline{X}_{\mathscr{C}_{2}}$.

Proof. By Definition 3.1, we have that
$C_{1x}=\bigcap\\{C_{i}|C_{i}(x)>0,C_{i}\in\mathscr{C}_{1},i\in I\\}$ and
$C_{2x}=\bigcap\\{C_{j}|C_{j}(x)>0,C_{j}\in\mathscr{C}_{2},j\in J\\}$. For any
$C_{i},$ where $i\in I$,
$C_{i}=\underline{C}_{i\mathscr{C}_{1}}=\underline{C}_{i\mathscr{C}_{2}}$. So
there exists at least $C_{j}\in\mathscr{C}_{2}$ such that $C_{j}\subseteq
C_{i}$ and $C_{j}(x)>0$. Hence, $C_{2x}\subseteq C_{1x}$. Similarly, we obtain
that $C_{1x}\subseteq C_{2x}$. Therefore,
$\overline{X}_{\mathscr{C}_{1}}=\overline{X}_{\mathscr{C}_{2}}$. $\Box$

From Theorem 3.18, we see that the lower and upper approximation operations
are not independent in the fuzzy covering approximation space. Concretely, the
lower approximation operation dominates the upper one.

###### Theorem 3.19

Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. Then
$\underline{C}_{\mathscr{C}_{1}}=\underline{C}_{\mathscr{C}_{2}}$ holds for
any $C\in\mathscr{C}_{1}\cup\mathscr{C}_{2}$ if and only if
$\underline{X}_{\mathscr{C}_{1}}=\underline{X}_{\mathscr{C}_{2}}$ for any
$X\subseteq U$.

Proof. Taking any $X\subseteq U$, by Definition 3.8, it follows that
$\underline{X}_{\mathscr{C}_{1}}=\bigcup\\{C_{i}|C_{i}\subseteq
X,C_{i}\in\mathscr{C}_{1},i\in I\\}$. For any
$C_{i}\subseteq\underline{X}_{\mathscr{C}_{1}}$, we have that
$C_{i}=\underline{C}_{i\mathscr{C}_{1}}=\underline{C}_{i\mathscr{C}_{2}}=\bigcup\\{C_{ij}|C_{ij}\in\mathscr{C}_{2},C_{ij}\subseteq
X,i\in I,j\in J\\}$. It implies that
$\underline{X}_{\mathscr{C}_{1}}\subseteq\underline{X}_{\mathscr{C}_{2}}$.
Analogously, it follows that
$\underline{X}_{\mathscr{C}_{2}}\subseteq\underline{X}_{\mathscr{C}_{1}}$.
Thereby, $\underline{X}_{\mathscr{C}_{1}}=\underline{X}_{\mathscr{C}_{2}}$ for
any $X\subseteq U$.

The converse is obvious by Definition 3.8. $\Box$

This result indicates that each elementary set in a fuzzy covering is
definable in the other fuzzy covering if and only if two fuzzy coverings of a
universe give the same lower approximations.

###  3.2 The fuzzy subcovering and its properties

It is well-known that the classical upper approximation based on neighborhoods
can be defined equivalently by using a family of subcoverings. In this
subsection, we propose the notion of fuzzy subcoverings and investigate the
relationship between the upper approximation based on neighborhoods and
subcoverings in the fuzzy covering approximation space.

###### Definition 3.20

Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, $X\subseteq U$,
and $\mathscr{C}^{\prime}\subseteq\mathscr{C}$. If
$X\subseteq\bigcup\\{C|C\in\mathscr{C}^{\prime}\\}$ , then
$\mathscr{C}^{\prime}$ is called a fuzzy subcovering of $X$.

In other words, the fuzzy subcovering of $X$ is a subset of $\mathscr{C}$
which covers $X$. Obviously, $\mathscr{C}$ is the maximum fuzzy subcovering
for $X\subseteq U$ if $X\subseteq\bigcup\mathscr{C}$. In this work, we denote
the set of all the fuzzy subcoverings of $X$ as $FC(X)$.

###### Theorem 3.21

Let $(U,\mathscr{C})$ be a fuzzy covering approximation space. Then
$\overline{X}_{\mathscr{C}}\subseteq\bigcap\\{\bigcup\\{C|C\in\mathscr{C}^{\prime}\\}|\mathscr{C}^{\prime}\in
FC(X)\\}$ holds for any $X\subseteq U$.

Proof. Taking any $X\subseteq U$, by Definition 3.8, it follows that
$\overline{X}_{\mathscr{C}}=(\bigcup\\{C_{x}|X(x)>0\text{ and
}\underline{X}_{\mathscr{C}}(x)=0\\})\cup\underline{X}_{\mathscr{C}}$. By
Proposition 3.9, it implies that $\underline{X}\subseteq
X\subseteq\bigcup\\{C|C\in\mathscr{C}^{\prime}\in FC(X)\\}$. Evidently,
$C_{x}\subseteq\bigcup\\{C|C\in\mathscr{C}^{\prime}\in FC(X)\\}$ for any $x\in
U$ satisfying $X(x)>0$. Thereby,
$\overline{X}_{\mathscr{C}}\subseteq\bigcap\\{\bigcup\\{C|C\in\mathscr{C}^{\prime}\\}|\mathscr{C}^{\prime}\in
FC(X)\\}$ holds for any $X\subseteq U$. $\Box$

However,
$\bigcap\\{\bigcup\\{C|C\in\mathscr{C}^{\prime}\\}|\mathscr{C}^{\prime}\in
FC(X)\\}\subseteq\overline{X}_{\mathscr{C}}$ does not hold generally. That is,
the upper approximation may not be represented with a family of fuzzy
subcoverings of $X$ as the classical covering approximation space, which is
shown by the following example.

###### Example 3.22

Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$,
$\mathscr{C}=\\{C_{24},C_{25},C_{26}\\}$, where
$C_{24}=\frac{0.2}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0.1}{x_{4}},$
$C_{25}=\frac{0.1}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.2}{x_{4}}$
and
$C_{26}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0.1}{x_{4}}$.
According to Definition 3.20, we obtain all fuzzy subcoverings of
$X=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0}{x_{4}}$ as
$FC(X)=\\{\\{C_{24}\\},\\{C_{26}\\},\\{C_{24},C_{25}\\},\\{C_{24},C_{26}\\},$
$\\{C_{25},C_{26}\\},$ $\\{C_{24},C_{25},C_{26}\\}\\}$. It follows that
$\bigcap\\{\bigcup\\{C|C\in\mathscr{C}^{\prime}\\}|\mathscr{C}^{\prime}\in
FC(X)\\}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0.1}{x_{4}}$,
but
$\overline{X}_{\mathscr{C}}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.1}{x_{4}}$.
Therefore,
$\bigcap\\{\bigcup\\{C|C\in\mathscr{C}^{\prime}\\}|\mathscr{C}^{\prime}\in
FC(X)\\}\nsubseteq\overline{X}_{\mathscr{C}}$.

According to Theorems 3.16 and 3.21, we have that
$\bigcup\\{C_{x}|X(x)>0\\}\subseteq\overline{X}_{\mathscr{C}}\subseteq\bigcap\\{\bigcup\\{C|C\in\mathscr{C}^{\prime}\\}|\mathscr{C}^{\prime}\in
FC(X)\\}$ for any $X\subseteq U$.

Sometimes, the fuzzy covering $\mathscr{C}$ of $U$ is a trivial subcovering of
$X\subseteq U$. Specially, we do not take $\mathscr{C}$ into account in the
following situation.

###### Proposition 3.23

Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $X\subseteq
U$. If $|FC(X)|\geq 2$, where $|FC(X)|$ stands for the cardinality of $FC(X)$,
then
$\bigcap\\{\bigcup\\{C|C\in\mathscr{C}^{\prime}\\}|\mathscr{C}^{\prime}\in
FC(X)\\}=\bigcap\\{\bigcup\\{C|C\in\mathscr{C}^{\prime}\\}|\mathscr{C}^{\prime}\in
FC(X)-\\{\mathscr{C}\\}\\}$.

Proof. Straightforward. $\Box$

###  3.3  The irreducible and reducible elements of a fuzzy covering

In this subsection, we provide the concepts of reducible and irreducible
elements to formally investigate the relationship among elementary elements of
a fuzzy covering. Although several theorems in this subsection are special
cases of [33], they don’t give their proofs. To better understand the
following results, we prove them concretely in the following.

###### Definition 3.24

[33] Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and
$C\in\mathscr{C}$. If $C$ can not be written as a union of some sets in
$\mathscr{C}-\\{C\\}$, then $C$ is called an irreducible element. Otherwise,
$C$ is called a reducible element.

It is obvious that the concept of the irreducible element in a fuzzy covering
approximation space is an extension of the notion of the irreducible element
in a covering approximation space, and the irreducible element can be used for
the definition of reducts of fuzzy coverings.

###### Proposition 3.25

[33] Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $C$ a
reducible element of $\mathscr{C}$. Then $\mathscr{C}-\\{C\\}$ is still a
fuzzy covering of $U$.

In other words, a fuzzy covering of a universe deleting all reducible elements
is a fuzzy covering, and the rest elements are irreducible.

###### Definition 3.26

[33] Let $(U,\mathscr{C})$ be a fuzzy covering approximation space. If every
element of $\mathscr{C}$ is an irreducible element, then $\mathscr{C}$ is
irreducible. Otherwise, $\mathscr{C}$ is reducible.

Next, we discuss the properties of reducible elements of a fuzzy covering.

###### Theorem 3.27

[33] Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, $C$ a
reducible element of $\mathscr{C}$, and $C_{0}\in\mathscr{C}-\\{C\\}$. Then
$C_{0}$ is a reducible element of $\mathscr{C}$ if and only if it is a
reducible element of $\mathscr{C}-\\{C\\}$.

Proof. We assume that $C_{0}$ is a reducible element of $\mathscr{C}$. It
follows that we can express $C_{0}$ as a union of subset of
$\mathscr{C}-\\{C_{0}\\}$, denoted as $C_{1},C_{2},...,C_{N}$. If there exists
no set which is equal to $C$ in $\\{C_{1},C_{2},...,C_{N}\\}$, then $C_{0}$ is
a reducible element of $\mathscr{C}-\\{C\\}$. If there is a set which is equal
to $C$ in $\\{C_{1},C_{2},...,C_{N}\\}$, taking $C_{1}=C$, then $C_{1}$ is the
union of some sets $\\{D_{1},D_{2},...,D_{M}\\}$ in $\mathscr{C}-\\{C\\}$.
Consequently, we obtain that $C_{0}=D_{1}\cup D_{2}\cup...D_{M}\cup
C_{2}\cup...\cup C_{N}$. Clearly, $D_{1},D_{2},...,D_{M},C_{2},...,C_{N}$ are
not equal to either $C_{0}$ or $C$. So $C$ is a reducible element of
$\mathscr{C}-\\{C\\}$.

Since $C_{0}$ is a reducible element of $\mathscr{C}-\\{C\\}$, it can be
expressed as a union of some sets in $\mathscr{C}-\\{C,C_{0}\\}$. We can
express it as a union of some sets in $\mathscr{C}-\\{C_{0}\\}$. Therefore,
$C_{0}$ is a reducible element of $\mathscr{C}$. $\Box$

Next, we investigate the relationship between the approximation operations and
the reducible elements in the fuzzy covering approximation space.

###### Theorem 3.28

[33] Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $C$ a
reducible element of $\mathscr{C}$. Then
$\underline{X}_{\mathscr{C}}=\underline{X}_{\mathscr{C}-\\{C\\}}$ holds for
any $X\subseteq U$.

Proof. Taking any $X\subseteq U$, by Definition 3.24, it follows that
$\underline{X}_{\mathscr{C}-\\{C\\}}\subseteq\underline{X}_{\mathscr{C}}\subseteq
X$. Moreover, there exist $C_{1},C_{2},...,C_{N}$ such that
$\underline{X}_{\mathscr{C}}=C_{1}\cup C_{2}\cup...\cup C_{N}$. If none of
$C_{1},C_{2},...,C_{N}$ is equal to $C$, then they belong to
$\mathscr{C}-\\{C\\}$. Consequently, $C_{1},C_{2},...,C_{N}$ are all the
subsets of $\underline{X}_{\mathscr{C}-\\{C\\}}$. If there is a set which is
equal to $C$, then we take $C=C_{1}$. Since $C$ is a reducible element of
$\mathscr{C}$, $C$ can be expressed as some sets in $\mathscr{C}-\\{C\\}$ such
that $C=D_{1}\cup D_{2}\cup...\cup D_{M}$. Hence,
$\underline{X}_{\mathscr{C}}=D_{1}\cup D_{2}\cup...\cup D_{M}\cup
C_{2}\cup...\cup C_{N}$. It implies that
$\underline{X}_{\mathscr{C}}\subseteq\underline{X}_{\mathscr{C}-\\{C\\}}$.
Therefore, $\underline{X}_{\mathscr{C}}=\underline{X}_{\mathscr{C}-\\{C\\}}$
holds for any $X\subseteq U$. $\Box$

In other words, the lower approximation of any $X\subseteq U_{1}$ in
$\mathscr{C}$ is the same as that in $\mathscr{C}-\\{C\\}$ if $C$ is
reducible.

###### Corollary 3.29

[33] Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $C$ a
reducible element of $\mathscr{C}$. Then
$\overline{X}_{\mathscr{C}}=\overline{X}_{\mathscr{C}-\\{C\\}}$ holds for any
$X\subseteq U$.

Proof. Straightforward from Theorems 3.27 and 3.28. $\Box$

In this sequel, we use $RED(\mathscr{C})$ to represent the set of all
irreducible elements of a fuzzy covering $\mathscr{C}$. It is easy to see that
$RED(\mathscr{C})=RED(RED(\mathscr{C}))$. Next, we study the relationship
between $RED(\mathscr{C})$ and the lower and upper approximation operations.

###### Corollary 3.30

Let $(U,\mathscr{C})$ be a fuzzy covering approximation space. Then
$\underline{X}_{\mathscr{C}}=\underline{X}_{RED(\mathscr{C})}$ holds for any
$X\subseteq U$.

Proof. Straightforward from Theorem 3.28. $\Box$

###### Corollary 3.31

Let $(U,\mathscr{C})$ be a fuzzy covering approximation space. Then
$\overline{X}_{\mathscr{C}}=\overline{X}_{RED(\mathscr{C})}$ holds for any
$X\subseteq U$.

Proof. Straightforward from Corollary 3.29. $\Box$

Based on Theorem 3.28, Corollaries 3.29, 3.30 and 3.31, we obtain the
following theorem.

###### Theorem 3.32

Let $U$ be a universe, $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ two irreducible
fuzzy coverings of $U$. If
$\underline{X}_{\mathscr{C}_{1}}=\underline{X}_{\mathscr{C}_{2}}$ for any
$X\subseteq U$. then the two fuzzy coverings are the same.

Proof. Taking any $C\in\mathscr{C}_{1}$, by Definition 3.8, it follows that
$\underline{C}_{\mathscr{C}_{1}}=C=\underline{C}_{\mathscr{C}_{2}}$.
Consequently, $C$ is the union of some sets of $\mathscr{C}_{2}$ such that
$C=C_{1}\cup C_{2}\cup...\cup C_{N}$. Similarly, there exist
$D_{i1},D_{i2},...,D_{iM(i)}\in\mathscr{C}_{1}$ such that $C_{i}=D_{i1}\cup
D_{i2}\cup...\cup D_{iM(i)}.$ Hence, $C=D_{11}\cup D_{12}\cup...\cup
D_{N1}\cup D_{N2}\cup...\cup D_{NM(N)}$. Since $C$ is irreducible, $C=D_{ij}$
for all $i,j$. It implies that $C$ is an element of $\mathscr{C}_{2}$. On the
other hand, any element of $\mathscr{C}_{2}$ is an element of
$\mathscr{C}_{1}$. Therefore, the two fuzzy coverings $\mathscr{C}_{1}$ and
$\mathscr{C}_{2}$ are the same. $\Box$

###### Corollary 3.33

Let $U$ be a universe, $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ two irreducible
fuzzy coverings of $U$. If
$\overline{X}_{\mathscr{C}_{1}}=\overline{X}_{\mathscr{C}_{2}}$ for any
$X\subseteq U$, then the two fuzzy coverings are the same.

Proof. The proof is similar to that in Theorem 3.32. $\Box$

###### Theorem 3.34

Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. Then
$\underline{X}_{\mathscr{C}_{1}}=\underline{X}_{\mathscr{C}_{2}}$ holds for
any $X\subseteq U$ if and only if $RED(\mathscr{C}_{1})=RED(\mathscr{C}_{2})$.

Proof. Since $\underline{X}_{\mathscr{C}_{1}}=\underline{X}_{\mathscr{C}_{2}}$
for any $X\subseteq U$,
$\underline{C}_{\mathscr{C}_{1}}=\underline{C}_{\mathscr{C}_{2}}$ for any
$C\in\mathscr{C}_{1}\cup\mathscr{C}_{2}$. Taking any $C\in
RED(\mathscr{C}_{1})$, it follows that $C=\bigcup\\{C_{i}|C_{i}\in
RED(\mathscr{C}_{2}),i\in
I\\}=\bigcup\\{\bigcup\\{C_{ij}|C_{ij}\in\mathscr{C}_{1},i\in I\\}|j\in J\\}$.
It implies that $C\in\mathscr{C}_{2}$. Hence, $RED(\mathscr{C}_{1})\subseteq
RED(\mathscr{C}_{2})$. Similarly, we obtain that
$RED(\mathscr{C}_{2})\subseteq RED(\mathscr{C}_{1})$. Therefore,
$RED(\mathscr{C}_{1})=RED(\mathscr{C}_{2})$.

The converse is obvious by Definitions 3.8 and 3.24. $\Box$

It can be seen from Theorem 3.34 that two fuzzy coverings of a universe
generate the same lower approximation if and only if there exist the same
irreducible elements in these fuzzy coverings.

To illustrate Theorem 3.34, we supply the following example.

###### Example 3.35

Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$, $\mathscr{C}_{1}=\\{C_{17},C_{18},$
$C_{19},C_{20},C_{27},C_{28}\\}$,
$\mathscr{C}_{2}=\\{C_{17},C_{18},C_{19},C_{20},C_{29},$ $C_{30}\\}$, where
$C_{27}=\frac{0.2}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.1}{x_{4}},$
$C_{28}=\frac{0.2}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0.1}{x_{4}},$
$C_{29}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0.2}{x_{4}}$
and
$C_{30}=\frac{0.1}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.2}{x_{4}}.$
Obviously, we obtain that
$RED(\mathscr{C}_{1})=RED(\mathscr{C}_{2})=\\{C_{17},C_{18},C_{19},C_{20}\\}$.

###### Corollary 3.36

Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. If
$\overline{X}_{\mathscr{C}_{1}}=\overline{X}_{\mathscr{C}_{2}}$ for any
$X\subseteq U$ if and only if $RED(\mathscr{C}_{1})=RED(\mathscr{C}_{2})$.

From Corollary 3.36, we see that they have the same irreducible elements if
and only if two fuzzy coverings of a universe generate the same upper
approximation.

###### Corollary 3.37

Let $(U,\mathscr{C}_{1})$ be a fuzzy covering approximation space,
$\mathscr{C}_{2}=\\{\bigcup_{C\in\mathscr{C}^{\prime}}C|\emptyset\neq\mathscr{C}^{\prime}\subseteq\mathscr{C}_{1}\\}$,
and $X\subseteq U$. Then
$\underline{X}_{\mathscr{C}_{1}}=\underline{X}_{\mathscr{C}_{2}}$ and
$\overline{X}_{\mathscr{C}_{1}}=\overline{X}_{\mathscr{C}_{2}}$.

Proof. By Definition 3.24, we observe that
$RED(\mathscr{C}_{1})=RED(\mathscr{C}_{2})$. Therefore,
$\underline{X}_{\mathscr{C}_{1}}=\underline{X}_{\mathscr{C}_{2}}$ and
$\overline{X}_{\mathscr{C}_{1}}=\overline{X}_{\mathscr{C}_{2}}$. $\Box$

We also investigate the relationship between the reducible elements and the
neighborhood operator in the fuzzy covering approximation space.

###### Theorem 3.38

Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $C$ a
reducible element of $\mathscr{C}$. Then $C_{x}$ in $\mathscr{C}-\\{C\\}$ is
the same as that in $\mathscr{C}$ for any $x\in U$.

Proof. By Definitions 3.1 and 3.24, we have that
$C_{x}=\bigcap\\{C_{i}|C_{i}(x)>0,C_{i}\in\mathscr{C}\\}=\bigcap\\{C_{i}|C_{i}(x)>0,C_{i}\in\mathscr{C}-\\{C\\}\\}$
for any $x\in U$. Therefore, $C_{x}$ in $\mathscr{C}-\\{C\\}$ is the same as
that in $\mathscr{C}$ for any $x\in U$. $\Box$

That is to say, if we delete some reducible elements in the fuzzy covering,
then it will not change the neighborhood $C_{x}$ for any $x\in U$.

###### Corollary 3.39

Let $(U,\mathscr{C})$ be a fuzzy covering approximation space. Then $C_{x}$ in
$RED(\mathscr{C})$ is the same as that in $\mathscr{C}$ for any $x\in U$.

Proof. Straightforward from Theorem 3.38. $\Box$

Corollary 3.39 indicates that $RED(\mathscr{C})$ and $\mathscr{C}$ generate
the same neighborhood $C_{x}$ for any $x\in U$ in the fuzzy covering
approximation space.

###### Corollary 3.40

Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. If
$RED(\mathscr{C}_{1})=RED(\mathscr{C}_{2})$, then $C_{x}$ in $\mathscr{C}_{1}$
is the same as that in $\mathscr{C}_{2}$ for any $x\in U$.

Proof. Straightforward from Corollary 3.39. $\Box$

By Corollary 3.40, if there exist the same irreducible elements in two fuzzy
coverings $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ of $U$, then they generate
the same neighborhood $C_{x}$ for any $x\in U$.

###### Theorem 3.41

Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. If
$\overline{C}_{\mathscr{C}_{1}}=\overline{C}_{\mathscr{C}_{2}}$ for any
$C\in\mathscr{C}_{1}\cup\mathscr{C}_{2}$, then
$\bigcup\\{C_{1x}|X(x)>0\\}=\bigcup\\{C_{2x}|X(x)>0\\}$ for any $X\subseteq
U$.

Proof. By Definition 3.1, we have that
$C_{1x}=\bigcap\\{C_{i}|C_{i}(x)>0,C_{i}\in\mathscr{C}_{1},i\in I\\}$ and
$C_{2x}=\bigcap\\{C_{j}|C_{j}(x)>0,C_{j}\in\mathscr{C}_{2},j\in J\\}$ for any
$x\in U$. Assume that there exists $x\in U$ such that $X(x)>0$ and $C_{1x}\neq
C_{2x}$. Without loss of generality, there is $y\in U$ such that
$(C_{1x})(y)>0$ and $(C_{2x})(y)=0$. Obviously, $y\neq x$. Hence, there exist
$C_{j}(y)=0$ and $C_{j}(x)>0$. But
$C_{j}=\overline{C}_{j\mathscr{C}_{2}}=\overline{C}_{j\mathscr{C}_{1}}\supseteq\bigcup\\{C_{1z}|C_{j}(z)>0\\}\supseteq
C_{1x}$. It implies that $C_{j}(y)>0$, which is a contradiction. Consequently,
$C_{1x}=C_{2x}$ for any $x\in U$. Therefore,
$\bigcup\\{C_{1x}|X(x)>0\\}=\bigcup\\{C_{2x}|X(x)>0\\}$ for any $X\subseteq
U$. $\Box$

Theorem 3.41 shows that two fuzzy coverings of a universe generate the same
neighborhood $C_{x}$ for any $x\in U$ if each elementary element has the same
lower approximation in two fuzzy coverings.

###  3.4 The non-intersectional and intersectional elements of fuzzy
coverings, the union and intersection operations on fuzzy coverings

For any universal set $U$, we denote $CC(U)$ as the set of all coverings of
$U$. It is well-known that the number of possible coverings for a set $U$ of
$n$ elements is

$|CC(U)|=\frac{1}{2}\sum^{n}_{k=0}(\frac{n}{k})2^{2^{n-k}},$

the first few of which are 1, 5, 109, 32297, 2147321017. Since $C(U)$ contains
a larger number of fuzzy coverings than $CC(U)$ in practice, it is of interest
to investigate the relationship between fuzzy coverings. In this subsection,
we introduce several operations on fuzzy coverings and study their basic
properties for facilitating the computation of fuzzy coverings.

###### Definition 3.42

Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and
$C\in\mathscr{C}$. If $C$ can not be written as an intersection of some sets
in $\mathscr{C}-\\{C\\}$, then $C$ is called a non-intersectional element.
Otherwise, $C$ is called an intersectional element.

For simplicity, we use $IS(\mathscr{C})$ to represent the set of all non-
intersectional elements of $\mathscr{C}$. It is easy to see that
$IS(\mathscr{C})=IS(IS(\mathscr{C}))$. Notice that the function
$IS:C(U)\longrightarrow C(U)$ that maps $\mathscr{C}$ to $IS(\mathscr{C})$ is
well-defined. Hence, we may view $IS$ as a unary operator on $C(U)$.

We employ an example to illustrate the non-intersectional and intersectional
elements in the following.

###### Example 3.43

Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$,
$\mathscr{C}=\\{C_{17},C_{18},C_{19},C_{20},C_{31}\\}$, where
$C_{31}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.1}{x_{4}}$.
By Definition 3.42, we have that
$IS(\mathscr{C})=\\{C_{17},C_{18},C_{19},C_{20}\\}$.

###### Proposition 3.44

Let $(U,\mathscr{C})$ be a fuzzy covering approximation space. Then
$\bigcup\\{C_{x}|X(x)>0\\}=\bigcup\\{C_{IS(\mathscr{C})x}|$ $X(x)>0\\}$ and
$\underline{X}_{IS(\mathscr{C})}\subseteq\underline{X}_{\mathscr{C}}$ for any
$X\subseteq U$.

Proof. By Definition 3.42, it follows that $C_{x}=C_{IS(\mathscr{C})x}$ for
any $x\in U$. Consequently,
$\bigcup\\{C_{x}|X(x)>0\\}=\bigcup\\{C_{IS(\mathscr{C})x}|X(x)>0\\}$ for any
$X\subseteq U$. Furthermore, since $IS(\mathscr{C})\subseteq\mathscr{C}$, we
have that
$\underline{X}_{IS(\mathscr{C})}\subseteq\underline{X}_{\mathscr{C}}$ for any
$X\subseteq U$. $\Box$

We observe that the neighborhood $C_{x}$ generated in the fuzzy covering
$\mathscr{C}$ is the same as that generated in all non-intersectional elements
of $\mathscr{C}$. On the other hand,
$\overline{X}_{\mathscr{C}}\subseteq\overline{X}_{IS(\mathscr{C})}$ does not
necessarily hold for any $X\subseteq U$. An example is given to illustrate
this point.

###### Example 3.45

Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$, $\mathscr{C}=\\{C_{32},C_{33},$
$C_{34},C_{35},C_{36},C_{37},C_{38},C_{39}\\}$, where
$C_{32}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0}{x_{3}}+\frac{0}{x_{4}},$
$C_{33}=\frac{0}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0}{x_{3}}+\frac{0}{x_{4}},$
$C_{34}=\frac{0.1}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0}{x_{3}}+\frac{0}{x_{4}},$
$C_{35}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0}{x_{4}},$
$C_{36}=\frac{0.4}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0}{x_{4}},$
$C_{37}=\frac{0.1}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0}{x_{3}}+\frac{0.1}{x_{4}},$
$C_{38}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.1}{x_{3}}+\frac{0.5}{x_{4}}$
and
$C_{39}=\frac{0}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.4}{x_{4}}.$
Evidently, $IS(\mathscr{C})=\\{C_{36},C_{37},C_{38},C_{39}\\}$. Taking
$X=\frac{0.4}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0}{x_{3}}+\frac{0}{x_{4}},$
according to Definitions 3.8 and 3.42, we obtain that
$\overline{X}_{\mathscr{C}}=\frac{0.1}{x_{1}}+\frac{0.2}{x_{2}}+\frac{0}{x_{3}}+\frac{0}{x_{4}}$
and
$\overline{X}_{IS(\mathscr{C})}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0}{x_{3}}+\frac{0}{x_{4}}.$
Thereby, $\overline{X}_{\mathscr{C}}\nsubseteq\overline{X}_{IS(\mathscr{C})}$.

Following, we present a theorem for the intersection element of a fuzzy
covering.

###### Theorem 3.46

Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, $C$ an
intersection element of $\mathscr{C}$, and $C_{0}\in\mathscr{C}-\\{C\\}$. Then
$C_{0}$ is an intersection element of $\mathscr{C}$ if and only if it is an
intersection element of $\mathscr{C}-\\{C\\}$.

Proof. The proof is similar to that in Theorem 3.27. $\Box$

Next, we present the notions of the union and intersection operations on fuzzy
coverings, and investigate their basic properties.

###### Definition 3.47

Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. If

$\mathscr{C}_{1}\cup\mathscr{C}_{2}=\\{C|C\in\mathscr{C}_{1}\text{ or
}C\in\mathscr{C}_{2}\\},$

then $\mathscr{C}_{1}\cup\mathscr{C}_{2}$ is called the union of
$\mathscr{C}_{1}$ and $\mathscr{C}_{2}$.

It is obvious that the union operation is to collect all elementary elements
in each fuzzy covering.

###### Proposition 3.48

Let $U$ be a non-empty universe of discourse,
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$, and $X\subseteq U$. Then
$\underline{X}_{\mathscr{C}_{i}}\subseteq\underline{X}_{\mathscr{C}_{1}\bigcup\mathscr{C}_{2}}$,
where $i=1,2.$

Proof. According to Definition 3.8, we have that
$\underline{X}_{\mathscr{C}_{i}}=\bigcup\\{C\in\mathscr{C}_{i}|C\subseteq
X\\}\subseteq(\bigcup\\{C\in\mathscr{C}_{1}|C\subseteq
X\\})\bigcup(\bigcup\\{C\in\mathscr{C}_{2}|C\subseteq
X\\})=\underline{X}_{\mathscr{C}_{1}\bigcup\mathscr{C}_{2}}$. Thereby,
$\underline{X}_{\mathscr{C}_{i}}\subseteq\underline{X}_{\mathscr{C}_{1}\bigcup\mathscr{C}_{2}}$,
where $i=1,2.$ $\Box$

The following example shows that the converse of Proposition 3.48 does not
hold generally.

###### Example 3.49

Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$, $\mathscr{C}_{1}=\\{C_{21},C_{22},$
$C_{23}\\}$, and
$\mathscr{C}_{2}=\\{C_{1x_{1}},C_{1x_{2}},C_{1x_{3}},C_{1x_{4}}\\}$. Taking
$X=\frac{0.2}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.6}{x_{3}}+\frac{0.1}{x_{4}}$.
By Definition 3.8, we have that
$\overline{X}_{\mathscr{C}_{1}}=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$,
$\overline{X}_{\mathscr{C}_{2}}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.5}{x_{4}}$,
$\underline{X}_{\mathscr{C}_{1}\bigcup\mathscr{C}_{2}}=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0}{x_{4}}$
and
$\overline{X}_{\mathscr{C}_{1}\bigcup\mathscr{C}_{2}}=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$.
Obviously,
$\overline{X}_{\mathscr{C}_{1}\bigcup\mathscr{C}_{2}}\nsubseteq\overline{X}_{\mathscr{C}_{2}}$.

###### Definition 3.50

Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. If

$\mathscr{C}_{1}\cap\mathscr{C}_{2}=\\{C_{1x}\cap C_{2x}|C_{ix}\in
Cov(\mathscr{C}_{i}),x\in U_{1},i=1,2\\},$

then $\mathscr{C}_{1}\cap\mathscr{C}_{2}$ is called the intersection of
$\mathscr{C}_{1}$ and $\mathscr{C}_{2}$.

It is obvious that $C_{1x}=\bigcap\\{C|C(x)>0,C\in\mathscr{C}_{1}\\}$ and
$C_{2x}=\bigcap\\{C^{\prime}|C^{\prime}(x)>0,C^{\prime}\in\mathscr{C}_{2}\\}$
for any $x\in U_{1}$. So $\mathscr{C}_{1}\bigcap\mathscr{C}_{2}$ is a fuzzy
covering of $U_{1}$. Furthermore, if we take the value of membership degree
from the set $\\{0,1\\}$, then Definition 3.50 is the same as that in
Definition 4.2 in [24].

It can be found that
$\overline{X}_{\mathscr{C}_{i}}\subseteq\overline{X}_{\mathscr{C}_{1}\bigcap\mathscr{C}_{2}}$
does not necessarily hold for any $X\subseteq U$ in the fuzzy covering
approximation space, where $i=1,2$. To illustrate this point, we give the
following example.

###### Example 3.51

Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$, $\mathscr{C}_{1}=\\{C_{21},C_{22},$
$C_{23}\\}$, and
$\mathscr{C}_{2}=\\{\frac{0.2}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.5}{x_{4}}\\}$.
According to Definition 3.1, we have that
$C_{1x_{1}}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0}{x_{4}}$,
$C_{1x_{2}}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0}{x_{4}}$,
$C_{1x_{3}}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.2}{x_{3}}+\frac{0}{x_{4}}$
and
$C_{1x_{4}}=\frac{0.1}{x_{1}}+\frac{0}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.5}{x_{4}}$.
Taking
$X=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$,
it implies that
$\underline{X}_{\mathscr{C}_{1}\bigcap\mathscr{C}_{2}}=\frac{0.1}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.5}{x_{4}}$,
$\underline{X}_{\mathscr{C}_{1}}=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.5}{x_{4}}$
and
$\underline{X}_{\mathscr{C}_{2}}=\frac{0.2}{x_{1}}+\frac{0.1}{x_{2}}+\frac{0.4}{x_{3}}+\frac{0.5}{x_{4}}$.
Clearly,
$\overline{X}_{\mathscr{C}_{1}}\nsubseteq\overline{X}_{\mathscr{C}_{1}\bigcap\mathscr{C}_{2}}$
and
$\overline{X}_{\mathscr{C}_{2}}\nsubseteq\overline{X}_{\mathscr{C}_{1}\bigcap\mathscr{C}_{2}}$.

By Definitions 3.4 and 3.50, we present the following proposition.

###### Proposition 3.52

Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. Then
$\mathscr{C}_{1}\bigcap\mathscr{C}_{2}=Cov(\mathscr{C}_{1}\bigcup\mathscr{C}_{2})$.

###### Definition 3.53

Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$. If there exists
$C^{\ast}\in\mathscr{C}_{2}$ such that $C\subseteq C^{\ast}$ for any
$C\in\mathscr{C}_{1}$, then $\mathscr{C}_{2}$ is said to be coarser than
$\mathscr{C}_{1}$, denoted as $\mathscr{C}_{1}\leq\mathscr{C}_{2}$.

In other words, there exists $C^{\ast}\in\mathscr{C}_{2}$ such that
$C^{\ast}(x)\leq C(x)$ for each $C\in\mathscr{C}_{1}$ and $x\in U$ if
$\mathscr{C}_{1}\leq\mathscr{C}_{2}$.

###### Example 3.54

Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$, $\mathscr{C}_{1}=\\{C_{21},C_{22},$
$C_{23}\\}$, and $\mathscr{C}_{2}=\\{C_{40},C_{41}\\}$, where
$C_{40}=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0}{x_{4}}$
and
$C_{2}=\frac{0.3}{x_{1}}+\frac{0}{x_{2}}+\frac{0.6}{x_{3}}+\frac{0.5}{x_{4}}$.
It is obvious that $\mathscr{C}_{2}$ is coarser than $\mathscr{C}_{1}$.

###### Proposition 3.55

Let $U$ be a non-empty universe of discourse, and
$\mathscr{C}_{1},\mathscr{C}_{2},\mathscr{C}_{3}\in C(U)$. Then

$(1)$ $\mathscr{C}_{1}\cup\mathscr{C}_{1}=\mathscr{C}_{1};$

$(2)$ $\mathscr{C}_{1}\cap\mathscr{C}_{1}\leq\mathscr{C}_{1};$

$(3)$ $\mathscr{C}_{1}\cup\mathscr{C}_{2}=\mathscr{C}_{2}\cup\mathscr{C}_{1};$

$(4)$ $\mathscr{C}_{1}\cap\mathscr{C}_{2}=\mathscr{C}_{2}\cap\mathscr{C}_{1};$

$(5)$
$(\mathscr{C}_{1}\cup\mathscr{C}_{2})\cup\mathscr{C}_{3}=\mathscr{C}_{1}\cup(\mathscr{C}_{2}\cup\mathscr{C}_{3});$

$(6)$
$(\mathscr{C}_{1}\cap\mathscr{C}_{2})\cap\mathscr{C}_{3}=\mathscr{C}_{1}\cap(\mathscr{C}_{2}\cap\mathscr{C}_{3});$

$(7)$
$\mathscr{C}_{1}\cup(\mathscr{C}_{1}\cap\mathscr{C}_{2})\leq\mathscr{C}_{1};$

$(8)$
$\mathscr{C}_{1}\cap(\mathscr{C}_{1}\cup\mathscr{C}_{2})\leq\mathscr{C}_{1}.$

Proof. Straightforward from Definitions 3.47, 3.50 and 3.53. $\Box$

###### Proposition 3.56

Let $U$ be a non-empty universe of discourse, and $C(U)$ the set of all fuzzy
coverings of $U$. Then $(C(U),\cap,\cup)$ is a lattice.

Proof. Given any $\mathscr{C}_{1},\mathscr{C}_{2}\in C(U)$, it is obvious that
$\mathscr{C}_{1}\cup\mathscr{C}_{2}\in C(U)$ and
$\mathscr{C}_{1}\cap\mathscr{C}_{2}\in C(U)$. Therefore, $(C(U),\cap,\cup)$ is
a lattice. $\Box$

We notice that $\\{\frac{1}{x_{1}}+\frac{1}{x_{2}}+...+\frac{1}{x_{n}}\\}$ is
the greatest element of $(C(U),\cap,\cup)$, but $(C(U),\cap,\cup)$ is not a
complete lattice necessarily, which is illustrated by the following example.

###### Example 3.57

Let $U=\\{x_{1},x_{2},x_{3}\\}$,
$C_{0}(U)=\\{\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{n},...\\}\subseteq
C(U)$, and
$\mathscr{C}_{n}=\\{\frac{\frac{1}{n}}{x_{1}}+\frac{\frac{1}{n}}{x_{2}}+\frac{\frac{1}{n}}{x_{3}}\\}$.
By Definition 3.50, it follows that $\bigcap
C_{0}(U)=\\{\frac{0}{x_{1}}+\frac{0}{x_{2}}+\frac{0}{x_{3}}\\}$. It is obvious
that $\bigcap C_{0}(U)\notin C(U)$. Consequently, $(C(U),\leq)$ is not an
intersection structure.

###### Proposition 3.58

Let $U$ be a non-empty universe of discourse. Then
$(U,C(U)\cup\\{\emptyset\\})$ is a topological space.

Proof. Straightforward from Definitions 3.47 and 3.50. $\Box$

At the end of this subsection, we provide two roughness measures of fuzzy sets
as follows.

###### Definition 3.59

Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $X\subseteq
U$. Then the roughness measure $\mu_{\mathscr{C}}(X)$ regarding $\mathscr{C}$
is defined as

$\displaystyle\mu_{\mathscr{C}}(X)=1-\frac{|\underline{X}_{\mathscr{C}}|}{|\overline{X}_{\mathscr{C}}|},$

where $|\underline{X}_{\mathscr{C}}|=\sum_{x\in
U}\underline{X}_{\mathscr{C}}(x)$ and $|\overline{X}_{\mathscr{C}}|=\sum_{x\in
U}\overline{X}_{\mathscr{C}}(x)$.

###### Definition 3.60

Let $(U,\mathscr{C})$ be a fuzzy covering approximation space, and $X\subseteq
U$. Then the $\alpha\beta-$roughness measure
$\mu^{\alpha\beta}_{\mathscr{C}}(X)$ with respect to $\mathscr{C}$ is defined
as

$\displaystyle\mu^{\alpha\beta}_{\mathscr{C}}(X)=1-\frac{|\underline{X}^{\alpha}_{\mathscr{C}}|}{|\overline{X}^{\beta}_{\mathscr{C}}|},$

where
$\underline{X}^{\alpha}_{\mathscr{C}}=\\{x|\underline{X}_{\mathscr{C}}(x)>\alpha,x\in
U\\}$,
$\overline{X}^{\beta}_{\mathscr{C}}=\\{x|\overline{X}_{\mathscr{C}}(x)>\beta,x\in
U\\}$ and $|\cdot|$ means the cardinality of the set.

An example is employed to illustrate Definitions 3.59 and 3.60 as follows.

###### Example 3.61

Let $U=\\{x_{1},x_{2},x_{3},x_{4}\\}$,
$\mathscr{C}=\\{C_{21},C_{22},C_{23}\\}$, and
$X=\frac{0.2}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.6}{x_{3}}+\frac{0.1}{x_{4}}$.
According to Definition 3.8, we have that
$\underline{X}_{\mathscr{C}}=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0}{x_{4}}$
and
$\overline{X}_{\mathscr{C}}=\frac{0.2}{x_{1}}+\frac{0.4}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}$.
It follows that
$\mu_{\mathscr{C}}(X)=1-\frac{0.2+0.4+0.5}{0.2+0.4+0.5+0.5}=0.3125$.
Furthermore, it is obvious that
$\underline{X}^{\alpha}_{\mathscr{C}}=\\{x_{3}\\}$ and
$\overline{X}^{\beta}_{\mathscr{C}}=\\{x_{2},x_{3},x_{4}\\}$ by taking
$\alpha=0.4$ and $\beta=0.2$. Subsequently, it follows that
$\mu^{\alpha\beta}_{\mathscr{C}}(X)=1-\frac{|\\{x_{3}\\}|}{|\\{x_{2},x_{3},x_{4}\\}|}=\frac{2}{3}$.

##  4 Consistent functions for fuzzy covering information systems

In [24], Wang et al. proposed the concept of consistent functions for
attribute reductions of covering information systems. But so far we have not
seen the similar work on fuzzy covering information systems. In this section,
we introduce the concepts of consistent functions, the fuzzy covering mappings
and inverse fuzzy covering mappings based on fuzzy coverings and examine their
basic properties. Additionally, several examples are employed to illustrate
our proposed notions.

As a generalization of the concept of consistent functions given in Definition
2.6, we introduce the notion of consistent functions for constructing
attribute reducts of fuzzy covering information systems.

###### Definition 4.1

Let $f$ be a mapping from $U_{1}$ to $U_{2}$,
$\mathscr{C}$=$\\{C_{1},C_{2},...,C_{N}\\}$ a fuzzy covering of $U_{1}$, and
$[x]$ the block of $U_{1}/IND(f)$ which contains $x$, where $U_{1}/IND(f)$
stands for the blocks of partition of $U_{1}$ by an equivalence relation
$IND(f)$ based on $f$. If $C_{i}(y)=C_{i}(z)$ $(1\leq i\leq N)$ for any
$y,z\in[x]$, then $f$ is called a consistent function with respect to
$\mathscr{C}$.

Unless stated otherwise, we take the equivalence relation
$IND(f)=\\{(x,y)|f(x)=f(y),x,y\in U_{1}\\}$ and $[x]=\\{y\in
U_{1}|f(x)=f(y),x,y\in U_{1}\\}$ when applying Definition 4.1 in this work.
Particularly, it is clear that our proposed function is the same as the
consistent function in [24] when the membership degree for any $x\in U_{1}$
has its value only from the set $\\{0,1\\}$. Thereby, the proposed model can
be viewed as an extension of that given in [24].

An example is employed to illustrate the concept of consistent functions in
the following.

###### Example 4.2

Consider the fuzzy covering approximation space $(U_{1},\mathscr{C}_{1})$ in
Example 3.5. Then, we take $U_{2}=\\{y_{1},y_{2}\\}$ and define a mapping
$f:U_{1}\longrightarrow U_{2}$ as

$f(x_{1})=f(x_{3})=y_{1};f(x_{2})=f(x_{4})=y_{2}.$

Obviously, $f$ is a consistent function with respect to $\mathscr{C}_{1}$.

Now we investigate the relationship between Definitions 2.7 and 4.1. If
$\\{R_{x}|x\in U_{1}\\}$ is a fuzzy covering of $U_{1}$, where
$R_{x}(y)=R(x,y)$ for any $x,y\in U_{1}$, then we can express Definition 2.7
as follows: let $U_{1}$ and $U_{2}$ be two universes, $f$ a mapping from
$U_{1}$ to $U_{2}$, $R\in\mathscr{F}(U_{1}\times U_{1})$, and $[x]_{f}=\\{y\in
U_{1}|f(x)=f(y)\\}$, $\\{[x]_{f}|x\in U_{1}\\}$ a partition on $U_{1}$. For
any $x,y\in U_{1}$, if $R(x,v)=R(x,t)$ for any two pairs
$(x,v),(x,t)\in[x]_{f}\times[y]_{f}$, then $f$ is said to be consistent with
respect to $R$. Consequently, the consistent function given in Definition 2.7
is the same as our proposed model if $\\{R_{x}|x\in U_{1}\\}$ is a fuzzy
covering of $U_{1}$ and $IND(f)=\\{(x,y)|f(x)=f(y),x,y\in U_{1}\\}$.

In the following, we investigate some conditions under which $\\{R_{x}|x\in
U_{1}\\}$ is a fuzzy covering of $U_{1}$.

###### Corollary 4.3

Let $R$ be a fuzzy relation on $U_{1}$. Then

$(1)$ if $R$ is $\alpha$-reflexive, then $\\{R_{x}|x\in U_{1}\\}$ is a fuzzy
covering of $U_{1}$, where $1>\alpha>0$;

$(2)$ if $R$ is reflexive, then $\\{R_{x}|x\in U_{1}\\}$ is a fuzzy covering
of $U_{1}$;

$(3)$ if $R$ is a fuzzy similarity, then $\\{R_{x}|x\in U_{1}\\}$ is a fuzzy
covering of $U_{1}$;

$(4)$ if $R$ is a fuzzy equivalence, then $\\{R_{x}|x\in U_{1}\\}$ is a fuzzy
covering of $U_{1}$.

Proof. $(1)$ If $R$ is $\alpha$-reflexive, then $R_{x}(x)\geq\alpha$ for any
$x\in U_{1}$. It follows that $(\bigcup\\{R_{x}|x\in U_{1}\\})(y)\geq\alpha$
for any $y\in U_{1}$. Therefore, $\\{R_{x}|x\in U_{1}\\}$ is a fuzzy covering
of $U_{1}$.

$(2)$ If $R$ is reflexive, then $R_{x}(x)=1$ for any $x\in U_{1}$. It implies
that $\bigcup\\{R_{x}|x\in U_{1}\\}=U_{1}$. Therefore, $\\{R_{x}|x\in
U_{1}\\}$ is a fuzzy covering of $U_{1}$.

$(3),(4)$ The proof is similar to that in Corollary 4.3(2). $\Box$

Additionally, we can construct a fuzzy relation by a fuzzy covering.

###### Corollary 4.4

Let $f$ be a mapping from $U_{1}$ to $U_{2}$,
$\mathscr{C}$=$\\{C_{1},C_{2},...,C_{N}\\}$ a fuzzy covering of $U_{1}$,
$R_{x}=C_{x}$ for any $x\in U_{1}$, and $\alpha=min\\{C_{x}(x)|x\in U_{1}\\}$.
Then

$(1)$ $R$ is a $\alpha$-reflexive relation;

$(2)$ $R$ is symmetric if $C_{x}(x)=C_{y}(y)$ for any $x,y\in U_{1}$;

$(3)$ $R$ is transitive;

$(4)$ $R$ is a fuzzy equivalence relation if $\alpha=1$.

Proof. Straightforward from Definition 3.1. $\Box$

By Corollaries 4.3 and 4.4, it is clear that there exists a relationship
between a fuzzy relation and a fuzzy covering. Since both Wang’s model[24] and
our proposed function are based on a fuzzy relation and a fuzzy covering,
respectively, by Corollaries 4.3 and 4.4, we can establish the relationship
between Definitions 2.7 and 4.1.

By means of Zadeh’s extension principle, we propose the concepts of the fuzzy
covering mapping and inverse fuzzy covering mapping.

###### Definition 4.5

Let $S_{1}=(U_{1},\mathscr{C}_{1})$ and $S_{2}=(U_{2},\mathscr{C}_{2})$ be
fuzzy covering approximation spaces, and $f$ a surjection from $U_{1}$ to
$U_{2}$, $f$ induces a mapping from $\mathscr{C}_{1}\text{ to
}\mathscr{C}_{2}$ and a mapping from $\mathscr{C}_{2}\text{ to
}\mathscr{C}_{1}$, that is

$\hat{f}:\mathscr{C}_{1}\longrightarrow\mathscr{C}_{2},C\mid\rightarrow\hat{f}(C)\in\mathscr{C}_{2},\forall
C\in\mathscr{C}_{1};$

$\hat{f}(C)(y)=\left\\{\begin{array}[]{ccc}\bigvee_{x\in
f^{-1}(y)}C(x),&{\rm}\hfil&f^{-1}(y)\neq\emptyset;\\\
0,&{\rm}\hfil&f^{-1}(y)=\emptyset;\end{array}\right.$

$\hat{f}^{-1}:\mathscr{C}_{2}\longrightarrow\mathscr{C}_{1},T\mid\rightarrow\hat{f}^{-1}(T)\in\mathscr{C}_{1},\forall
T\in\mathscr{C}_{2};$

$f^{-1}(T)(x)=T(f(x)),x\in U_{1}.$
Then $\hat{f}$ and $\hat{f}^{-1}$ are called the fuzzy covering mapping and
the inverse fuzzy covering mapping induced by $f$, respectively. In
convenience, we denote $\hat{f}$ and $\hat{f}^{-1}$ as $f$ and $f^{-1}$,
respectively.

By Definition 4.5, we observe that $\hat{f}$ and $\hat{f}^{-1}$ will be
reduced to Definition 4.1 in [24] if the membership degree takes values from
the set $\\{0,1\\}$. The following theorem discusses the problem of fuzzy set
operations under a consistent function $f$.

###### Theorem 4.6

Let $f$ be a mapping from $U_{1}$ to $U_{2}$,
$\mathscr{C}$=$\\{C_{1},C_{2},...,C_{N}\\}$ a fuzzy covering of $U_{1}$, and
$C_{i},C_{j}\in\mathscr{C}$. Then

$(1)$ $f(C_{i}\cap C_{j})\subseteq f(C_{i})\cap f(C_{j})$;

$(2)$ $f(C_{i}\cup C_{j})=f(C_{i})\cup f(C_{j})$;

$(3)$ If $f$ is a consistent function with respect to $\mathscr{C}$, then
$f(C_{i}\cap C_{j})=f(C_{i})\cap f(C_{j})$.

Proof. (1) By Definition 4.5, we obtain that $f(C_{i}\cap C_{j})(f(x))=0$ when
$(C_{i}\cap C_{j})(x)=0$ for $x\in U_{1}$. Moreover, by Definition 4.5, it
follows that $f(C_{i}\cap C_{j})(y)=\bigvee_{x^{\prime}\in
f^{-1}(y)}(C_{i}\cap C_{j})(x^{\prime})=\bigvee_{x^{\prime}\in
f^{-1}(y)}(C_{i}(x^{\prime})\wedge
C_{j}(x^{\prime}))\leq\bigvee_{x^{\prime}\in
f^{-1}(y)}C_{i}(x^{\prime})\wedge\bigvee_{x^{\prime}\in
f^{-1}(y)}C_{j}(x^{\prime})=(f(C_{i})\cap f(C_{j}))(y)$. Consequently,
$f(C_{i}\cap C_{j})\subseteq f(C_{i})\cap f(C_{j})$.

(2) According to Definition 4.5, we have that $f(C_{i}\cup C_{j})(f(x))=0$
when $(C_{i}\cup C_{j})(x)=0$ for $x\in U_{1}$. Furthermore, by Definition
4.5, it follows that $f(C_{i}\cup C_{j})(y)=\bigvee_{x^{\prime}\in
f^{-1}(y)}(C_{i}\cup C_{j})(x^{\prime})=\bigvee_{x^{\prime}\in
f^{-1}(y)}(C_{i}(x^{\prime})\vee C_{j}(x^{\prime}))=\bigvee_{x^{\prime}\in
f^{-1}(y)}C_{i}(x^{\prime})\vee\bigvee_{x^{\prime}\in
f^{-1}(y)}C_{j}(x^{\prime})=(f(C_{i})\cup f(C_{j}))(y)$. Therefore,
$f(C_{i}\cup C_{j})=f(C_{i})\cup f(C_{j})$.

(3) By Theorem 4.6(1), it is obvious that $f(C_{i}\cap C_{j})\subseteq
f(C_{i})\cap f(C_{j})$. So we only need to prove that $f(C_{i})\cap
f(C_{j})\subseteq f(C_{i}\cap C_{j}).$ Suppose that $y\in U_{2}$, there exists
$x\in U_{1}$ such that $f(x)=y$. Based on Definitions 4.1 and 4.5, we have
that $(f(C_{i})\cap f(C_{j}))(y)=\bigvee_{x^{\prime}\in
f^{-1}(y)}C_{i}(x^{\prime})\wedge\bigvee_{x^{\prime}\in
f^{-1}(y)}C_{j}(x^{\prime})=C_{i}(x^{\prime})\wedge
C_{j}(x^{\prime})\subseteq\bigvee_{x^{\prime}\in
f^{-1}(y)}(C_{i}(x^{\prime})\wedge C_{j}(x^{\prime}))=f(C_{i}\cap C_{j})(y)$.
Thereby, $f(C_{i}\cap C_{j})=f(C_{i})\cap f(C_{j})$. $\Box$

Theorem 4.6 shows that the mapping $f$ preserves some fuzzy set operations,
especially it preserves the intersection operation of fuzzy sets if $f$ is
consistent.

To illustrate Theorem 4.6, we give an example below.

###### Example 4.7

Consider $S=(U_{1},\mathscr{C}_{1})$ in Example 3.5 and the consistent
function $f$ in Example 4.2. Then we observe that $f(C_{1}\cap
C_{2})=f(C_{1})\cap f(C_{2})$, $f(C_{1}\cap C_{3})=f(C_{1})\cap f(C_{3})$ and
$f(C_{2}\cap C_{3})=f(C_{2})\cap f(C_{3})$.

By Theorem 4.6, we obtain the following corollary.

###### Corollary 4.8

Let $f$ be a mapping from $U_{1}$ to $U_{2}$, and
$\mathscr{C}$=$\\{C_{1},C_{2},...,C_{N}\\}$ a fuzzy covering of $U_{1}$. If
$f$ is a consistent function with respect to $\mathscr{C}$, then
$f(\bigcap_{i=1}^{N}C_{i})=\bigcap_{i=1}^{N}f(C_{i})$.

Subsequently, we investigate the properties of the inverse mapping of a
consistent function.

###### Theorem 4.9

Let $f$ be a mapping from $U_{1}$ to $U_{2}$,
$\mathscr{C}$=$\\{C_{1},C_{2},...,C_{N}\\}$ a fuzzy covering of $U_{1}$, and
$C_{i}\in\mathscr{C}$. Then

$(1)$ $C_{i}\subseteq f^{-1}(f(C_{i}))$;

$(2)$ If $f$ is a consistent function with respect to $\mathscr{C}$, then
$f^{-1}(f(C_{i}))=C_{i}$.

Proof. (1) According to Definition 4.5, we have that
$f^{-1}(f(C_{i}))(x)=f(C_{i})(f(x))$. Taking $y=f(x)$, it follows that
$f(C_{i})(f(x))=f(C_{i})(y)=\bigvee_{x^{\prime}\in
f^{-1}(y)}C_{i}(x^{\prime})\geq C_{i}(x).$ Therefore, $C_{i}\subseteq
f^{-1}(f(C_{i}))$.

(2) By Definition 4.5, we see that $f^{-1}(f(C_{i}(x)))=f(C_{i})(f(x))$.
Assume that $y=f(x)$, it follows that
$f(C_{i})(f(x))=f(C_{i})(y)=\bigvee_{x^{\prime}\in
f^{-1}(y)}C_{i}(x^{\prime}).$ According to Definitions 4.1 and 4.5, it implies
that $C_{i}(x^{\prime})=C_{i}(x)$ for any $x^{\prime}\in f^{-1}(y)$.
Consequently, $\bigvee_{x^{\prime}\in f^{-1}(y)}C_{i}(x^{\prime})=C_{i}(x)$.
Hence, $f^{-1}(f(C_{i}))(x)=C_{i}(x).$ Thereby, $C_{i}=f^{-1}(f(C_{i}))$.
$\Box$

We give an example to illustrate Theorem 4.9 in the following.

###### Example 4.10

Consider $S=(U_{1},\mathscr{C}_{1})$ in Example 3.5 and the consistent
function $f$ in Example 4.2. Then we see that
$f^{-1}(f(C_{1}))(x_{i})=C_{1}(x_{i}),$
$f^{-1}(f(C_{2}))(x_{i})=C_{2}(x_{i}),$ and
$f^{-1}(f(C_{3}))(x_{i})=C_{3}(x_{i}),i=1,2,3,4.$ Therefore,
$f^{-1}(f(C_{1}))=C_{1},$ $f^{-1}(f(C_{2}))=C_{2},$ and
$f^{-1}(f(C_{3}))=C_{3}.$

By Theorem 4.9, we have the following corollary.

###### Corollary 4.11

Let $f$ be a mapping from $U_{1}$ to $U_{2}$, and
$\mathscr{C}$=$\\{C_{1},C_{2},...,C_{N}\\}$ a fuzzy covering of $U_{1}$. If
$f$ is a consistent function with respect to $\mathscr{C}$, then
$f^{-1}(f(\bigcap_{i=1}^{N}C_{i}))=\bigcap_{i=1}^{N}C_{i}$.

We also explore the properties of a consistent function on a family of fuzzy
coverings.

###### Theorem 4.12

Let $f$ be a mapping from $U_{1}$ to $U_{2}$, and
${\mathscr{C}_{1},\mathscr{C}_{2}}\in C(U_{1})$. If $f$ is a consistent
function with respect to $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$,
respectively, then $f$ is consistent with respect to
$\mathscr{C}_{1}\bigcap\mathscr{C}_{2}$.

Proof. Based on Definition 4.1, we have that $C_{ix}(y)=C_{ix}(z)$ for any
$y,z\in[x]$, where $i=1,2$. It follows that $C_{1x}(y)\wedge
C_{2x}(y)=C_{1x}(z)\wedge C_{2x}(z)$ for any $y,z\in[x]$. Hence, $(C_{1x}\cap
C_{2x})(y)=(C_{1x}\cap C_{2x})(z)$ for any $y,z\in[x]$. Therefore, $f$ is
consistent with respect to $\mathscr{C}_{1}\bigcap\mathscr{C}_{2}$. $\Box$

The following example is employed to illustrate Theorem 4.12.

###### Example 4.13

Consider $S=(U_{1},\Delta)$ in Example 3.7. We take $U_{2}=\\{y_{1},y_{2}\\}$
and define a mapping $f:U_{1}\rightarrow U_{2}$ as follows:

$f(x_{1})=f(x_{2})=y_{1},f(x_{3})=f(x_{4})=y_{2}.$

It is obvious that $f$ is a consistent function with respect to
$\mathscr{C}_{1}$ and $\mathscr{C}_{2}$, respectively. By Definition 4.1, we
observe that $f$ is consistent with respect to
$\mathscr{C}_{1}\cap\mathscr{C}_{2}$.

Based on Theorem 4.12, we obtain the following corollary.

###### Corollary 4.14

Let $\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}\in C(U_{1})$, and $f$
a mapping from $U_{1}$ to $U_{2}$. If $f$ is a consistent function with
respect to any $\mathscr{C}_{i}$ $(1\leq i\leq m)$, then $f$ is consistent
with respect to $\bigcap_{i=1}^{m}\mathscr{C}_{i}$.

Now, we introduce two concepts for fuzzy covering approximation spaces.

###### Definition 4.15

Let $f$ be a mapping from $U_{1}$ to $U_{2}$,
$\mathscr{C}_{1}=\\{C_{11},C_{12},...,C_{1N}\\}\in C(U_{1})$, and
$\mathscr{C}_{2}=\\{T_{21},T_{22},...,T_{2M}\\}\in C(U_{2})$. Then
$f(\mathscr{C}_{1})$ and $f(\mathscr{C}_{2})$ are defined by

$\displaystyle f(\mathscr{C}_{1})$ $\displaystyle=$
$\displaystyle\\{f(C_{1i}),C_{1i}\in\mathscr{C}_{1},1\leq i\leq N\\};$
$\displaystyle f^{-1}(\mathscr{C}_{2})$ $\displaystyle=$
$\displaystyle\\{f^{-1}(T_{2j}),T_{2j}\in\mathscr{C}_{2},1\leq j\leq M\\}.$

###### Theorem 4.16

Let $U$ be a non-empty universe of discourse, and $\mathscr{C}\in C(U)$. If
$f$ is a consistent function with respect to $\mathscr{C}$, then
$f^{-1}(f(\mathscr{C}))=\mathscr{C}.$

Proof. By Theorem 4.9, it follows that $f^{-1}(f(C_{i}))=C_{i}$ for any
$C_{i}\in\mathscr{C}$. Therefore, $f^{-1}(f(\mathscr{C}))=\mathscr{C}.$ $\Box$

Obviously, Examples 3.5 and 4.2 can illustrate Theorem 4.16. Then we get the
following corollary.

###### Corollary 4.17

Let $\mathscr{C}_{i}\in C(U)$, and $\Delta=\\{\mathscr{C}_{i}|i=1,2,...,m\\}$.
If $f$ is a consistent function with respect to any
$\mathscr{C}_{i}\in\Delta$, then $f^{-1}(f(\bigcap\Delta))=\bigcap\Delta.$

At the end of this section, we discuss the fuzzy covering operations under a
consistent function.

###### Theorem 4.18

Let $f$ be a mapping from $U_{1}$ to $U_{2}$, and $\mathscr{C}_{1}$,
$\mathscr{C}_{2}\in C(U_{1})$. Then we have

$(1)$ $f(\mathscr{C}_{1}\bigcap\mathscr{C}_{2})\subseteq
f(\mathscr{C}_{1})\bigcap f(\mathscr{C}_{2})$;

$(2)$ If $f$ is a consistent function with respect to $\mathscr{C}_{1}$ and
$\mathscr{C}_{2}$, respectively, then
$f(\mathscr{C}_{1}\bigcap\mathscr{C}_{2})=f(\mathscr{C}_{1})\bigcap
f(\mathscr{C}_{2})$.

Proof. (1) According to Definitions 4.1 and 4.5, it is obvious that
$f(\mathscr{C}_{1}\bigcap\mathscr{C}_{2})\subseteq f(\mathscr{C}_{1})\bigcap
f(\mathscr{C}_{2})$.

(2) Evidently, we only need to prove that $f(\mathscr{C}_{1})\bigcap
f(\mathscr{C}_{2})\subseteq f(\mathscr{C}_{1}\bigcap\mathscr{C}_{2})$. Assume
that $C_{x}$ is the minimal element containing $x$ in
$\mathscr{C}_{1}\bigcap\mathscr{C}_{2}$, $C_{1x}$ is the minimal element
containing $x$ in $Cov(\mathscr{C}_{1})$, and $C_{2x}$ is the minimal element
containing $x$ in $Cov(\mathscr{C}_{2})$ for any $x\in U_{1}$. By Definition
3.50, it follows that $C_{x}=C_{1x}\bigcap C_{2x}$. According to Theorem 4.12,
it implies that $f$ is a consistent function with respect to
$\mathscr{C}_{1}\bigcap\mathscr{C}_{2}$. Consequently, we obtain that
$f(C_{x})=f(C_{1x})\cap f(C_{2x})$. By Definition 3.1, we have that
$C_{x}(x)>0$ for any $x\in U_{1}$. It follows that $f(C_{x})(f(x))>0$. Hence,
$(f(C_{1x})\cap f(C_{2x}))(f(x))>0.$ Suppose that $f(C_{1x})\cap f(C_{2x})$ is
not the minimal subset containing $f(x)$ in
$f(\mathscr{C}_{1}\bigcap\mathscr{C}_{2})$. Then there exists $x_{0}\in U_{1}$
such that $f(C_{ix_{0}})(f(x))>0$ and $f(C_{1x})\cap f(C_{2x})\cap
f(C_{ix_{0}})\subset f(C_{1x})\cap f(C_{2x}),$ it means that $(f(C_{1x})\cap
f(C_{2x})\cap f(C_{ix_{0}}))(f(x))>0.$ Thereby, there exist $u,v$ and $w$ such
that $C_{1x}(u)>0,C_{2x}(v)>0,C_{ix_{0}}(w)>0$ and $f(u)=f(v)=f(w)=f(x)$.
According to Theorem 4.6, we have that $f(C_{1x})\cap f(C_{2x})=f(C_{1x}\cap
C_{2x})\subseteq f(C_{ix_{0}})$ and $f(C_{1x})\cap f(C_{2x})\cap
f(C_{ix_{0}})=f(C_{1x})\cap f(C_{2x}),$ it implies that $f(C_{1x})\cap
f(C_{2x})$ is the minimal subset containing $f(x)$ in
$f(\mathscr{C}_{1}\bigcap\mathscr{C}_{2})$. Based on the above statement, it
follows that $f(\mathscr{C}_{1})\bigcap f(\mathscr{C}_{2})\subseteq
f(\mathscr{C}_{1}\bigcap\mathscr{C}_{2})$. Therefore,
$f(\mathscr{C}_{1}\bigcap\mathscr{C}_{2})=f(\mathscr{C}_{1})\bigcap
f(\mathscr{C}_{2})$. $\Box$

Based on Theorem 4.18, we have the following corollary.

###### Corollary 4.19

Let $\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}$ be fuzzy coverings
of $U_{1}$, and $f$ a mapping from $U_{1}$ to $U_{2}$. If $f$ is a consistent
function with respect to
$\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}$, respectively, then
$f(\bigcap_{i=1}^{m}\mathscr{C}_{i})=\bigcap_{i=1}^{m}f(\mathscr{C}_{i})$.

##  5 Data compressions of fuzzy covering information systems and dynamic
fuzzy covering information systems

In this section, we further investigate data compressions of fuzzy covering
information systems and dynamic fuzzy covering information systems.

###  5.1 Data compression of fuzzy covering information systems

In this subsection, the concepts of an induced fuzzy covering information
system and homomorphisms between fuzzy covering information systems are
introduced for data compression of the fuzzy covering information system. Then
the algorithm of constructing attribute reducts of fuzzy covering information
systems is provided. An example is finally employed to illustrate the proposed
concepts and algorithm.

###### Definition 5.1

Let $f$ be a surjection from $U_{1}$ to $U_{2}$,
$\Delta_{1}$=$\\{\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}\\}$ a
family of fuzzy coverings of $U_{1}$, and
$f(\Delta_{1})$=$\\{f(\mathscr{C}_{1}),f(\mathscr{C}_{2}),...,f(\mathscr{C}_{m})\\}$.
Then $(U_{1},\Delta_{1})$ is referred to as a fuzzy covering information
system and $(U_{2},f(\Delta_{1}))$ is called the $f$-induced fuzzy covering
information system of $(U_{1},\Delta_{1})$.

Definition 5.1 shows that we can induce a new fuzzy covering information
system under a surjection.

Based on Definitions 4.1 and 5.1, we propose the notion of a homomorphism
between two fuzzy covering information systems.

###### Definition 5.2

Let $f$ be a surjection from $U_{1}$ to $U_{2}$,
$\Delta_{1}$=$\\{\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}\\}$ a
family of fuzzy coverings of $U_{1}$, and
$f(\Delta_{1})$=$\\{f(\mathscr{C}_{1}),f(\mathscr{C}_{2}),...,f(\mathscr{C}_{m})\\}$.
If $f$ is consistent with respect to any $\mathscr{C}_{i}\in\Delta_{1}$
$(1\leq i\leq m)$ on $U_{1}$, then $f$ is called a homomorphism from
$(U_{1},\Delta_{1})$ to $(U_{2},f(\Delta_{1}))$.

We provide the concept of reducts of fuzzy covering information systems in the
following.

###### Definition 5.3

Let $(U_{1},\Delta_{1})$ be a fuzzy covering information system, and
$\mathscr{C}_{i}\in\Delta_{1}$ $(1\leq i\leq m)$. If
$\bigcap\\{\Delta_{1}-\mathscr{C}_{i}\\}=\bigcap\Delta_{1}\ $, then
$\mathscr{C}_{i}$ is called superfluous. Otherwise, $\mathscr{C}_{i}$ is
called indispensable. The collection of all indispensable elements in
$\Delta_{1}$, denoted as Core($\Delta_{1}$), is called the core of
$\Delta_{1}$. $P\subseteq\Delta_{1}$ is called a reduct of $\Delta_{1}$ if $P$
satisfies: $\bigcap P=\bigcap\Delta_{1}$ and
$\bigcap\\{P-\mathscr{C}\\}\neq\bigcap\Delta_{1}$ for any $\mathscr{C}\in P.$

Now we present the following theorem which shows that the reducts of fuzzy
covering information systems can be preserved under a homomorphism.

###### Theorem 5.4

Let $f$ be a surjection from $U_{1}$ to $U_{2}$,
$\Delta_{1}$=$\\{\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}\\}$ a
family of fuzzy coverings of $U_{1}$, and
$f(\Delta_{1})$=$\\{f(\mathscr{C}_{1}),f(\mathscr{C}_{2}),...,f(\mathscr{C}_{m})\\}$.
If $f$ is a homomorphism from $(U_{1},\Delta_{1})$ to $(U_{2},f(\Delta_{1}))$,
then $P\subseteq\Delta_{1}$ is a reduct of $\Delta_{1}$ if and only if $f(P)$
is a reduct of $f(\Delta_{1})$.

Proof. Suppose $P$ is a reduct of $\Delta_{1}$. It follows that $\bigcap
P=\bigcap\Delta_{1}$. Hence, $f(\bigcap P)=f(\bigcap\Delta_{1})$. Then we
obtain that $\bigcap f(P)=\bigcap f(\Delta_{1})$ since $f$ is a homomorphism
from $(U_{1},\Delta_{1})$ to $(U_{2},f(\Delta_{1}))$. Assume that there exists
$\mathscr{C}\in P$ such that $\bigcap(f(P)-f(\mathscr{C}))=\bigcap f(P)$. It
implies that $\bigcap(f(P)-f(\mathscr{C}))=\bigcap f(P-\mathscr{C})$. Hence,
we see that $\bigcap f(\Delta_{1})=\bigcap f(P-\mathscr{C})$. It follows that
$f^{-1}(\bigcap f(\Delta_{1}))=f^{-1}(\bigcap f(P-\mathscr{C}))$. We obtain
that $\bigcap\Delta_{1}=\bigcap(P-\mathscr{C})$, which contradicts that $P$ is
a reduct of $\Delta_{1}$. So $f(P)$ is a reduct of $f(\Delta_{1})$.

On the other hand, we assume that $f(P)$ is a reduct of $f(\Delta_{1})$. It
follows that $\bigcap f(\Delta_{1})=\bigcap f(P)$. Since $f$ is a homomorphism
from $(U_{1},\Delta_{1})$ to $(U_{2},f(\Delta_{1}))$, we obtain that
$f(\bigcap\Delta_{1})=f(\bigcap P)$. It implies that
$\bigcap\Delta_{1}=\bigcap P$. Assume that there exists $\mathscr{C}\in P$
satisfying $\bigcap\Delta_{1}=\bigcap(P-\mathscr{C})$, it follows that
$f(\bigcap\Delta_{1})=f(\bigcap(P-\mathscr{C}))$. Obviously, $\bigcap
f(\Delta_{1})=\bigcap f(P-\mathscr{C})=\bigcap(f(P)-f(\mathscr{C}))$, which is
a contradiction. Therefore, $P\subseteq\Delta_{1}$ is a reduct of
$\Delta_{1}$. $\Box$

By Theorem 5.4, we obtain the following corollary.

###### Corollary 5.5

Let $f$ be a surjection from $U_{1}$ to $U_{2}$,
$\Delta_{1}$=$\\{\mathscr{C}_{1},\mathscr{C}_{2},...,\mathscr{C}_{m}\\}$ a
family of fuzzy coverings of $U_{1}$, and
$f(\Delta_{1})$=$\\{f(\mathscr{C}_{1}),f(\mathscr{C}_{2}),...,f(\mathscr{C}_{m})\\}$.
If $f$ is a homomorphism from $(U_{1},\Delta_{1})$ to $(U_{2},f(\Delta_{1}))$,
then

$(1)$ $\mathscr{C}$ is indispensable in $\Delta_{1}$ if and only if
$f(\mathscr{C})$ is indispensable in $f(\Delta_{1})$;

$(2)$ $\mathscr{C}$ is superfluous in $\Delta_{1}$ if and only if
$f(\mathscr{C})$ is superfluous in $f(\Delta_{1})$;

$(3)$ The image of the core of $\Delta_{1}$ is the core of $f(\Delta_{1})$,
and the inverse image of the core of $f(\Delta_{1})$ is the core of the
original image.

Proof. Straightforward from Definition 5.3 and Theorem 5.4. $\Box$

From Corollary 5.5, we see that the attribute reductions of the original fuzzy
covering information system and image system are equivalent to each other
under the condition of a homomorphism.

###### Definition 5.6

Let $(U_{1},\mathscr{C}_{1})$ be a fuzzy covering approximation space, the
equivalence relation $R_{\mathscr{C}_{1}}=\\{(x,y)|C_{x}=C_{y},x,y\in
U_{1}\\}$, and $U_{1}/R_{\mathscr{C}_{1}}=\\{R_{\mathscr{C}_{1}}(x)|x\in
U_{1}\\}$. Then $U_{1}/R_{\mathscr{C}_{1}}$ is called the partition based on
$\mathscr{C}_{1}$.

For the sake of convenience, we denote $U_{1}/R_{\mathscr{C}_{1}}$ as
$U_{1}/\mathscr{C}_{1}$ simply.

Following, we employ Table 2 to show the partition based on each fuzzy
covering for the fuzzy covering information system $(U_{1},\Delta_{1})$, where
$P_{ix_{j}}$ stands for the block containing $x_{j}$ in the partition
$U_{1}/R_{\mathscr{C}_{i}}$. It is easy to see that
$P_{\Delta_{1}x_{j}}=\bigcap_{1\leq i\leq m}P_{ix_{j}}$, where
$P_{\Delta_{1}x_{j}}$ denotes the block containing $x_{j}$ in the partition
based on $\Delta_{1}$.

Subsequently, we propose the main feature of the algorithm to construct
attribute reducts of fuzzy covering information systems. It shows how to
construct a homomorphism and compress a large-scale information system into a
small one under the condition of the homomorphism.

###### Algorithm 5.7

Let $U_{1}=\\{x_{1},...,x_{n}\\}$, and
$\Delta_{1}$=$\\{\mathscr{C}_{1},\mathscr{C}_{2},$ $...,\mathscr{C}_{m}\\}$ a
family of fuzzy coverings of $U_{1}$. Then

Step 1. Input the fuzzy covering information system $(U_{1},\Delta_{1})$;

Step 2. Computing the partition $U_{1}/\mathscr{C}_{i}$ $(1\leq i\leq m)$ and
obtain $U_{1}/\Delta_{1}=\\{C_{i}|1\leq i\leq K\\}$;

Step 3. Define $f:U_{1}\rightarrow U_{2}$ as follows: $f(x)=y_{l}$, $x\in
C_{l}$, where $1\leq l\leq K$ and $U_{2}=\\{y_{1},y_{2},...,y_{K}\\}$;

Step 4. Compute
$f(\Delta_{1})$=$\\{f(\mathscr{C}_{1}),f(\mathscr{C}_{2}),...,f(\mathscr{C}_{m})\\}$
and obtain $(U_{2},f(\Delta_{1}))$;

Step 5. Construct attribute reducts of $(U_{2},f(\Delta_{1}))$ and obtain a
reduct
$\\{f(\mathscr{C}_{i1}),f(\mathscr{C}_{i2}),...,f(\mathscr{C}_{ik})\\};$

Step 6. Obtain a reduct
$\\{\mathscr{C}_{i1},\mathscr{C}_{i2},...,\mathscr{C}_{ik}\\}$ of
$(U_{1},\Delta_{1})$ and output the results.

Remark. In Example 5.1[24], Wang et al. obtained the partition
$U_{1}/\Delta_{1}$ by only computing $\Delta_{x}$ for any $x\in U_{1}$. But we
get $U_{1}/\Delta_{1}$ by computing $U_{1}/\mathscr{C}_{i}$ for any
$\mathscr{C}_{i}\in\Delta_{1}$ in Algorithm 5.7. By using the proposed
approach, we can compress the dynamic fuzzy covering information system on the
basis of data compression of the original system with lower time complexity,
which is illustrated in Subsection 5.2.

Now, we employ a car evaluation problem to illustrate Algorithm 5.7.

###### Example 5.8

Suppose that $U_{1}=\\{x_{1},x_{2},...,x_{8}\\}$ is a set of eight cars,
$C_{1}=\\{price,structure,size,appearance\\}$ is a set of attributes. The
domains of $price$, $structure$, $size$ and $appearance$ are $\\{high,middle,$
$low\\}$, $\\{excellent,ordinary,poor\\}$, $\\{big,middle,small\\}$ and
$\\{beautiful,fair,ugly\\}$, respectively. In this example, we do not list
their evaluation reports for simplicity. According to the four specialists’
evaluation reports, we obtain the following fuzzy coverings of $U_{1}$ as
$\Delta_{1}=\\{\mathscr{C}_{price},\mathscr{C}_{structure},\mathscr{C}_{size},\mathscr{C}_{appearance}\\}$,
$\mathscr{C}_{price},\mathscr{C}_{structure},\mathscr{C}_{size}$ and
$\mathscr{C}_{appearance}$ are based on $price$, $structure$, $size$ and
$appearance$, respectively, where

$\displaystyle\mathscr{C}_{price}$ $\displaystyle=$
$\displaystyle\\{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{0.5}{x_{3}}+\frac{1}{x_{4}}+\frac{0.5}{x_{5}}+\frac{1}{x_{6}}+\frac{1}{x_{7}}+\frac{1}{x_{8}},\frac{0.5}{x_{1}}+\frac{0.5}{x_{2}}+\frac{0.5}{x_{3}}+\frac{1}{x_{4}}+\frac{0.5}{x_{5}}+\frac{0.5}{x_{6}}+\frac{1}{x_{7}}+\frac{1}{x_{8}},$
$\displaystyle\frac{0}{x_{1}}+\frac{0}{x_{2}}+\frac{1}{x_{3}}+\frac{0.5}{x_{4}}+\frac{1}{x_{5}}+\frac{1}{x_{6}}+\frac{0.5}{x_{7}}+\frac{0.5}{x_{8}}\\};$
$\displaystyle\mathscr{C}_{structure}$ $\displaystyle=$
$\displaystyle\\{\frac{0}{x_{1}}+\frac{0}{x_{2}}+\frac{1}{x_{3}}+\frac{0}{x_{4}}+\frac{1}{x_{5}}+\frac{0}{x_{6}}+\frac{0}{x_{7}}+\frac{0}{x_{8}},\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{0.5}{x_{3}}+\frac{1}{x_{4}}+\frac{0.5}{x_{5}}+\frac{1}{x_{6}}+\frac{1}{x_{7}}+\frac{1}{x_{8}},$
$\displaystyle\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{0.5}{x_{3}}+\frac{0.5}{x_{4}}+\frac{0.5}{x_{5}}+\frac{0}{x_{6}}+\frac{0.5}{x_{7}}+\frac{0.5}{x_{8}}\\};$
$\displaystyle\mathscr{C}_{size}$ $\displaystyle=$
$\displaystyle\\{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\frac{0}{x_{4}}+\frac{1}{x_{5}}+\frac{1}{x_{6}}+\frac{0}{x_{7}}+\frac{0}{x_{8}},\frac{0.5}{x_{1}}+\frac{0.5}{x_{2}}+\frac{1}{x_{3}}+\frac{0.5}{x_{4}}+\frac{1}{x_{5}}+\frac{0.5}{x_{6}}+\frac{0.5}{x_{7}}+\frac{0.5}{x_{8}},$
$\displaystyle\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\frac{1}{x_{4}}+\frac{1}{x_{5}}+\frac{0.5}{x_{6}}+\frac{1}{x_{7}}+\frac{1}{x_{8}}\\};$
$\displaystyle\mathscr{C}_{appearance}$ $\displaystyle=$
$\displaystyle\\{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{0.5}{x_{3}}+\frac{1}{x_{4}}+\frac{0.5}{x_{5}}+\frac{1}{x_{6}}+\frac{1}{x_{7}}+\frac{1}{x_{8}},\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{0.5}{x_{3}}+\frac{1}{x_{4}}+\frac{0.5}{x_{5}}+\frac{1}{x_{6}}+\frac{1}{x_{7}}+\frac{1}{x_{8}},$
$\displaystyle\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\frac{1}{x_{4}}+\frac{1}{x_{5}}+\frac{0.5}{x_{6}}+\frac{1}{x_{7}}+\frac{1}{x_{8}}\\}.$

By Definition 2.8, we see that $(U_{1},\Delta_{1})$ is a fuzzy covering
information system. Furthermore, according to Definitions 3.1, 3.6 and 5.6, we
obtain the following results:

$\displaystyle U_{1}/\mathscr{C}_{price}$ $\displaystyle=$
$\displaystyle\\{\\{x_{1},x_{2}\\},\\{x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}\\};$
$\displaystyle U_{1}/\mathscr{C}_{structure}$ $\displaystyle=$
$\displaystyle\\{\\{x_{1},x_{2},x_{4},x_{7},x_{8}\\},\\{x_{3},x_{5}\\},\\{x_{6}\\}\\};$
$\displaystyle U_{1}/\mathscr{C}_{size}$ $\displaystyle=$
$\displaystyle\\{\\{x_{1},x_{2},x_{3},x_{5},x_{6}\\},\\{x_{4},x_{7},x_{8}\\}\\};$
$\displaystyle U_{1}/\mathscr{C}_{appearance}$ $\displaystyle=$
$\displaystyle\\{\\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}\\}.$

The partitions
$U_{1}/\mathscr{C}_{price},U_{1}/\mathscr{C}_{structure},U_{1}/\mathscr{C}_{size}$
and $U_{1}/\mathscr{C}_{appearance}$ are shown in Table 3. Then we obtain that
$U_{1}/\Delta_{1}=\\{\\{x_{1},x_{2}\\},\\{x_{3},x_{5}\\},\\{x_{4},x_{7},x_{8}\\},\\{x_{6}\\}\\}.$
Thus we take $U_{2}=\\{y_{1},y_{2},y_{3},y_{4}\\}$ and define a mapping
$f:U_{1}\longrightarrow U_{2}$ as follows:

$f(x_{1})=f(x_{2})=y_{1};f(x_{3})=f(x_{5})=y_{2};f(x_{4})=f(x_{7})=f(x_{8})=y_{3};f(x_{6})=y_{4}.$

According to the function $f$, we obtain that
$f(\Delta_{1})=\\{f(\mathscr{C}_{price}),f(\mathscr{C}_{structure}),f(\mathscr{C}_{size}),f(\mathscr{C}_{appearance})\\}$,
where

$\displaystyle f(\mathscr{C}_{price})$ $\displaystyle=$
$\displaystyle\\{\frac{1}{y_{1}}+\frac{0.5}{y_{2}}+\frac{1}{y_{3}}+\frac{1}{y_{4}},\frac{0.5}{y_{1}}+\frac{0.5}{y_{2}}+\frac{1}{y_{3}}+\frac{0.5}{y_{4}},\frac{0}{y_{1}}+\frac{1}{y_{2}}+\frac{0.5}{y_{3}}+\frac{1}{y_{4}}\\};$
$\displaystyle f(\mathscr{C}_{structure})$ $\displaystyle=$
$\displaystyle\\{\frac{0}{y_{1}}+\frac{1}{y_{2}}+\frac{0}{y_{3}}+\frac{0}{y_{4}},\frac{1}{y_{1}}+\frac{0.5}{y_{2}}+\frac{1}{y_{3}}+\frac{1}{y_{4}},\frac{1}{y_{1}}+\frac{0.5}{y_{2}}+\frac{0.5}{y_{3}}+\frac{0}{y_{4}}\\};$
$\displaystyle f(\mathscr{C}_{size})$ $\displaystyle=$
$\displaystyle\\{\frac{1}{y_{1}}+\frac{1}{y_{2}}+\frac{0}{y_{3}}+\frac{1}{y_{4}},\frac{0.5}{y_{1}}+\frac{1}{y_{2}}+\frac{0.5}{y_{3}}+\frac{0.5}{y_{4}},\frac{1}{y_{1}}+\frac{1}{y_{2}}+\frac{1}{y_{3}}+\frac{0.5}{y_{4}}\\};$
$\displaystyle f(\mathscr{C}_{appearance})$ $\displaystyle=$
$\displaystyle\\{\frac{1}{y_{1}}+\frac{0.5}{y_{2}}+\frac{1}{y_{3}}+\frac{1}{y_{4}},\frac{1}{y_{1}}+\frac{0.5}{y_{2}}+\frac{1}{y_{3}}+\frac{1}{y_{4}},\frac{1}{y_{1}}+\frac{1}{y_{2}}+\frac{1}{y_{3}}+\frac{0.5}{y_{4}}\\}.$

According to Definition 5.1, we obtain the $f$-induced fuzzy covering
information system $(U_{2},f(\Delta_{1}))$ of $(U_{1},\Delta_{1})$. Clearly,
the size of $(U_{2},f(\Delta_{1}))$ is relatively smaller than that of
$(U_{1},\Delta_{1})$. Then, by Definitions 5.1, 5.2 and 5.3, we have the
following results:

(1) $f$ is a homomorphism from $(U_{1},\Delta_{1})$ to
$(U_{2},f(\Delta_{1}))$;

(2) $f(\mathscr{C}_{appearance})$ is superfluous in $f(\Delta_{1})$ if and
only if $\mathscr{C}_{appearance}$ is superfluous in $\Delta_{1}$;

(3)
$\\{f(\mathscr{C}_{price}),f(\mathscr{C}_{structure}),f(\mathscr{C}_{size})\\}$
is a reduct of $f(\Delta_{1})$ if and only if
$\\{\mathscr{C}_{price},\mathscr{C}_{structure},\mathscr{C}_{size}\\}$ is a
reduct of $\Delta_{1}$.

From Example 5.8, we see that the image system $(U_{2},f(\Delta_{1}))$ has
relatively smaller size than the original system $(U_{1},\Delta_{1})$. But
their attribute reductions are equivalent to each other under the condition of
a homomorphism.

From the practical viewpoint, it may be difficult to construct attribute
reducts of a large-scale fuzzy covering information system directly. However,
we can compress it into a relatively smaller fuzzy covering information system
under the condition of a homomorphism and conduct the attribute reductions on
the image system. Therefore, the notion of a homomorphism may provide a more
efficient approach to dealing with large-scale fuzzy covering information
systems.

###  5.2 Data compression of dynamic fuzzy covering information systems

In Subsection 5.1, we derive a partition based on each fuzzy covering shown in
Table 2, which is useful for data compression of dynamic fuzzy covering
information systems. In the following, we discuss how to compress two types of
dynamic fuzzy covering information systems by utilizing the compression of the
original fuzzy covering information system.

Type 1: Adding a family of fuzzy coverings. By adding a fuzzy covering
$\mathscr{C}_{m+1}$ to the fuzzy covering information system
$(U_{1},\Delta_{1})$, we obtain the dynamic fuzzy covering information system
$(U_{1},\Delta)$, where $\Delta=\Delta_{1}\cup\\{\mathscr{C}_{m+1}\\}$. There
are three steps to compress the dynamic fuzzy covering information system
$(U_{1},\Delta)$. First, we obtain the partition $U_{1}/\mathscr{C}_{m+1}$ in
the sense of Definition 5.6 and get Table 4 by adding
$U_{1}/\mathscr{C}_{m+1}$ into Table 2. Then we derive the partition
$U_{1}/\Delta$ based on $U_{1}/\mathscr{C}_{i}$ $(1\leq i\leq m+1)$.
Afterwards, we define the homomorphism $f$ based on $U_{1}/\Delta$ as Example
5.8 and compress $(U_{1},\Delta)$ into a small-scale fuzzy covering
information system $(f(U_{1}),f(\Delta))$. Furthermore, the same process can
be applied to the dynamic fuzzy covering information system when adding a
family of fuzzy coverings.

Type 2: Deleting a family of fuzzy coverings. We obtain the dynamic fuzzy
covering information system $(U_{1},\Delta)$ when deleting the fuzzy covering
$\mathscr{C}_{k}\in\Delta_{1}$, where
$\Delta=\Delta_{1}-\\{\mathscr{C}_{k}\\}$. To compress the dynamic fuzzy
covering information system $(U_{1},\Delta)$, we first derive Table 5 by
canceling the partition $U_{1}/\mathscr{C}_{k}$ in Table 2\. Then we obtain
the partition $U_{1}/\Delta$ based on $U_{1}/\mathscr{C}_{i}$ $(1\leq i\leq
k-1,k+1\leq i\leq m)$ and define the homomorphism $f$ as Example 5.8.
Afterwards, $(U_{1},\Delta)$ is compressed into a small-scale fuzzy covering
information system $(f(U_{1}),f(\Delta))$. Moreover, we can compress the
dynamic fuzzy covering information system when deleting a family of fuzzy
coverings using the same approach.

In practice, it may be very costly or even intractable to construct the
compression of the dynamic fuzzy covering information system as the original
fuzzy covering information system. Thus the proposed approach based on the
compression of the original fuzzy covering information system may provide a
more efficient approach to dealing with data compression of dynamic fuzzy
covering information systems.

##  6 Conclusion and further research

In this paper, we have presented some new operations on fuzzy coverings and
investigated their properties in detail. Particularly, the lower and upper
approximation operations based on fuzzy coverings have been introduced for the
fuzzy covering approximation space. Then we have constructed a consistent
function for the communication between fuzzy covering information systems, and
pointed out that a homomorphism is a special fuzzy covering mapping between
the two fuzzy covering information systems. In addition, we have proved that
attribute reductions of the original system and image system are equivalent to
each other under the condition of a homomorphism. We have also applied the
proposed approach to attribute reductions of fuzzy covering information
systems and dynamic fuzzy covering information systems.

In future, we will further study the fuzzy covering information systems by
extending the covering rough sets and apply the proposed method to feature
selections of fuzzy covering information systems. Furthermore, we will discuss
the data compression of dynamic relation information systems and dynamic fuzzy
relation information systems. Especially, we will apply an incremental
updating scheme to maintain the compression dynamically and avoid unnecessary
computations by utilizing the compression of the original system.

##  Acknowledgments

We would like to thank the anonymous reviewers very much for their helpful
comments and valuable suggestions. This work is supported by the National
Natural Science Foundation of China (NO. 11071061) and the National Basic
Research Program of China (NO. 2010CB334706).

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Table 1: An incomplete information system. $U$ |  $structure$ |  $color$ |  $price$
---|---|---|---
$x_{1}$ |  $bad$ |  $good$ |  $low$
$x_{2}$ |  $\ast$ |  $good$ |  $high$
$x_{3}$ |  $good$ |  $bad$ |  $high$
$x_{4}$ |  $bad$ |  $bad$ |  $\ast$
$x_{5}$ |  $good$ |  $\ast$ |  $low$
$x_{6}$ |  $\ast$ |  $bad$ |  $\ast$

Table 2: The partitions based on each fuzzy covering $\mathscr{C}_{i}$ $(1\leq i\leq m)$ and $\Delta_{1}$, respectively. $U_{1}$ |  $\mathscr{C}_{1}$ |  $\mathscr{C}_{2}$ |  . |  . |  . |  $\mathscr{C}_{m}$ |  $\Delta_{1}$
---|---|---|---|---|---|---|---
$x_{1}$ |  $P_{1x_{1}}$ |  $P_{2x_{1}}$ |  . |  . |  . |  $P_{mx_{1}}$ |  $P_{\Delta_{1}x_{1}}$
$x_{2}$ |  $P_{1x_{2}}$ |  $P_{2x_{2}}$ |  . |  . |  . |  $P_{mx_{2}}$ |  $P_{\Delta_{1}x_{2}}$
. |  . |  . |  . |  . |  . |  . |  .
. |  . |  . |  . |  . |  . |  . |  .
. |  . |  . |  . |  . |  . |  . |  .
$x_{n}$ |  $P_{1x_{n}}$ |  $P_{2x_{2}}$ |  . |  . |  . |  $P_{mx_{n}}$ |  $P_{\Delta_{1}x_{n}}$

Table 3: The partitions based on $\mathscr{C}_{prize},\mathscr{C}_{structure},\mathscr{C}_{size},\mathscr{C}_{appearance}$ and $\Delta_{1}$, respectively. $U_{1}$ | $\mathscr{C}_{price}$ | $\mathscr{C}_{structure}$ | $\mathscr{C}_{size}$ | $\mathscr{C}_{appearance}$ | $\Delta_{1}$
---|---|---|---|---|---
$x_{1}$ | $\\{x_{1},x_{2}\\}$ | $\\{x_{1},x_{2},x_{4},x_{7},x_{8}\\}$ | $\\{x_{1},x_{2},x_{3},x_{5},x_{6}\\}$ | $U_{1}$ | $\\{x_{1},x_{2}\\}$
$x_{2}$ | $\\{x_{1},x_{2}\\}$ | $\\{x_{1},x_{2},x_{4},x_{7},x_{8}\\}$ | $\\{x_{1},x_{2},x_{3},x_{5},x_{6}\\}$ | $U_{1}$ | $\\{x_{1},x_{2}\\}$
$x_{3}$ | $\\{x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}$ | $\\{x_{3},x_{5}\\}$ | $\\{x_{1},x_{2},x_{3},x_{5},x_{6}\\}$ | $U_{1}$ | $\\{x_{3},x_{5}\\}$
$x_{4}$ | $\\{x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}$ | $\\{x_{1},x_{2},x_{4},x_{7},x_{8}\\}$ | $\\{x_{4},x_{7},x_{8}\\}$ | $U_{1}$ | $\\{x_{4},x_{7},x_{8}\\}$
$x_{5}$ | $\\{x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}$ | $\\{x_{3},x_{5}\\}$ | $\\{x_{1},x_{2},x_{3},x_{5},x_{6}\\}$ | $U_{1}$ | $\\{x_{3},x_{5}\\}$
$x_{6}$ | $\\{x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}$ | $\\{x_{6}\\}$ | $\\{x_{1},x_{2},x_{3},x_{5},x_{6}\\}$ | $U_{1}$ | $\\{x_{6}\\}$
$x_{7}$ | $\\{x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}$ | $\\{x_{1},x_{2},x_{4},x_{7},x_{8}\\}$ | $\\{x_{4},x_{7},x_{8}\\}$ | $U_{1}$ | $\\{x_{4},x_{7},x_{8}\\}$
$x_{8}$ | $\\{x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}$ | $\\{x_{1},x_{2},x_{4},x_{7},x_{8}\\}$ | $\\{x_{4},x_{7},x_{8}\\}$ | $U_{1}$ | $\\{x_{4},x_{7},x_{8}\\}$

Table 4: The partitions based on each fuzzy covering $\mathscr{C}_{i}$ $(1\leq i\leq m+1)$ and $\Delta$, respectively. $U_{1}$ |  $\mathscr{C}_{1}$ |  $\mathscr{C}_{2}$ |  . |  . |  . |  $\mathscr{C}_{m}$ |  $\mathscr{C}_{m+1}$ |  $\Delta$
---|---|---|---|---|---|---|---|---
$x_{1}$ |  $P_{1x_{1}}$ |  $P_{2x_{1}}$ |  . |  . |  . |  $P_{mx_{1}}$ |  $P_{(m+1)x_{1}}$ |  $P_{\Delta x_{1}}$
$x_{2}$ |  $P_{1x_{2}}$ |  $P_{2x_{2}}$ |  . |  . |  . |  $P_{mx_{2}}$ |  $P_{(m+1)x_{2}}$ |  $P_{\Delta x_{2}}$
. |  . |  . |  . |  . |  . |  . |  . |  .
. |  . |  . |  . |  . |  . |  . |  . |  .
. |  . |  . |  . |  . |  . |  . |  . |  .
$x_{n}$ |  $P_{1x_{n}}$ |  $P_{2x_{2}}$ |  . |  . |  . |  $P_{mx_{n}}$ |  $P_{(m+1)x_{n}}$ |  $P_{\Delta x_{n}}$

Table 5: The partitions based on each fuzzy covering $\mathscr{C}_{i}$ $(1\leq i\leq k-1,k+1\leq i\leq m)$ and $\Delta$, respectively. $U_{1}$ | $\mathscr{C}_{1}$ | $\mathscr{C}_{2}$ | . | . | . | $\mathscr{C}_{k-1}$ | $\mathscr{C}_{k+1}$ | . | . | . | $\mathscr{C}_{m}$ | $\Delta$
---|---|---|---|---|---|---|---|---|---|---|---|---
$x_{1}$ | $P_{1x_{1}}$ | $P_{2x_{1}}$ | . | . | . | $P_{(k-1)x_{1}}$ | $P_{(k+1)x_{1}}$ | . | . | . | $P_{mx_{1}}$ | $P_{\Delta x_{1}}$
$x_{2}$ | $P_{1x_{2}}$ | $P_{2x_{2}}$ | . | . | . | $P_{(k-1)x_{2}}$ | $P_{(k+1)x_{2}}$ | . | . | . | $P_{mx_{2}}$ | $P_{\Delta x_{2}}$
. | . | . | . | . | . | . | . | . | . | . | . | .
. | . | . | . | . | . | . | . | . | . | . | . | .
. | . | . | . | . | . | . | . | . | . | . | . | .
$x_{n}$ | $P_{1x_{n}}$ | $P_{2x_{n}}$ | . | . | . | $P_{(k-1)x_{n}}$ | $P_{(k+1)x_{n}}$ | . | . | . | $P_{mx_{n}}$ | $P_{\Delta x_{n}}$

Potential referees

(1) Jerzy W. Grzymala-Busse, E-mail address<EMAIL_ADDRESS>Affiliations:
Department of Electrical Engineering and Computer Science, University of
Kansas, Lawrence, U. S. A.

(2) Yiyu Yao, E-mail address<EMAIL_ADDRESS>Affiliations: Department of
Computer Science, University of Regina, Regina, Saskatchewan, Canada.

(3) Tsau Young Lin, E-mail address<EMAIL_ADDRESS>Affiliations:
Department of Computer Science, San Jose State University, San Jose, U. S. A.

(4) Weizhi Wu, E-mail address<EMAIL_ADDRESS>Affiliations: School of
Mathematics, Physics and Information Science, Zhejiang Ocean University,
Zhoushan, Zhejiang 316000, P. R. China.

(5) Wei Yao, E-mail address<EMAIL_ADDRESS>Affiliations: Department of
Mathematics, Hebei University of Science and Technology, Shijiazhuang 050018,
P. R. China.

(6) Ping Zhu, E-mail address<EMAIL_ADDRESS>Affiliations: School of
Science, Beijing University of Posts and Telecommunications, Beijing 100876,
P. R. China.