cc -reduction in a fully fuzzy information system

Abstract

Attribute reduction is an important issue in knowledge discovery and data mining. Applying rough set theory, information systems based on crisp (or explicit) features can be easily done attribute reductions, but there exists rarely deeper discussion about attribute reduction in fuzzy information systems (i.e. information systems based on vague or undefinable features). A fully fuzzy information system is an information system where each of its attributes determines a fuzzy set on the object set, this paper investigates cc-reduction in this information system. The concept of class-consistent is first introduced. Then, the class-consistent relation cc(P) induced by the given attribute subset P in a fully fuzzy information system is proposed. Next, the class-consistent reduction (for short, cc-reduction) in a fully fuzzy information system is proposed and studied along with its corresponding algorithm. Moreover, considering that homomorphism is a kind of tools to study relationships between two fully fuzzy information systems, invariant characterizations of fully fuzzy information systems under homomorphisms are obtained. Finally, a numerical experiment is employed to illustrate the practical significance and possible applications of cc-reduction in a fully fuzzy information system.

Keywords

Rough set theory fully fuzzy information system class-consistent relation homomorphism algorithm characterization

1 Introduction

In 1982, Pawlak [11] proposed rough set theory. This theory is an important tool for the study of intelligent systems described by uncertain and incomplete information. The main idea is to make decision or classification rules through knowledge reduction under the condition of keeping the classification ability unchanged. So far, rough set theory has been applied to machine learning, decision analysis, process control, pattern recognition, inductive reasoning and data mining [12 –15].

Information systems based on rough set theory was also introduced by Pawlak. They may reveal large databases and knowledge discovery process mathematically. It is well known that attributes in the same information system are not equally important, even some of them are redundant. Attribute reductions in of a given information system mean deleting irrelevant or unimportant attributes under the condition of keeping the classification ability of this information system. As one of the core contents of rough set theory, attribute reduction has attracted a great deal of attention.

In real life, fuzziness is a universal phenomenon. Thus, various kinds of fuzzy information systems have been studied by many scholars. For example, Zhang et al. [4] studied knowledge reduction with fuzzy decision information systems, Guan et al. [4] gave knowledge reduction methods in fuzzy objective information systems, Huang [5] investigated graded dominance interval-based fuzzy objective information systems, Huang et al. [7] proposed dominance-based rough set model in intuitionistic fuzzy information systems, Sun et al. [17] presented fuzzy rough set theory for interval-valued fuzzy information systems, Huang et al. [6] introduced a dominance relation in an interval-valued intuitionistic fuzzy information system and established a dominance-based rough set model for this fuzzy information system, Cheng et al. [1] considered rule extraction based on granulation order in interval-valued fuzzy information system, Feng et al. [2] studied belief functions on general intuitionistic fuzzy information systems, Lang et al. [9] researched homomorphisms-based attribute reduction of dynamic fuzzy covering information systems, Zhang et al. [20] obtained generalized dominance rough set models for the dominance intuitionistic fuzzy information systems. Zhang et al. [21] considered information structures and uncertainty measures in a fully fuzzy information system.

Fuzzy set was proposed by Zadeh [19] as an extension of classical set to depict fuzziness. A fully fuzzy information system is an information system where each of its attributes determines a fuzzy set on the object set. So far, however, attribute reduction in a fully fuzzy information system has not been reported. This paper will address this topic.

Given a fully fuzzy information system (U, A). For a ∈ A and two objects x, y of U, a (x) and a (y) is the information function values of the objects x and y on the attribute a, respectively. But, a (x) and a (y) are also the membership degrees of x and y to the fuzzy set a, respectively. It should be noted that a (x) and a (y) are two numbers in [0, 1]. Then, “a (x) = a (y) " is actually difficult to be achieved. For this reason, we may consider the concept of “class-consistent" and then propose the class-consistent relation on the object set U. The aim of this paper is to give an attempt to study attribute reduction based on class-consistent relations (for short, cc-reduction) for a fully fuzzy information system. And the work process of the paper is displayed in Fig. 1.

Fig.1

The work process of the paper.

The remaining part of this paper is organized as follows. In Section 2, we recall some concepts about rough sets and fully fuzzy information systems, and propose class-consistent, class-preserving mappings, class-consistent relations and homomorphisms. In Section 3, we introduce cc-reduction in a fully fuzzy information system and give its properties. In Section 4, we put forward an algorithm on cc-reduction in a fully fuzzy information system. In Section 5, we obtain some invariant characterizations of fully fuzzy information systems under homomorphisms. In Section 6, we give a numerical experiment on the Yeast data set. In Section 7, we summarize this paper.

2 Preliminaries

In this section, we briefly recall basic concepts about rough sets and fully fuzzy information systems, introduce class-consistent, class-preserving mappings, class-consistent relations and homomorphisms, and give the related properties.

Throughout this paper, U and V denote two non-empty finite sets, 2^U expresses the family of all subsets of U and |X| means the cardinality of X ∈ 2^U.

Put $U = {x_{1}, x_{2}, \dots, x_{n}}, V = {y_{1}, y_{2}, \dots, y_{m}} .$

2.1 Rough sets

Rough set theory was initiated by Pawlak [11, 12] for dealing with vagueness and granularity in information systems.

Given that R is an equivalence relation on U. Then the pair (U, R) is called a Pawlak approximation space. For each subset X of U, one can define the following rough approximations: $\underline{R} (X) = {x \in U : [x]_{R} \subseteq X},$ $\bar{R} (X) = {x \in U : [x]_{R} \cap X \neq \emptyset} .$ Then $\underline{R} (X)$ and $\bar{R} (X)$ are called the Pawlak lower approximation and the Pawlak upper approximation of X, respectively.

The Pawlak boundary region Bnd_R (X) of X is defined as the difference set $\bar{R} (X) - \underline{R} (X)$ . Obviously, $\underline{R} (X) \subseteq X \subseteq \bar{R} (X)$ .

A set is Pawlak rough if its boundary region is not empty; otherwise, it is definable. Thus, X is Pawlak rough if $\underline{R} (X) \neq \bar{R} (X)$ .

Definition 2.1. ([11]) Let U be a finite set of objects and A a finite set of attributes. Then the pair (U, A) is called an information system, if each attribute a of A determines an information function a : U → V_a, where V_a is the information function value set of the attribute a.

Let (U, A) be an information system. If A = C ∪ D where C is a set of conditional attributes and D is a set of decision attributes, then (U, A) is called a decision information system.

Let (U, A) be an information system. Given P ⊆ A. Then an equivalence relation (or indiscernibility relation) IND (P) can be defined by $(x, y) \in IND (P) \Leftrightarrow \forall a \in P, a (x) = a (y) .$

It is noticed that the original study of information systems is based on this equivalence relation (or indiscernibility relation).

Example 2.2. Table 1 depicts an information system about cars where U = {x₁, x₂, x₃, x₄, x₅, x₆} is the set of cars and A = {Price, Mileage, Size, Max-speed} = {a₁, a₂, a₃, a₄} is the set of attributes.

Table 1
An information system about cars

Price Mileage Size Max-speed

x ₁ high low full low

x ₂ low very low full low

x ₃ low high compact low

x ₄ high high full high

x ₅ high very-high full high

x ₆ low high full very high

	Price	Mileage	Size	Max-speed
x ₁	high	low	full	low
x ₂	low	very low	full	low
x ₃	low	high	compact	low
x ₄	high	high	full	high
x ₅	high	very-high	full	high
x ₆	low	high	full	very high

Pick P = {a₃, a₄}. It should be noted that

IND ({a₃}) = ▵ ∪ {(x₁, x₂) , (x₂, x₁) , (x₁, x₄) , (x₄, x₁) , (x₁, x₅) , (x₅, x₁) , (x₁, x₆) , (x₆, x₁) , (x₂, x₄) , (x₄, x₂) , (x₂, x₅) , (x₅, x₂) , (x₂, x₆) , (x₆, x₂) , (x₄, x₅) , (x₅, x₄) , (x₄, x₆) , (x₆, x₄) , (x₅, x₆) , (x₆, x₅)},

IND ({a₄}) = ▵ ∪ {(x₁, x₂) , (x₂, x₁) , (x₁, x₃) , (x₃, x₁) , (x₂, x₃) , (x₃, x₂) , (x₄, x₅) , (x₅, x₄)}.

Then IND (P) = ▵ ∪ {(x₁, x₂) , (x₂, x₁) , (x₄, x₅) , (x₅, x₄)}.

2.2 Fully fuzzy information systems

Fuzzy sets are extensions of classical sets [19]. A fuzzy set P on U is defined as a function assigning to a value P (x) ∈ [0, 1] for each element x ∈ U and P (x) is called the membership degree of x to the fuzzy set P.

Definition 2.3. ([11, 24]) Given that (U, A) is an information system. Then (U, A) is referred to as a fuzzy information system, if there exists a ∈ A such that V_a = [0, 1] (i.e., the attribute a determines a fuzzy set on U or the attribute a is fuzzy).

If each attribute determines a fuzzy set on U, then (U, A) is called a fully fuzzy information system.

Here, a attribute is not naturally fuzzy. In fact, it originates from the information function values of objects under the attribute rather than a attribute itself that brings about the fuzzy indiscernibility.

Let (U, C ∪ D) be a decision information system. If each attribute of C is fuzzy, then (U, C ∪ D) is called a fuzzy condition information system; if each attribute of D is fuzzy, then (U, C ∪ D) is called a fuzzy objective information system; if each attribute of C ∪ D is fuzzy, then (U, C ∪ D) is called a fuzzy condition and fuzzy objective information system.

Obviously, a fuzzy condition and fuzzy objective information system is a fully fuzzy information system.

In this paper, we discuss fully fuzzy information systems whether they are decision information systems or non-decision information systems.

Given a fully fuzzy information system (U, A). Then every attribute a ∈ A determines a fuzzy set (which is still denoted by a) in the object set U. For two objects x, y of U, a (x) and a (y) are the information function values of the objects x and y on the attribute a, respectively. But a (x) and a (y) are also the membership degrees of the objects x and y to the fuzzy set a, respectively. It should be noted that a (x) and a (y) are two number in the unit interval [0, 1]. Then, “a (x) = a (y)" is actually very difficult to be achieved. Thus, we can not consider IND (P) in the fully fuzzy information system (U, A). As to a fuzzy objective information system, we usually deal with it by means of the inclusion degree (see [24]).

In this paper, we attempt to deal with fully fuzzy information systems from another viewpoint. More specifically, given a given fully fuzzy information system, since the range of values of every information function is the unit interval [0, 1], we may divide the unit interval into finitely many subinterval which are mutually disjoint and assume that the two information values in the same subinterval provide the same information. This reason leads to the following concept of “class-consistent".

Definition 2.4. Suppose k ∈ N. Given a, b ∈ [0, 1]. If a = b = 0 or $a, b \in (0, \frac{1}{10^{k}})$ or $a = b = \frac{1}{10^{k}}$ or ··· ·· · or $a = b = \frac{10^{k} - 1}{10^{k}}$ or $a, b \in (\frac{10^{k} - 1}{10^{k}}, 1)$ or a = b = 1, then a and b are said to be class-consistent, and k is said to be a threshold value. We denote it by a ≈ _kb.

In this paper, we pick k = 1.

In order to study “class-consistent" mentioned above, let us consider the following mapping.

Definition 2.5. A mapping f : [0, 1] → [0, 1] is called class-preserving, if for any a, b ∈ [0, 1], $a \approx_{1} b \Leftrightarrow f (a) \approx_{1} f (b) .$

Example 2.6. Define a mapping f : [0, 1] → [0, 1] by $f (x) = 1 - x (\forall x \in [0, 1]) .$ Then f is class-preserving.

Definition 2.7. Let (U, A) be a fully fuzzy information system. For any P ⊆ A, denote $cc (P) = {(x, y) \in U \times U : \forall a \in P, a (x) \approx_{1} a (y)} .$ Then cc (P) is called the class-consistent relation on U induced by P.

It is worth mentioning that the purpose of the above definition is to deal with a fully fuzzy information system by using a discrete method.

Obviously, the class-consistent relation cc (P) is an equivalence relation on U, cc (P) = ⋂ _a∈Pcc ({a}) .

In particular, if cc (P) = δ, then cc (P) is called a universal relation on U; if cc (P) =▵, then cc (P) is called an identity relation on U.

In this paper, we stipulate that cc (∅) = δ.

For P ⊆ A and x ∈ U, denote $[x]_{cc (P)} = {y \in U : (x, y) \in cc (P)},$ $U / cc (P) = {_{cc (P)} : x \in U} .$

The universe U may be divided into some disjoint classes by the equivalence relation cc (P), which is said to be a quotient set. This quotient set is namely U/cc (P).

For P ⊆ A, X ∈ 2^U is characterized by $\bar{P} (X)$ and $\underline{P} (X)$ , where $\underline{P} (X) = \cup {Y \in U / cc (P) : Y \subseteq X},$ $\bar{P} (X) = \cup {Y \in U / cc (P) : Y \cap X \neq \emptyset} .$

It is easy to prove that $\underline{P} (X) = {x \in U : [x]_{cc (P)} \subseteq X},$ $\bar{P} (X) = {x \in U : [x]_{cc (P)} \cap X \neq \emptyset} .$

It is worth mentioning that the partition or quotient set U/cc (P) of U induced by the equivalence relation cc (P) provides information granules for describing the concept X ∈ 2^U.

Theorem 2.8. Let (U, A) be a fully fuzzy information system. Then for P, Q ⊆ A, the following conditions are equivalent:

(1) cc (P) = cc (Q);

(2) ∀ i, [x_i] _cc(P) = [x_i] _cc(Q);

(3) U/cc (P) = U/cc (Q).

Proof. This is obvious. □

Theorem 2.9. Given that (U, A) is a fully fuzzy information system. Then for P, Q ⊆ A, the following conditions are equivalent:

(1) cc (P) ⊆ cc (Q);

(2) ∀ i, [x_i] _cc(P) ⊆ [x_i] _cc(Q);

(3) U/cc (P) refines U/cc (Q), i.e., for each X ∈ U/cc (P), there exists Y ∈ U/cc (Q) such that X ⊆ Y.

Proof. (1) ⇒ (2) ⇒ (3) is obvious.

(3) ⇒ (1). ∀ (x, y) ∈ cc (P), we have x, y ∈ [x] _cc(P). Since U/cc (P) refines U/cc (Q), there exists Y ∈ U/cc (Q) such that [x] _cc(P) ⊆ Y. Then x, y ∈ Y.

It should be noted that Y ∈ U/cc (Q). Then there exists x^* ∈ U such that [x^*] _cc(Q) = Y. This implies x, y ∈ [x^*] _cc(Q).

Since cc (Q) is an equivalence relation on U, we have [x^*] _cc(Q) = [x] _cc(Q) = [y] _cc(Q). Obviously, y ∈ [y] _cc(Q). This implies y ∈ [x] _cc(Q). Then (x, y) ∈ cc (Q).

Thus cc (P) ⊆ cc (Q). □

Theorem 2.10. Suppose that (U, A) is a fully fuzzy information system. Given P, Q ⊆ A. Denote $σ (U / cc (P)) = {⋃_{x \in X} [x]_{cc (P)} : X \in 2^{U}},$

U/cc (Q) = {D₁, D₂, ⋯ , D_r} .

Then the following conditions are equivalent:

(1) cc (P) ⊆ cc (Q);

(2) ∀ j, D_j ∈ σ (U/cc (P));

(3) ∀ j, $\underline{P} (D_{j}) = D_{j}$ ;

(4) $⋃_{j = 1}^{r} \underline{P} (D_{j}) = U$ ;

Proof. (1) ⇒ (2). ∀ x ∈ D_j, we have D_j = [x] _cc(Q). Since cc (P) ⊆ cc (Q), by Theorem 2.9, [x] _cc(P) ⊆ [x] _cc(Q). Then {x} ⊆ [x] _cc(P) ⊆ D_j. So

$D_{j} = ⋃_{x \in D_{j}} {x} \subseteq ⋃_{x \in D_{j}} [x]_{cc (P)} \subseteq D_{j} .$

Thus D_j = ⋃ _{x∈D_j} [x] _cc(P) ∈ σ (U/cc (P)).

(2) ⇒ (3). Since D_j ∈ σ (U/R), there exists X ∈ 2^U such that D_j = ⋃ _x∈X [x] _cc(P).

Thus $\underline{P} (D_{j}) = \underline{P} (⋃_{x \in X} [x]_{cc (P)}) = ⋃_{x \in X} [x]_{cc (P)} = D_{j}$ .

(3) ⇒ (4). This is obvious.

(4) ⇒ (1). ∀ i, ∀ y ∈ [x_i] _cc(P). Since $⋃_{j = 1}^{r} \underline{P} (D_{j}) = U$ , there exists j such that $x_{i} \in \underline{P} (D_{j})$ . Then [x_i] _cc(P) ⊆ D_j. So x_i, y ∈ D_j. Put D_j = [x^*] _cc(Q). x_i ∈ D_j implies [x_i] _cc(Q) = [x^*] _cc(Q) = D_j. Then y ∈ [x_i] _cc(Q).

So ∀ i, [x_i] _cc(P) ⊆ [x_i] _cc(Q).

By Theorem 2.9, cc (P) ⊆ cc (Q). □

2.3 Homomorphisms between fully fuzzyinformation systems

Communication between information systems is an important issue. In mathematics, it can be viewed as a mapping between information systems. The notion of homomorphisms between information systems as a kind of tools to study communication between information systems with rough sets was introduced by Grzymala-Busse in [3, 10]. As explained in [16], homomorphisms allow us to translate the information contained in one granular world into the granularity of another granular world, and thus a communication mechanism can be provided for exchanging information with other granular worlds. Li et al. [8] studied invariant characterizations of information systems under homomorphisms.

Definition 2.11. ([8]) Suppose that (U, A) and (V, B) are two information systems. Given that h₁ : U → V, h₂ : A → B and h₃ : U^* → V^* are three mappings where

U^* = {a (x) : x ∈ U, a ∈ A} ,

V^* = {b (y) : y ∈ V, b ∈ B} .

Then the triple h = (h₁, h₂, h₃) is called a homomorphism from (U, A) to (V, B), if for all x ∈ U and a ∈ A, $h_{3} (a (x)) = h_{2} (a) (h_{1} (x)) .$

The above definition may also apply to fully fuzzy information systems.

Definition 2.12. Let (U, A) and (V, B) be two fully fuzzy information systems. Given that h₁ : U → V, h₂ : A → B and h₃ : U^* → V^* are three mappings where

U^* = {a (x) : x ∈ U, a ∈ A} ⊊ [0, 1] ,

V^* = {b (y) : y ∈ V, b ∈ B} ⊊ [0, 1] .

Then the triple h = (h₁, h₂, h₃) is called a homomorphism from (U, A) to (V, B), if h₃ is class-preserving and for any x ∈ U, a ∈ A, $h_{3} (a (x)) \approx_{1} h_{2} (a) (h_{1} (x)) .$ We write $(U, A) \sim_{h} (V, B)$

3 cc-reduction in a fully fuzzy informationsystem

Given a fully fuzzy information system (U, A), each subset P of A determines the equivalence relation (or indiscernibility relation) cc (P). cc (P) or U/cc (P) represents the classification ability of the subsystem (U, P). If cc (P) = cc (A), then the subsystem (U, P) and the system (U, A) have the same classification abilities. A class-consistent reduction means reducing the number of the attributes in (U, A) to the minimum without distorting the classification ability of (U, A), which plays a vital role in saving expensive tests and time when addressing decision-making problems.

Definition 3.1. Let (U, A) be a fully fuzzy information system. Given P ⊆ A.

(1) P is called a class-consistent coordinate (for short, cc-coordinate) subset of A, if cc (A) = cc (P).

(2) a ∈ P is called cc-independent in P, if cc (P - {a}) ≠ cc (P); P is called a cc-independent subset of A, if for each a ∈ P, a is cc-independent in P.

(3) P is called a class-consistent reduction (for short, cc-reduction) of A, if P is both cc-coordinate and cc-independent.

In this paper, the family of all cc-coordinate subsets (resp., cc-reductions) of A is denoted by co^cc (A) (resp., red^cc (A)).

Obviously,

P ∈ red^cc (A) ⇔ P ∈ co^cc (A) and ∀ a ∈ P, P - {a} ∉ co^cc (A) .

Proposition 3.2. Given that (U, A) is a fully fuzzy information system. Then there always exists a cc-reduction of A.

Proof. Suppose that ∀ a ∈ A, A - {a} ∉ co^cc (A). Then A ∈ red^cc (A).

Suppose that ∃ a₁ ∈ A, A - {a₁} ∈ co^cc (A). Then, we consider A - {a₁}. Again suppose that ∀ a ∈ A - {a₁}, (A - {a₁}) - {a} ∉ co^cc (A). Then A - {a₁} ∈ red^cc (A). Again suppose that ∃ a₂ ∈ A - {a₁}, (A - {a₁}) - {a₂} ∈ co^cc (A). Then, we consider A - {a₁, a₂}. Repeat this process. Since A is finite, we can find a cc-reduction of A.

Thus, there always exists a cc-reduction of A. □

Definition 3.3 Let (U, A) be a fully fuzzy information system. For any x, y ∈ U, put $D^{cc} (x, y) = {a \in A : a (x) ≉_{1} a (y)} .$ Then

(1) D^cc (x, y) is called the cc-discernibility set on x and y.

(2) $D^{cc} (A) = (d_{ij})_{n \times n}$ is called the cc-discernibility matrix where U = {x₁, x₂, ·· · , x_n} and d_ij = D^cc (x_i, x_j) (1 ≤ i, j ≤ n).

Example 3.4. Table 2 depicts a fully fuzzy information system where U = {x₁, x₂, x₃, x₄, x₅, x₆} and A = {a₁, a₂, a₃, a₄}.

Table 2
An information system about cars

         a₁         a₂         a₃         a₄

x₁         0.55         0.84         0.55         0.32

x₂         0.58         0.82         0.58         0.43

x₃         0.58         0.82         0.58         0.43

x₄         0.52         0.85         0.52         0.61

x₅         0.22         0.88         0.22         0.36

x₆         0.27         0.77         0.27         0.47

	a₁	a₂	a₃	a₄
x₁	0.55	0.84	0.55	0.32
x₂	0.58	0.82	0.58	0.43
x₃	0.58	0.82	0.58	0.43
x₄	0.52	0.85	0.52	0.61
x₅	0.22	0.88	0.22	0.36
x₆	0.27	0.77	0.27	0.47

The cc-discernibility matrix $D^{cc} (A)$ of A is obtained as follows: $(\begin{matrix} \emptyset & {a_{2}} & A & A & {a_{2}} & {a_{1}, a_{4}} \\ {a_{2}} & \emptyset & {a_{1}, a_{3}, a_{4}} & {a_{1}, a_{3}, a_{4}} & \emptyset & {a_{1}, a_{2}, a_{4}} \\ A & {a_{1}, a_{3}, a_{4}} & \emptyset & \emptyset & {a_{1}, a_{3}, a_{4}} & {a_{2}, a_{3}} \\ A & {a_{1}, a_{3}, a_{4}} & \emptyset & \emptyset & {a_{1}, a_{3}, a_{4}} & {a_{2}, a_{3}} \\ {a_{2}} & \emptyset & {a_{1}, a_{3}, a_{4}} & {a_{1}, a_{3}, a_{4}} & \emptyset & {a_{1}, a_{2}, a_{4}} \\ {a_{1}, a_{4}} & {a_{1}, a_{2}, a_{4}} & {a_{2}, a_{3}} & {a_{2}, a_{3}} & {a_{1}, a_{2}, a_{4}} & \emptyset \end{matrix})$

Proposition 3.5. Given that (U, A) is a fully fuzzy information system. The for any x, y, z ∈ U,

(1) D^cc (x, x) =∅.

(2) D^cc (x, y) = D^cc (y, x).

(3) D^cc (x, y) ⊆ D^cc (x, z) ∪ D^cc (z, y).

Proof. (1) and (2) are obvious.

(3) Suppose that D^cc (x, y) notsubseteqD^cc (x, z) ∪ D^cc (z, y). Then D^cc (x, y) - D^cc (x, z) ∪ D^cc (z, y)≠∅. Pick $a \in D^{cc} (x, y) - D^{cc} (x, z) \cup D^{cc} (z, y) .$

a ∈ D^cc (x, y) implies (x, y) ∉ cc ({a}).

Since a ∉ D^cc (x, z) ∪ D^cc (z, y), we have a ∉ D^cc (x, z) and a ∉ D^cc (z, y). Then (x, z) ∈ cc ({a}) and (z, y) ∈ cc ({a}).

Thus D^cc (x, y) ⊆ D^cc (x, z) ∪ D^cc (z, y). □

Corollary 3.6. Let (U, A) be a fully fuzzy information system. Then d is a distance function on U where $d (x, y) = | D^{cc} (x, y) | (x, y \in U) .$

Proof. This holds by Proposition 3.5. □

Proposition 3.7. Given that (U, A) is a fully fuzzy information system. Then $P \in {co}^{cc} (A) \Leftrightarrow if (x, y) \notin cc (A), then P \cap D^{cc} (x, y) \neq \emptyset .$

Proof. “⇒”. Let (x, y) ∉ cc (A). Since P ∈ co^cc (A), we have cc (P) = cc (A) . Then (x, y) ∉ cc (P) . So there exists a ∈ P such that a (x) _notapprox1a (y). This implies a ∈ D^cc (x, y). Then a ∈ P ∩ D^cc (x, y).

Thus P ∩ D^cc (x, y) ≠ ∅ .

“⟸". Suppose P ∉ co^cc (A). Then cc (P) ≠ cc (A) . This implies cc (P) - cc (A) ≠ ∅ . Pick $(x, y) \in cc (P) - cc (A) .$

Since (x, y) ∉ cc (A), we have P ∩ D^cc (x, y) ≠ ∅ .

It should be noted that (x, y) ∈ cc (P). Then for each a ∈ P, a (x) ≈ ₁a (y). So a ∉ D^cc (x, y) . Thus P ∩ D^cc (x, y) = ∅ . This is a contradiction.

Hence P ∈ co^cc (A). □

cc-discernibility sets can easily determine cc-reductions.

Theorem 3.8. Let (U, A) be a fully fuzzy information system. Then P ∈ red^cc (A) ⇔ (i) if (x, y) ∉ cc (A), then P∩ D^cc (x, y) ≠ ∅ ;

(ii) ∀ a ∈ P, ∃ (x_a, y_a) ∈ cc (A), (P - {a})∩ D^cc (x_a, y_a) = ∅.

Proof. This holds by Proposition 3.7. □

Definition 3.9. Given that (U, A) is a fully fuzzy information system. Put ${core}^{cc} (A) = ⋂_{P \in {red}^{cc} (A)} P .$ Then core^cc (A) is called the cc-core of A. Moreover,

(1) a ∈ A is called a necessary cc-attribute, if a ∈ core^cc (A).

(2) a ∈ A is called a relatively necessary cc-attribute, if a ∈ ⋃ _{P∈red^cc(A)}P - core^cc (A).

(3) a ∈ A is called an unnecessary cc-attribute, if a ∈ A - ⋃ _{P∈red^cc(A)}P.

Proposition 3.10. Let (U, A) be a fully fuzzy information system. Then $| {red}^{cc} (A) | = 1 \Leftrightarrow {core}^{cc} (A) \in {red}^{cc} (A) .$

Proof. “⇒” is obvious.

“⟸". Denote red^cc (A) = {P_k : 1 ≤ k ≤ n}. We only need to prove n = 1.

Suppose n ≥ 2. Since core^cc (A) ∈ red^cc (A), there exists i such that core^cc (A) = P_i. Pick j ≠ i. Then $P_{i} = ⋂_{k = 1}^{n} P_{k} \subseteq P_{j}$ . Put P_i ≠ P_j. Thus P_i ⊂ P_j. Since P_j ∈ red^cc (A), we have P_i ∉ co^cc (A). Then P_i ∉ red^cc (A). This is a contradiction.

Thus n = 1. □

cc-discernibility sets can easily determine the cc-core.

Proposition 3.11. Given that (U, A) is a fully fuzzy information system. Then the following conditions are equivalent:

(1) a is a necessary cc-attribute;

(2) a is cc-independent in A;

(3) ∃ x, y ∈ U, D^cc (x, y) = {a}.

Proof.

(1) ⇒ (2). Let a be a necessary cc-attribute. Suppose that a is not cc-independent in A. Then $cc (A - {a}) = cc (A) .$ This implies A - {a} ∈ co^cc (A). Consider A - {a}, by Proposition 3.2, ∃ P ⊆ A - {a}, P ∈ red^cc (A).

P ⊆ A - {a} implies a ∉ P. Then a is not a necessary cc-attribute. This is a contradiction.

(2) ⇒ (1). Let a be cc-independent in A. Suppose that a is not a necessary cc-attribute. Then ∃ P ∈ red^cc (A), a ∉ P. So P ⊆ A - {a} ⊆ A. This implies $cc (P) \supseteq cc (A - {a}) \supseteq cc (A) .$

Since P ∈ red^cc (A), we have cc (P) = cc (A). Then cc (A - {a}) = cc (A) . So a is not cc-independent in A. This is a contradiction.

(2) ⇒ (3). Since a is cc-independent in A, we have cc (A - {a}) ≠ cc (A). Then cc (A - {a}) - cc (A)≠ ∅. Pick $(x, y) \in cc (A - {a}) - cc (A) .$ Denote A = {a₁, a₂, …, a_n}. Then ∃ j, a = a_j. So $(x, y) \in ⋂_{1 \leq i \leq n, i \neq j} cc ({a_{i}}) - ⋂_{1 \leq i \leq n} cc ({a_{i}}) .$ This implies that (x, y) ∉ cc ({a_j}) and (x, y) ∈ cc ({a_i}) (i ≠ j) .

Then a_j ∈ D^cc (x, y), a_i ∉ D^cc (x, y) (i ≠ j) .

Thus D^cc (x, y) = {a_j} = {a}.

(3) ⇒ (2). Since ∃ x, y ∈ U, D^cc (x, y) = {a}, we have $(x, y) \notin cc ({a}), (x, y) \in cc ({a^{'}}) (a^{'} \neq a) .$

Then (x, y) ∈ cc (A - {a}). But (x, y) ∉ cc (A).

Thus cc (A - {a}) ≠ cc (A) . Hence a is cc-independent in A. □

Proposition 3.12. Let (U, A) be a fully fuzzy information system. Denote $R (a) = ⋃_{P \in {co}^{cc} (A)} cc (P - {a}) .$ Then the following conditions are equivalent:

(1) a is an unnecessary cc-attribute;

(2) ∀ P ∈ co^cc (A), P - {a}∈ co^cc (A) ;

(3) R (a) = cc (A);

(4) R (a) ⊆ cc ({a}).

Proof. (1) ⇒ (2). Given P ∈ co^cc (A). By Proposition 3.2, ∃ C ⊆ P, C ∈ red^cc (A). Since a is an unnecessary cc-attribute, we have a ∉ C, which implies C ⊆ A - {a}. Then $C \subseteq P \cap (A - {a}) = P - {a} \subseteq P .$ We have $cc (C) \supseteq cc (P - {a}) \supseteq cc (P) .$

It should be noted that P ∈ co^cc (A) and C ∈ red^cc (A). Then cc (P) = cc (A) = cc (C).

Thus cc (P - {a}) = cc (A) .

Hence P - {a} ∈ co^cc (A) .

(2) ⇒ (3) ⇒ (4) are obvious.

(4) ⇒ (1). Suppose that a is not an unnecessary cc-attribute. Then ∃ P ∈ red^cc (A), a ∈ P. This implies P - {a} ⊂ P. Since P ∈ red^cc (A), we have P - {a} ∉ co^cc (A). Then cc (P - {a}) - cc (A) ≠ ∅ . P ∈ red^cc (A) implies cc (P) = cc (A). Then $cc (P - {a}) - cc (P) \neq \emptyset .$

Pick (x, y) ∈ cc (P - {a}) - cc (P). It should be noted that cc (P) = cc (P - {a}) ∩ cc ({a}). Then (x, y) ∉ cc ({a}).

Since P ∈ co^cc (A) and R (a) ⊆ cc ({a}), we have cc (P - {a}) ⊆ cc ({a}). Then (x, y) ∈ cc ({a}). This is a contradiction. □

Theorem 3.13. Given that (U, A) is a fully fuzzy information system. Then

(1) a is a necessary cc-attribute ⇔ A - {a} ∉ co^cc (A).

(2) a is a relatively necessary cc-attribute ⇔ A - {a} ∈ co^cc (A) and R (a) notsubseteqcc ({a}).

(3) a is an unnecessary cc-attribute ⇔ ∀ P ∈ co^cc (A), P - {a} ∈ co^cc (A) .

Proof. This holds by Propositions 3.11 and 3.12. □

Example 3.14. In Example 3.4, we have

(1) a₂ is a necessary cc-attribute;

(2) a₁ and a₄ are two relatively necessary cc-attributes;

(3) a₁, a₃ and a₄ are three unnecessary cc-attributes.

4 An algorithm on cc-reduction in a fullyfuzzy information system

It is more convenient to calculate all cc-reductions and the cc-core in a fully fuzzy information system by using the following cc-discernibility function.

Below, we give an algorithm on cc-reduction in a fully fuzzy information system by using mathematical logic.

“⋁”(disjunction), “⋀”(conjunction), “⟶”(implication), “↔”(biimplication) are propositional connectives in mathematical logic. They are read as “or”, “and”, “if-then”, “if and only if”, respectively.

Let (U, A) be a fully fuzzy information system. ∀ a ∈ A, we specify a Boolean variable “a”. If D^cc (x, y) = {a₁, a₂, ·· · , a_k} with x, y ∈ U, then we specify a Boolean function a₁ ∨ a₂ ∨ ·· · ∨ a_k.

Denote $⋁ {a_{1}, a_{2}, \cdot \cdot \cdot, a_{k}} or ⋁_{i = 1}^{k} a_{i} = a_{1} \lor a_{2} \lor \cdot \cdot \cdot \lor a_{k},$ $⋀ {a_{1}, a_{2}, \cdot \cdot \cdot, a_{k}} or ⋀_{i = 1}^{k} a_{i} = a_{1} \land a_{2} \land \cdot \cdot \cdot \land a_{k} .$

We stipulate that ∨ ∅ =1 and ∧ ∅ =0 where 0 and 1 are two Boolean constants.

Definition 4.1. Let (U, A) be a fully fuzzy information system. Then the cc-discernibility matrix of A, denoted by $D^{cc} (A) = (d_{ij})_{n \times n}$ , is define as $Δ^{cc} (A) = ⋀ (⋁ d_{ij}) .$

Example 4.2. In Example 3.4, we have

▵ (A) = a₂ ∧ (a₁ ∨ a₂ ∨ a₃ ∨ a₄) ∧ (a₁ ∨ a₄) ∧ (a₁ ∨ a₃ ∨ a₄) ∧ (a₁ ∨ a₂ ∨ a₄) ∧ (a₂ ∨ a₃) .

Denote $L (A) = {⋁ d_{ij} : 1 \leq i, j \leq n} .$

A binary relation “≤” on L (A) is defined as follows: $⋁ d_{ij} \leq ⋁ d_{kl} \Leftrightarrow d_{ij} \subseteq d_{kl} for any ⋁ d_{ij}, ⋁ d_{kl} \in L (A) .$

For any ⋁d_ij, ⋁ d_kl ∈ L (A), we denote $(⋁ d_{ij}) ⨆ (⋁ d_{kl}) = ⋁ (d_{ij} \cup d_{kl}),$ $(⋁ d_{ij}) ⊓ (⋁ d_{kl}) = ⋁ (d_{ij} \cap d_{kl}) .$

Proposition 4.3. (L (A) , ≤) is a poset.

Proof. (1) ⋁d_ij ≤ ⋁ d_ij for any ⋁d_ij ∈ L (A).

(2) Given ⋁d_ij, ⋁ d_kl ∈ L (A). Suppose that ⋁d_ij ≤ ⋁ d_kl and ⋁d_kl ≤ ⋁ d_ij. Then d_ij ⊆ d_kl and d_kl ⊆ d_ij. This implies that d_ij = d_kl. So ⋁d_ij = ⋁ d_kl.

(3) Given ⋁d_ij, ⋁ d_kl, ⋁ d_hv ∈ L (A). Suppose that ⋁d_ij ≤ ⋁ d_kl and ⋁d_kl ≤ ⋁ d_hv. Then d_ij ⊆ d_kl and d_kl ⊆ d_hv. This implies that d_ij ⊆ d_hv. So ⋁d_ij ≤ ⋁ d_hv.

Thus (L (A) , ≤) is a poset. □

Proposition 4.4 Let (U, A) be a fully fuzzy information system. Given U = {x₁, x₂, ·· · , x_n}. If {d_ij : 1 ≤ i, j ≤ n} is a topology on A, then (L (A) , ≤ , ⊔ , ⊓) is a lattice with top element and bottom element.

Proof. Denote τ = {d_ij : 1 ≤ i, j ≤ n}. By Proposition 4.3, (L (A) , ≤) is a poset. For ⋁d_ij, ⋁ d_kl ∈ L (A), since τ is a topology on A, we have d_ij ∪ d_kl ∈ τ, d_ij ∩ d_kl ∈ τ. This implies $(⋁ d_{ij}) ⊔ (⋁ d_{kl}) = ⋁ (d_{ij} \cup d_{kl}) \in L (A),$ $(⋁ d_{ij}) ⊓ (⋁ d_{kl}) = ⋁ (d_{ij} \cap d_{kl}) \in L (A) .$ Obviously, 1_L(A) = ∨ A, 0_L(A) =∨ ∅.

Thus (L (A) , ≤ , ⊔ , ⊓) is a lattice with top element and bottom element. □

Example 4.5. In Example 3.4, (⋁ d₂₃) ⊓ (⋁ d₆₃) = a₃ ∉ L (A). Then (L (A) , ≤ , ⊔ , ⊓) is not a lattice.

Definition 4.6. Given a fully fuzzy information system (U, A). If $Δ^{cc} (A) = ⋁_{k = 1}^{q} (⋀_{l = 1}^{p_{k}} a_{kl})$ , where every P_k = {a_kl : l ≤ p_k} ⊆ A has not repetitive elements, then $⋁_{k = 1}^{q} (⋀_{l = 1}^{p_{k}} a_{kl})$ is called the standard minimum formula of Δ^cc (A). We denote it by $Δ_{*}^{cc} (A)$ . That is, $Δ_{*}^{cc} (A) = ⋁_{k = 1}^{q} (⋀_{l = 1}^{p_{k}} a_{kl}) .$

Example 4.7. In Example 3.4, we have $a_{2} \leq a_{1} \lor a_{2} \lor a_{3} \lor a_{4}, a_{2} \leq a_{1} \lor a_{2} \lor a_{4},$ $a_{2} \leq a_{2} \lor a_{3}, a_{1} \lor a_{4} \leq a_{1} \lor a_{3} \lor a_{4} .$

Obviously, $a_{2} \land (a_{1} \lor a_{2} \lor a_{3} \lor a_{4}) = a_{2}, a_{2} \land (a_{1} \lor a_{2} \lor a_{4}) = a_{2},$ $a_{2} \land (a_{2} \lor a_{3}) = a_{2}, (a_{1} \lor a_{4}) \land (a_{1} \lor a_{3} \lor a_{4}) = a_{1} \lor a_{4} .$ Then

Δ^cc (A) = a₂ ∧ (a₁ ∨ a₂ ∨ a₃ ∨ a₄) ∧ (a₁ ∨ a₄) ∧ (a₁ ∨ a₃ ∨ a₄) ∧ (a₁ ∨ a₂ ∨ a₄) ∧ (a₂ ∨ a₃)

= a₂ ∧ (a₁ ∨ a₄)

= (a₁ ∧ a₂) ∨ (a₂ ∧ a₄) . Thus $Δ_{*}^{cc} (A) = (a_{1} \land a_{2}) \lor (a_{2} \land a_{4})$ .

Theorem 4.8. Given that (U, A) is a fully fuzzy information system. If $Δ_{*}^{cc} (A) = ⋁_{k = 1}^{q} (⋀_{l = 1}^{p_{k}} a_{kl})$ is the standard minimum formula of Δ^cc (A), then red^cc (A) = {P_k : k ≤ q} where P_k = {a_kl : l ≤ p_k}.

Proof. (1) Given P_{k
₀} ∈ {P_k : k ≤ q}.

(i) Clearly, $Δ_{*}^{cc} (A) = ⋁_{k = 1}^{q} (⋀_{l = 1}^{p_{k}} a_{kl}) = ⋁_{k = 1}^{q} (⋀ P_{k})$ . Then $⋀ P_{k_{0}} ⟶ Δ_{*}^{cc} (A)$ .

Since $Δ_{*}^{cc} (A) = Δ^{cc} (A) = ⋀ (⋁ d_{ij})$ , we have $Δ_{*}^{cc} (A) \Leftrightarrow ⋁ d_{ij} for any 1 \leq i, j \leq n .$

Then ∀ x, y ∈ U, ⋀P_{k
₀} ⟶ ⋁ D^cc (x, y).

So ∀ (x, y) ∉ cc (A), ⋀P_{k
₀} ⟶ ⋁ D^cc (x, y).

Now ⋀P_{k
₀} ⇔ a_{k
₀
l} for any l ≤ p_{k
₀} and ⋁D^cc (x, y) ↔ a for some a ∈ D^cc (x, y). Then ∀ (x, y) ∉ cc (A), a_{k
₀
l} for any l ≤ p_{k
₀} ⟶ a for some a ∈ D^cc (x, y).

So ∀ (x, y) ∉ cc (A), there exists l₀ ≤ p_{k
₀} such that a = a_{k
₀
l
₀}, i.e., a ∈ P_{k
₀} ∩ D^cc (x, y).

Thus ∀ (x, y) ∉ cc (A), P_{k
₀}∩ D^cc (x, y) ≠ ∅.

By Proposition 3.7, P_{k
₀} ∈ co^cc (A).

(ii) To prove P_{k
₀} ∈ red^cc (A), by Theorem 3.8, we only need to show that $\forall a \in P_{k_{0}}, \exists (x_{a}, y_{a}) \in cc (A), (P_{k_{0}} - {a}) \cap D^{cc} (x_{a}, y_{a}) = \emptyset .$

Suppose that ∃ a₀ ∈ P_{k
₀} such that (P_{k
₀} - {a₀})∩ D^cc (x, y) ≠ ∅ for any (x, y) ∉ cc (A). Pick a_xy ∈ (P_{k
₀} - {a₀}) ∩ D^cc (x, y). Then ⋀ (P_{k
₀} - {a₀}) ⟶ a_xy and a_xy ⟶ ⋁ D^cc (x, y).

Thus ∀ (x, y) ∉ cc (A), ⋀ (P_{k
₀} - {a₀}) ⟶ ⋁ D^cc (x, y) .

∀ (x, y) ∈ cc (A), we have D^cc (x, y) =∅. Then ⋀ (P_{k
₀} - {a₀}) ⟶ ⋁ D^cc (x, y) .

It follows that ∀ x, y ∈ U, $⋀ (P_{k_{0}} - {a_{0}}) ⟶ ⋁ D^{cc} (x, y) .$

Since $Δ_{*}^{cc} (A)$ contains all true explanations of Δ^cc (A), we have P_{k
₀} - {a₀} ∈ {P_k : k ≤ q}. Then

(⋀ P_{k
₀}) ⋁ (⋀ (P_{k
₀} - {a₀}))

= ((⋀ (P_{k
₀} - {a₀})) ⋀ {a₀}) ⋁ ((⋀ (P_{k
₀} - {a₀})) ⋀1)

= (⋀ (P_{k
₀} - {a₀})) ⋀ ({a₀} ⋁1)

= (⋀ (P_{k
₀} - {a₀})) ⋀1

= ⋀ (P_{k
₀} - {a₀}).

So P_{k
₀} ∉ {P_k : k ≤ q}. This is a contradiction.

Thus P_{k
₀} ∈ red^cc (A). This shows that red^cc (A) ⊇ {P_k : k ≤ q}.

(2) Let P ∈ red^cc (A). Then P ∈ co^cc (A). By Proposition 3.7, P∩ D^cc (x, y) ≠ ∅ for any (x, y) ∉ cc (A).

Similar to the proof of (1) (ii), we can show that P ∈ {P_k : k ≤ q} .

Thus red^cc (A) ⊆ {P_k : k ≤ q}.

Hence red^cc (A) = {P_k : k ≤ q} . □

Algorithm 4.9. Given a fully fuzzy information system (U, A). Then an algorithm on cc-reduction of A is shown as follows:

Example 4.10. (Continued from Example 3.4) In Step 1, we input a fully fuzzy information system (U, A).

In Step 2, we give the cc-discernibility matrix $D^{cc} (A)$ of A.

In Step 3, we get

Δ^cc (A) = a₂ ∧ (a₁ ∨ a₂ ∨ a₃ ∨ a₄) ∧ (a₁ ∨ a₄) ∧ (a₁ ∨ a₃ ∨ a₄) ∧ (a₁ ∨ a₂ ∨ a₄) ∧ (a₂ ∨ a₃) .

In Step 4, we gain $Δ_{*}^{cc} (A) = (a_{1} \land a_{2}) \lor (a_{2} \land a_{4}) .$

In Step 5, we obtain red^cc (A) = {{a₁, a₂} , {a₂, a₄}} and core^cc (A) = {a₂}.

5 Invariant characterizations of fully fuzzyinformation systems underhomomorphisms

Proposition 5.1. Suppose (U, A) ∼ _h (V, B). Given P ⊆ A. Then $[h_{1} (x)]_{cc (h_{2} (P))} = h_{1} ([x]_{cc (P)}) .$

Proof. Obviously, $[x]_{cc (P)} = {u \in U : \forall a \in P, a (x) \approx_{1} a (u)},$ $[h_{1} (x)]_{cc (h_{2} (P))} = {v \in V : \forall a \in P, h_{2} (a) (h_{1} (x))$ $\approx_{1} h_{2} (a) (v)} .$

Suppose that v ∈ [h₁ (x)] _cc(h₂(P)). Then ∀ a ∈ P, h₂ (a) (h₁ (x)) ≈ ₁h₂ (a) (v) . Let h₁ (u) = v. Then ∀ a ∈ P, h₂ (a) (h₁ (x)) ≈ ₁h₂ (a) (h₁ (u)).

It should be noted that (U, A) ∼ _h (V, B). Then ∀ a ∈ P, $h_{2} (a) (h_{1} (x)) \approx_{1} h_{3} (a (x)), h_{2} (a) (h_{1} (u)) \approx_{1} h_{3} (a (u)) .$

So ∀ a ∈ P, h₃ (a (x)) ≈ ₁h₃ (a (u)) .

Since h₃ is class-preserving, ∀ a ∈ P, a (x) ≈ ₁a (u). Then u ∈ [x] _cc(P). This implies that v ∈ h₁ ([x] _cc(P)).

Conversely, suppose that v ∈ h₁ ([x] _cc(P)). Then ∃ u ∈ U, v = h₁ (u), u ∈ [x] _cc(P). So ∀ a ∈ P, a (x) ≈ ₁a (u). Since h₃ is class-preserving, ∀ a ∈ P, h₃ (a (x)) ≈ ₁h₃ (a (u)) .

It should be noted that (U, A) ∼ _h (V, B). Then ∀ a ∈ P, $h_{2} (a) (h_{1} (x)) \approx_{1} h_{3} (a (x)), h_{2} (a) (h_{1} (u)) \approx_{1} h_{3} (a (u)) .$

Then ∀ a ∈ P, h₂ (a) (h₁ (x)) ≈ ₁h₂ (a) (h₁ (u)) = h₂ (a) (v). This implies that v ∈ [h₁ (x)] _cc(h₂(P)).

Therefore, [h₁ (x)] _cc(h₂(P)) = h₁ ([x] _cc(P)). □

Theorem 5.2. Suppose (U, A) ∼ _h (V, B).

(1) If P ∈ co^cc (A), then h₂ (P) ∈ co^cc (B).

(2) If h₁ is injective, h₂ (P) ∈ co^cc (B), then P ∈ co^cc (A).

Proof. (1) ∀ j, let h₁ (x_i) = y_j. Since P ∈ co^cc (A), we have cc (P) = cc (A) . This implies that [x_i] _cc(P) = [x_i] _cc(A). Then h₁ ([x_i] _cc(P)) = h₁ ([x_i] _cc(A)) . By Proposition 5.1, $[h_{1} (x_{i})]_{cc (h_{2} (P))} = [h_{1} (x_{i})]_{cc (h_{2} (A))} .$ Thus ∀ j, [y_j] _cc(h₂(P)) = [y_j] _cc(B).

By Theorem 2.8, cc (h₂ (P)) = cc (B).

Therefore, h₂ (P) ∈ co^cc (B).

(2) ∀ i, let y_j = h₁ (x_i). Since h₂ (P) ∈ co^cc (B), we have $cc (h_{2} (P)) = cc (B) = cc (h_{2} (A)) .$

This implies [y_j] _cc(h₂(P)) = [y_j] _cc(h₂(A)), i.e., $[h_{1} (x_{i})]_{cc (h_{2} (P))} = [h_{1} (x_{i})]_{cc (h_{2} (A))} .$

By Proposition 5.1, $h_{1} ([x_{i}]_{cc (P)}) = h_{1} ([x_{i}]_{cc (A)}) .$

Since h₁ is injective, ∀ i, [x_i] _cc(P) = [x_i] _cc(A).

By Theorem 2.8, cc (P) = cc (A).

Therefore, P ∈ co^cc (A). □

Corollary 5.3. Suppose (U, A) ∼ _h (V, B). If h₁ is injective, then

(1) P ∈ co^cc (A) ⇔ h₂ (P) ∈ co^cc (B) .

(2) h₂ (co^cc (A)) = co^cc (B) , where h₂ (co^cc (A)) = {h₂ (P) : P ∈ co^cc (A)} .

Proof. (1) This holds by Theorem 5.2.

(2) This follows from (1). □

Theorem 5.4 Suppose (U, A) ∼ _h (V, B). If h₁ and h₂ are injective, then $P \in {red}^{cc} (A) \Leftrightarrow h_{2} (P) \in {red}^{cc} (B) .$

Proof. “⇒". Since P ∈ red^cc (A), we have P ∈ co^cc (A). By Theorem 5.2, h₂ (P) ∈ co^cc (B).

∀ b ∈ h₂ (P), let h₂ (a) = b. Since P ∈ red^cc (A), we have P - {a} ∉ co^cc (A).

By Corollary 5.3, h₂ (P - {a}) ∉ co^cc (B) . Since h₂ is injective, we have $h_{2} (P - {a}) = h_{2} (P) - {h_{2} (a)} = h_{2} (P) - {b} .$

Then h₂ (P) - {b} ∉ co^cc (B) .

Hence, h₂ (P) ∈ red^cc (B).

“⟸". Since h₂ (P) ∈ red^cc (B), we have h₂ (P) ∈ co^cc (B). By Corollary 5.3, P ∈ co^cc (A).

∀ a ∈ P, let b = h₂ (a). Then b ∈ h₂ (P). Since h₂ (P) ∈ red^cc (B), we have h₂ (P) - {b} ∉ co^cc (B).

It should be noted that h₂ is injective. Then $h_{2} (P) - {b} = h_{2} (P) - {h_{2} (a)} = h_{2} (P - {a}) .$

This implies h₂ (P - {a}) ∉ co^cc (B) .

By Corollary 5.3, P - {a} ∉ co^cc (A) .

Hence P ∈ red^cc (A). □

Corollary 5.5. Suppose (U, A) ∼ _h (V, B). If h₁ and h₂ are injective, then $h_{2} ({red}^{cc} (A)) = {red}^{cc} (B),$ where h₂ (red^cc (A)) = {h₂ (P) : P ∈ red^cc (A)} .

Proof. This holds by Theorem 5.4. □

Theorem 5.6. Suppose (U, A) ∼ _h (V, B). If h₁ and h₂ are injective, then $a \in {core}^{cc} (A) \Leftrightarrow h_{2} (a) \in {core}^{cc} (B) .$

Proof. “ ⇒ ". Since a ∈ core^cc (A), by Theorem 3.13, A - {a} ∉ co^cc (A).

By Corollary 5.3, h₂ (A - {a}) ∉ co^cc (B).

Since h₂ is injective, we have $h_{2} (A - {a}) = h_{2} (A) - {h_{2} (a)} = B - {h_{2} (a)} .$

Then B - {h₂ (a)} ∉ co^cc (B) .

By Theorem 3.13, h₂ (a) ∈ core^cc (B).

“ ⟸ ". Since h₂ (a) ∈ core^cc (B), by Theorem 3.13, B - {h₂ (a)} ∉ co^cc (B), i.e., h₂ (A) - {h₂ (a)} ∉ co^cc (B).

It should be noted that h₂ is injective. Then $h_{2} (A) - {h_{2} (a)} = h_{2} (A - {a}) .$ This implies h₂ (A - {a}) ∉ co^cc (B).

By Corollary 5.3, A - {a} ∉ co^cc (A).

by Theorem 3.13, a ∈ core^cc (A). □

Corollary 5.7. Suppose (U, A) ∼ _h (V, B). If h₁ and h₂ are injective, then $h_{2} ({core}^{cc} (A)) = {core}^{cc} (B) .$

Proof. This holds by Theorem 5.6. □

Theorem 5.8. Suppose (U, A) ∼ _h (V, B). If h₁ and h₂ are injective, then

a is an unnecessary cc - attribute ⇔ h₂ (a) is an unnecessary cc - attribute .

Proof. “ ⇒ ". ∀ Q ∈ co^cc (B), by Corollary 5.3(2), there exists P ∈ co^cc (B) such that Q = h₂ (P).

Since a is an unnecessary cc-attribute, by Theorem 3.13(3), P - {a} ∈ co^cc (A). By Theorem 5.2(1), h₂ (P - {a}) ∈ co^cc (B).

Since h₂ is injective, we have $h_{2} (P - {a}) = h_{2} (P) - {h_{2} (a)} = Q - {h_{2} (a)} .$

Then Q - {h₂ (a)} ∈ co^cc (B).

By Theorem 3.13(3), h₂ (a) is an unnecessary cc-attribute.

“ ⟸ ". ∀ P ∈ co^cc (A), By Theorem 5.2(1), h₂ (P) ∈ co^cc (B).

Since h₂ (a) is an unnecessary cc-attribute, by Theorem 3.13(3), h₂ (P) - {h₂ (a)} ∈ co^cc (A).

Since h₂ is injective, we have $h_{2} (P - {a}) = h_{2} (P) - {h_{2} (a)} .$

Then h₂ (P - {a}) ∈ co^cc (B). By Theorem 5.2(2), P - {a} ∈ co^cc (A).

By Theorem 3.13(3), a is an unnecessary cc-attribute.

□

Corollary 5.9. Suppose (U, A) ∼ _h (V, B). If h₁ and h₂ are injective, then

a is a relatively necessary cc-attribute ⇔ h₂ (a) is a relatively necessary cc-attribute.

Proof. This holds by Theorems 5.6 and 5.8. □

6 A numerical experiment

In order to illustrate the practical significance and possible applications for cc-reduction in a fully fuzzy information system, we present a numerical experiment in this section.

Example 6.1 The following Table 3 shows the Yeast data set (see Fig. 2(A,B)) that comes from UCI Repository of machine learning databases. Its experiment data is a portion randomly selected from the Yeast data set. Table 3 can be expressed as the fully fuzzy information system (U, A) of Yeast where U = {u₁, u₂, u₃, u₄, u₅, u₆, u₇, u₈, u₉, u₁₀, u₁₁, u₁₂, u₁₃, u₁₄, . 8ptu₁₅, . 8ptu₁₆, . 8ptu₁₇, . 8ptu₁₈, . 8ptu₁₉, . 8ptu₂₀, . 8ptu₂₁, . 8ptu₂₂, . 8ptu₂₃, . 8ptu₂₄, u₂₅} is the set of Yeast and A = {mcg = “ McG-eoch′s method for signal sequence recognition", gvh=“von Heijne’s method for signal sequence recognition", alm=“Score of the ALOM membrane spanning region prediction program", mit=“Score of discriminant analysis of the amino acid content of the N-terminal region (20 residues long) of mitochondrial and non-mitochondrial proteins", erl=“Presence of “HDEL" substring (thought to act as a signal for retention in the endoplasmic reticulum lumen)", pox=“Peroxisomal targeting signal in the C-terminus", vac=“Score of discriminant analysis of the amino acid content of vacuolar and extracellular proteins", nuc=“Score of discriminant analysis of nuclear localization signals of nuclear and non-nuclear proteins" } = {a₁, a₂, a₃, a₄, a₅, a₆, a₇, a₈} is the set of attributes.

Fig.2

Yeast.

Table 3

The information system (U, A) of Yeast

U	a ₁	a ₂	a ₃	a ₄	a ₅	a ₆	a ₇	a ₈
u ₁	0.85	0.56	0.33	0.38	1.00	0.00	0.55	0.25
u ₂	0.34	0.31	0.60	0.37	0.50	0.00	0.50	0.30
u ₃	0.59	0.53	0.47	0.26	0.50	0.00	0.53	0.31
u ₄	0.67	0.63	0.34	0.41	0.50	0.00	0.60	0.33
u ₅	0.70	0.73	0.38	0.36	0.50	0.00	0.54	0.22
u ₆	0.54	0.51	0.54	0.52	0.50	0.00	0.47	0.25
u ₇	0.53	0.50	0.54	0.56	0.50	0.00	0.49	0.41
u ₈	0.84	0.63	0.38	0.23	0.50	0.00	0.53	0.22
u ₉	0.89	0.77	0.36	0.34	0.50	0.00	0.44	0.22
u ₁₀	0.48	0.43	0.45	0.17	0.50	0.00	0.48	0.37
u ₁₁	0.36	0.35	0.47	0.22	0.50	0.00	0.44	0.31
u ₁₂	0.57	0.49	0.48	0.16	0.50	0.00	0.51	0.22
u ₁₃	0.43	0.52	0.53	0.52	1.00	0.00	0.49	0.22
u ₁₄	0.38	0.49	0.54	1.00	0.50	0.00	0.50	0.27
u ₁₅	0.43	0.32	0.56	0.27	0.50	0.00	0.47	0.27
u ₁₆	0.84	0.87	0.45	0.54	0.50	0.00	0.28	0.22
u ₁₇	0.27	0.56	0.56	0.18	0.50	0.00	0.52	0.22
u ₁₈	0.89	0.82	0.60	0.49	0.50	0.00	0.55	0.22
u ₁₉	0.86	0.92	0.50	0.37	1.00	0.00	0.53	0.32
u ₂₀	0.37	0.55	0.29	0.32	0.50	0.00	0.51	0.22
u ₂₁	0.87	0.83	0.57	0.36	0.50	0.00	0.36	0.22
u ₂₂	0.43	0.45	0.54	0.14	0.50	0.00	0.50	0.31
u ₂₃	0.52	0.48	0.54	0.28	0.50	0.00	0.54	0.22
u ₂₄	0.77	0.60	0.42	0.33	0.50	0.00	0.57	0.22
u ₂₅	0.76	0.74	0.56	0.34	0.50	0.00	0.51	0.30

The cc-discernibility matrix $D^{cc} (A)$ is obtained as follows:

$D^{cc} (u_{1}, u_{1}) = \emptyset$ ;

$D^{cc} (u_{2}, u_{1}) = {a_{1}, a_{2}, a_{3}, a_{5}, a_{8}}, D^{cc} (u_{2}, u_{2}) = \emptyset$ ;

$D^{cc} (u_{3}, u_{1}) = {a_{1}, a_{3}, a_{4}, a_{5}, a_{8}}, D^{cc} (u_{3}, u_{2}) = {a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{3}, u_{3}) = \emptyset$ ;

$D^{cc} (u_{4}, u_{1}) = {a_{1}, a_{2}, a_{4}, a_{5}, a_{7}, a_{8}}, D^{cc} (u_{4}, u_{2}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{4}, u_{3}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{4}, u_{4}) = \emptyset$ ;

$D^{cc} (u_{5}, u_{1}) = {a_{1}, a_{2}, a_{5}}, D^{cc} (u_{5}, u_{2}) = {a_{1}, a_{2}, a_{3}, a_{8}}, D^{cc} (u_{5}, u_{3}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{5},$ $u_{4}) = {a_{1}, a_{2}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{5}, u_{5}) = \emptyset$ ;

$D^{cc} (u_{6}, u_{1}) = {a_{1}, a_{3}, a_{4}, a_{5}, a_{7}}, D^{cc} (u_{6}, u_{2}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{6}, u_{3}) = {a_{3}, a_{4}, a_{7},$ $a_{8}}, D^{cc} (u_{6}, u_{4}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc}$ $(u_{6}, u_{5}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{6}, u_{6}) = \emptyset$ ;

$D^{cc} (u_{7}, u_{1}) = {a_{1}, a_{3}, a_{4}, a_{5}, a_{7}, a_{8}}, D^{cc} (u_{7}, u_{2})$ $= {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{7}, u_{3}) = {a_{3}, a_{4}, a_{7},$ $a_{8}}, D^{cc} (u_{7}, u_{4}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc}$ $(u_{7}, u_{5}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{7}, u_{6}) = {a_{8}}, D^{cc} (u_{7}, u_{7}) = \emptyset$ ;

$D^{cc} (u_{8}, u_{1}) = {a_{2}, a_{4}, a_{5}}, D^{cc} (u_{8}, u_{2}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{8}, u_{3}) = {a_{1}, a_{2}, a_{3}, a_{8}}, D^{cc} (u_{8}, u_{4}) = {a_{1}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{8}, u_{5}) = {a_{1}, a_{2}, a_{4}}, D^{cc} (u_{8}, u_{6}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{8}, u_{7}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{8}, u_{8}) = \emptyset$ ;

$D^{cc} (u_{9}, u_{1}) = {a_{2}, a_{5}, a_{7}}, D^{cc} (u_{9}, u_{2}) = {a_{1}, a_{2}, a_{3}, a_{7}, a_{8}}, D^{cc} (u_{9}, u_{3}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{9}, u_{4}) = {a_{1}, a_{2}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{9}, u_{5}) = {a_{1}, a_{7}}, D^{cc} (u_{9}, u_{6}) = {a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{9}, u_{7}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{9}, u_{8}) = {a_{2}, a_{4}, a_{7}}, D^{cc} (u_{9}, u_{9}) = \emptyset$ ;

$D^{cc} (u_{10}, u_{1}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{7}, a_{8}}, D^{cc} (u_{10}, u_{2}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{10}, u_{3}) = {a_{1}, a_{2}, a_{4}, a_{7}}, D^{cc} (u_{10}, u_{4}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{10}, u_{5}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{10}, u_{6}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{10}, u_{7}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{10}, u_{8}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{10}, u_{9}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{10}, u_{10}) = \emptyset$ ;

$D^{cc} (u_{11}, u_{1}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{7}, a_{8}}, D^{cc} (u_{11}, u_{2}) = {a_{3}, a_{4}, a_{7}}, D^{cc} (u_{11}, u_{3}) = {a_{1}, a_{2}, a_{7}}, D^{cc} (u_{11}, u_{4}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{11}, u_{5}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{11}, u_{6}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{11}, u_{7}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{11}, u_{8}) = {a_{1}, a_{2}, a_{3}, a_{7}, a_{8}}, D^{cc} (u_{11}, u_{9}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{11}, u_{10}) = {a_{1}, a_{2}, a_{4}}, D^{cc} (u_{11}, u_{11}) = \emptyset$ ;

$D^{cc} (u_{12}, u_{1}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{5}}, D^{cc} (u_{12}, u_{2}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{12}, u_{3}) = {a_{2}, a_{4}, a_{8}}, D^{cc} (u_{12}, u_{4}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{12}, u_{5}) = {a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{12}, u_{6}) = {a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{12}, u_{7}) = {a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{12}, u_{8}) = {a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{12}, u_{9}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{12}, u_{10}) = {a_{1}, a_{7}, a_{8}}, D^{cc} (u_{12}, u_{11}) = {a_{1}, a_{2}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{12}, u_{12}) = \emptyset$ ;

$D^{cc} (u_{13}, u_{1}) = {a_{1}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{13}, u_{2}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{7}, a_{8}}, D^{cc} (u_{13}, u_{3}) = {a_{1}, a_{3}, a_{4}, a_{5}, a_{7}, a_{8}}, D^{cc} (u_{13}, u_{4}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{7}, a_{8}}, D^{cc} (u_{13}, u_{5}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{7}}, D^{cc} (u_{13}, u_{6}) = {a_{1}, a_{5}}, D^{cc} (u_{13}, u_{7}) = {a_{1}, a_{5}, a_{8}}, D^{cc} (u_{13}, u_{8}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{7}}, D^{cc} (u_{13}, u_{9}) = {a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{13}, u_{10}) = {a_{2}, a_{3}, a_{4}, a_{5}, a_{8}}, D^{cc} (u_{13}, u_{11}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{13}, u_{12}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{7}}, D^{cc} (u_{13}, u_{13})$ =∅;

$D^{cc} (u_{14}, u_{1}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{5}}, D^{cc} (u_{14}, u_{2}) = {a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{14}, u_{3}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{14}, u_{4}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{14}, u_{5}) = {a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{14}, u_{6}) = {a_{1}, a_{2}, a_{4}, a_{7}}, D^{cc} (u_{14}, u_{7}) = {a_{1}, a_{2}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{14}, u_{8}) = {a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{14}, u_{9}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{14}, u_{10}) = {a_{1}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{14}, u_{11}) = {a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{14}, u_{12}) = {a_{1}, a_{3}, a_{4}}, D^{cc} (u_{14}, u_{13}) = {a_{1}, a_{2}, a_{4}, a_{5}, a_{7}}, D^{cc} (u_{14}, u_{14}) = \emptyset$ ;

$D^{cc} (u_{15}, u_{1}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{7}}, D^{cc} (u_{15},$ $u_{2}) = {a_{1}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{15}, u_{3}) = {a_{1}, a_{2},$ $a_{3}, a_{7}, a_{8}}, D^{cc} (u_{15}, u_{4}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}},$ $D^{cc} (u_{15}, u_{5}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{15}, u_{6}) =$ ${a_{1}, a_{2}, a_{4}}, D^{cc} (u_{15}, u_{7}) = {a_{1}, a_{2}, a_{4}, a_{8}},$ $D^{cc} (u_{15}, u_{8}) = {a_{1}, a_{2}, a_{3}, a_{7}}, D^{cc} (u_{15}, u_{9}) =$ ${a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{15}, u_{10}) = {a_{2}, a_{3}, a_{4}, a_{8}},$ $D^{cc} (u_{15}, u_{11}) = {a_{1}, a_{3}, a_{8}}, D^{cc} (u_{15}, u_{12}) = {a_{1},$ $a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{15}, u_{13}) = {a_{2}, a_{4}, a_{5}}, D^{cc} (u_{15}, u_{14}) = {a_{1}, a_{2}, a_{4}, a_{7}}, D^{cc} (u_{15}, u_{15}) = \emptyset$ ;

$D^{cc} (u_{16}, u_{1}) = {a_{2}, a_{3}, a_{4}, a_{5}, a_{7}}, D^{cc} (u_{16}, u_{2})$ $= {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{16}, u_{3}) = {a_{1}, a_{2},$ $a_{4}, a_{7}, a_{8}}, D^{cc} (u_{16}, u_{4}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}},$ $D^{cc} (u_{16}, u_{5}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{16}, u_{6}) =$ ${a_{1}, a_{2}, a_{3}, a_{7}}, D^{cc} (u_{16}, u_{7}) = {a_{1}, a_{2}, a_{3}, a_{7}, a_{8}},$ $D^{cc} (u_{16}, u_{8}) = {a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{16}, u_{9}) =$ ${a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{16}, u_{10}) = {a_{1}, a_{2}, a_{4}, a_{7},$ $a_{8}}, D^{cc} (u_{16}, u_{11}) = {a_{1}, a_{2}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{16},$ $u_{12}) = {a_{1}, a_{2}, a_{4}, a_{7}}, D^{cc} (u_{16}, u_{13}) = {a_{1}, a_{2},$ $a_{3}, a_{5}, a_{7}}, D^{cc} (u_{16}, u_{14}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}},$ $D^{cc} (u_{16}, u_{15}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{16}, u_{16}) = \emptyset$ ;

$D^{cc} (u_{17}, u_{1}) = {a_{1}, a_{3}, a_{4}, a_{5}}, D^{cc} (u_{17}, u_{2}) =$ ${a_{1}, a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{17}, u_{3}) = {a_{1}, a_{3}, a_{4}, a_{8}},$ $D^{cc} (u_{17}, u_{4}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{17}, u_{5})$ $= {a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{17}, u_{6}) = {a_{1}, a_{4}, a_{7}},$ $D^{cc} (u_{17}, u_{7}) = {a_{1}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{17}, u_{8}) =$ ${a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{17}, u_{9}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}},$ $D^{cc} (u_{17}, u_{10}) = {a_{1}, a_{2}, a_{3}, a_{7}, a_{8}}, D^{cc} (u_{17}, u_{11})$ $= {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{17}, u_{12}) = {a_{1}, a_{2},$ $a_{3}}, D^{cc} (u_{17}, u_{13}) = {a_{1}, a_{4}, a_{5}, a_{7}}, D^{cc} (u_{17}, u_{14})$ $= {a_{1}, a_{2}, a_{4}}, D^{cc} (u_{17}, u_{15}) = {a_{1}, a_{2}, a_{4}, a_{7}},$ $D^{cc} (u_{17}, u_{16}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{17}, u_{17})$ =∅;

$D^{cc} (u_{18}, u_{1}) = {a_{2}, a_{3}, a_{4}, a_{5}}, D^{cc} (u_{18}, u_{2}) =$ ${a_{1}, a_{2}, a_{4}, a_{8}}, D^{cc} (u_{18}, u_{3}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{8}},$ $D^{cc} (u_{18}, u_{4}) = {a_{1}, a_{2}, a_{3}, a_{7}, a_{8}}, D^{cc} (u_{18}, u_{5}) =$ ${a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{18}, u_{6}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}},$ $D^{cc} (u_{18}, u_{7}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{18}, u_{8})$ $= {a_{2}, a_{3}, a_{4}}, D^{cc} (u_{18}, u_{9}) = {a_{2}, a_{3}, a_{4}, a_{7}},$ $D^{cc} (u_{18}, u_{10}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{18},$ $u_{11}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{18}, u_{12}) =$ ${a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{18}, u_{13}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{5},$ $a_{7}}, D^{cc} (u_{18}, u_{14}) = {a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{18}, u_{15})$ $= {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{18}, u_{16}) = {a_{3}, a_{4}, a_{7}},$ $D^{cc} (u_{18}, u_{17}) = {a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{18}, u_{18}) = \emptyset$ ;

$D^{cc} (u_{19}, u_{1}) = {a_{2}, a_{3}, a_{8}}, D^{cc} (u_{19}, u_{2}) = {a_{1},$ $a_{2}, a_{3}, a_{5}}, D^{cc} (u_{19}, u_{3}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{5}},$ $D^{cc} (u_{19}, u_{4}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{7}}, D^{cc} (u_{19}, u_{5})$ $= {a_{1}, a_{2}, a_{3}, a_{5}, a_{8}}, D^{cc} (u_{19}, u_{6}) = {a_{1}, a_{2}, a_{4},$ $a_{5}, a_{7}, a_{8}}, D^{cc} (u_{19}, u_{7}) = {a_{1}, a_{2}, a_{4}, a_{5}, a_{7}, a_{8}},$ $D^{cc} (u_{19}, u_{8}) = {a_{2}, a_{3}, a_{4}, a_{5}, a_{8}}, D^{cc} (u_{19}, u_{9}) =$ ${a_{2}, a_{3}, a_{5}, a_{7}, a_{8}}, D^{cc} (u_{19}, u_{10}) = {a_{1}, a_{2}, a_{3}, a_{4},$ $a_{5}, a_{7}}, D^{cc} (u_{19}, u_{11}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{7}},$ $D^{cc} (u_{19}, u_{12}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{8}}, D^{cc} (u_{19},$ $u_{13}) = {a_{1}, a_{2}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{19}, u_{14}) = {a_{1}, a_{2},$ $a_{4}, a_{5}, a_{8}}, D^{cc} (u_{19}, u_{15}) = {a_{1}, a_{2}, a_{4}, a_{5}, a_{7}, a_{8}},$ $D^{cc} (u_{19}, u_{16}) = {a_{2}, a_{3}, a_{4}, a_{5}, a_{7}, a_{8}}, D^{cc} (u_{19},$ $u_{17}) = {a_{1}, a_{2}, a_{4}, a_{5}, a_{8}}, D^{cc} (u_{19}, u_{18}) = {a_{2}, a_{3},$ $a_{4}, a_{5}, a_{8}}, D^{cc} (u_{19}, u_{19}) = \emptyset$ ;

$D^{cc} (u_{20}, u_{1}) = {a_{1}, a_{3}, a_{5}}, D^{cc} (u_{20}, u_{2}) = {a_{2},$ $a_{3}, a_{8}}, D^{cc} (u_{20}, u_{3}) = {a_{1}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{20},$ $u_{4}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{20}, u_{5}) = {a_{1},$ $a_{2}, a_{3}}, D^{cc} (u_{20}, u_{6}) = {a_{1}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{20},$ $u_{7}) = {a_{1}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{20}, u_{8}) = {a_{1}, a_{2},$ $a_{3}, a_{4}}, D^{cc} (u_{20}, u_{9}) = {a_{1}, a_{2}, a_{3}, a_{7}}, D^{cc} (u_{20},$ $u_{10}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{20}, u_{11}) =$ ${a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{20}, u_{12}) = {a_{1}, a_{2}, a_{3}, a_{4}},$ $D^{cc} (u_{20}, u_{13}) = {a_{1}, a_{3}, a_{4}, a_{5}, a_{7}}, D^{cc} (u_{20}, u_{14})$ $= {a_{2}, a_{3}, a_{4}}, D^{cc} (u_{20}, u_{15}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}},$ $D^{cc} (u_{20}, u_{16}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{20}, u_{17})$ $= {a_{1}, a_{3}, a_{4}}, D^{cc} (u_{20}, u_{18}) = {a_{1}, a_{2}, a_{3}, a_{4}},$ $D^{cc} (u_{20}, u_{19}) = {a_{1}, a_{2}, a_{3}, a_{5}, a_{8}}, D^{cc} (u_{20}, u_{20})$ =∅;

$D^{cc} (u_{21}, u_{1}) = {a_{2}, a_{3}, a_{5}, a_{7}}, D^{cc} (u_{21}, u_{2}) = {a_{1}, a_{2}, a_{3}, a_{7}, a_{8}}, D^{cc} (u_{21}, u_{3}) = {a_{1}, a_{2}, a_{3}, a_{4},$ $a_{7}, a_{8}}, D^{cc} (u_{21}, u_{4}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}},$ $D^{cc} (u_{21}, u_{5}) = {a_{1}, a_{2}, a_{3}, a_{7}}, D^{cc} (u_{21}, u_{6}) = {a_{1},$ $a_{2}, a_{4}, a_{7}}, D^{cc} (u_{21}, u_{7}) = {a_{1}, a_{2}, a_{4}, a_{7}, a_{8}},$ $D^{cc} (u_{21}, u_{8}) = {a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{21}, u_{9}) =$ ${a_{2}, a_{3}, a_{7}}, D^{cc} (u_{21}, u_{10}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7},$ $a_{8}}, D^{cc} (u_{21}, u_{11}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc}$ $(u_{21}, u_{12}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{21}, u_{13}) =$ ${a_{1}, a_{2}, a_{4}, a_{5}, a_{7}}, D^{cc} (u_{21}, u_{14}) = {a_{1}, a_{2}, a_{4},$ $a_{7}}, D^{cc} (u_{21}, u_{15}) = {a_{1}, a_{2}, a_{4}, a_{7}}, D^{cc} (u_{21}, u_{16})$ $= {a_{3}, a_{4}, a_{7}}, D^{cc} (u_{21}, u_{17}) = {a_{1}, a_{2}, a_{4}, a_{7}},$ $D^{cc} (u_{21}, u_{18}) = {a_{3}, a_{4}, a_{7}}, D^{cc} (u_{21}, u_{19}) = {a_{2},$ $a_{5}, a_{7}, a_{8}}, D^{cc} (u_{21}, u_{20}) = {a_{1}, a_{2}, a_{3}, a_{7}}, D^{cc}$ (u₂₁, u₂₁) =∅;

$D^{cc} (u_{22}, u_{1}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{8}}, D^{cc} (u_{22},$ $u_{2}) = {a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{22}, u_{3}) = {a_{1}, a_{2}, a_{3},$ $a_{4}}, D^{cc} (u_{22}, u_{4}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{22},$ $u_{5}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{22}, u_{6}) = {a_{1}, a_{2},$ $a_{4}, a_{7}, a_{8}}, D^{cc} (u_{22}, u_{7}) = {a_{1}, a_{2}, a_{4}, a_{7}, a_{8}}, D^{cc}$ $(u_{22}, u_{8}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{22}, u_{9}) = {a_{1},$ $a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{22}, u_{10}) = {a_{3}, a_{7}}, D^{cc}$ $(u_{22}, u_{11}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{22}, u_{12}) =$ ${a_{1}, a_{3}, a_{8}}, D^{cc} (u_{22}, u_{13}) = {a_{2}, a_{5}, a_{4}, a_{7}, a_{8}},$ $D^{cc} (u_{22}, u_{14}) = {a_{1}, a_{4}, a_{8}}, D^{cc} (u_{22}, u_{15}) = {a_{2},$ $a_{4}, a_{7}, a_{8}}, D^{cc} (u_{22}, u_{16}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}},$ $D^{cc} (u_{22}, u_{17}) = {a_{1}, a_{2}, a_{4}, a_{8}}, D^{cc} (u_{22}, u_{18}) =$ ${a_{1}, a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{22}, u_{19}) = {a_{1}, a_{2}, a_{4},$ $a_{5}}, D^{cc} (u_{22}, u_{20}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{22},$ $u_{21}) = {a_{1}, a_{2}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{22}, u_{22}) = \emptyset$ ;

$D^{cc} (u_{23}, u_{1}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{5}}, D^{cc} (u_{23}, u_{2})$ $= {a_{1}, a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{23}, u_{3}) = {a_{2}, a_{3}, a_{8}},$ $D^{cc} (u_{23}, u_{4}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{23}, u_{5})$ $= {a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{23}, u_{6}) = {a_{2}, a_{4}, a_{7}}, D^{cc}$ $(u_{23}, u_{7}) = {a_{2}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{23}, u_{8}) = {a_{1},$ $a_{2}, a_{3}}, D^{cc} (u_{23}, u_{9}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc}$ $(u_{23}, u_{10}) = {a_{1}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{23}, u_{11}) =$ ${12378}, D^{cc} (u_{23}, u_{12}) = {34}, D^{cc} (u_{23}, u_{13}) =$ ${12457}, D^{cc} (u_{23}, u_{14}) = {a_{1}, a_{4}}, D^{cc} (u_{23}, u_{15}) =$ ${a_{1}, a_{2}, a_{7}}, D^{cc} (u_{23}, u_{16}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}},$ $D^{cc} (u_{23}, u_{17}) = {a_{1}, a_{2}, a_{4}}, D^{cc} (u_{23}, u_{18}) = {a_{1},$ $a_{2}, a_{3}, a_{4}}, D^{cc} (u_{23}, u_{19}) = {a_{1}, a_{2}, a_{4}, a_{5}, a_{8}},$ $D^{cc} (u_{23}, u_{20}) = {a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{23}, u_{21}) =$ ${a_{1}, a_{2}, a_{4}, a_{7}}, D^{cc} (u_{23}, u_{22}) = {a_{1}, a_{4}, a_{8}}, D^{cc}$ (u₂₃, u₂₃) =∅;

$D^{cc} (u_{24}, u_{1}) = {a_{1}, a_{2}, a_{3}, a_{5}}, D^{cc} (u_{24}, u_{2}) =$ ${a_{1}, a_{2}, a_{3}, a_{8}}, D^{cc} (u_{24}, u_{3}) = {a_{1}, a_{2}, a_{4}, a_{8}},$ $D^{cc} (u_{24}, u_{4}) = {a_{1}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{24}, u_{5}) =$ ${a_{2}, a_{3}}, D^{cc} (u_{24}, u_{6}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}},$ $D^{cc} (u_{24}, u_{7}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{24}, u_{8})$ $= {a_{1}, a_{3}, a_{4}}, D^{cc} (u_{24}, u_{9}) = {a_{1}, a_{2}, a_{3}, a_{7}}, D^{cc}$ $(u_{24}, u_{10}) = {a_{1}, a_{2}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{24}, u_{11}) =$ ${a_{1}, a_{2}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{24}, u_{12}) = {a_{1}, a_{2}, a_{4}},$ $D^{cc} (u_{24}, u_{13}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{7}}, D^{cc} (u_{24},$ $u_{14}) = {a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{24}, u_{15}) = {a_{1}, a_{2},$ $a_{3}, a_{4}, a_{7}}, D^{cc} (u_{24}, u_{16}) = {a_{1}, a_{2}, a_{4}, a_{7}}, D^{cc}$ $(u_{24}, u_{17}) = {a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{24}, u_{18}) = {a_{1},$ $a_{2}, a_{3}, a_{4}}, D^{cc} (u_{24}, u_{19}) = {a_{1}, a_{2}, a_{3}, a_{5}, a_{8}},$ $D^{cc} (u_{24}, u_{20}) = {a_{1}, a_{2}, a_{3}}, D^{cc} (u_{24}, u_{21}) = {a_{1},$ $a_{2}, a_{3}, a_{7}}, D^{cc} (u_{24}, u_{22}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{8}},$ $D^{cc} (u_{24}, u_{23}) = {a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc} (u_{24}, u_{24}) = \emptyset$ ;

$D^{cc} (u_{25}, u_{1}) = {a_{1}, a_{2}, a_{3}, a_{5}, a_{8}}, D^{cc} (u_{25}, u_{2})$ $= {a_{1}, a_{2}, a_{3}}, D^{cc} (u_{25}, u_{3}) = {a_{1}, a_{2}, a_{3}, a_{4}}, D^{cc}$ $(u_{25}, u_{4}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{25}, u_{5}) = {a_{3},$ $a_{8}}, D^{cc} (u_{25}, u_{6}) = {a_{1}, a_{2}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{25},$ $u_{7}) = {a_{1}, a_{2}, a_{4}, a_{7}, a_{8}}, D^{cc} (u_{25}, u_{8}) = {a_{1}, a_{2},$ $a_{3}, a_{4}, a_{8}}, D^{cc} (u_{25}, u_{9}) = {a_{1}, a_{3}, a_{7}, a_{8}}, D^{cc} (u_{25},$ $u_{10}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{25}, u_{11}) = {a_{1},$ $a_{2}, a_{3}, a_{4}, a_{7}}, D^{cc} (u_{25}, u_{12}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{8}},$ $D^{cc} (u_{25}, u_{13}) = {a_{1}, a_{2}, a_{4}, a_{5}, a_{7}, a_{8}}, D^{cc} (u_{25},$ $u_{14}) = {a_{1}, a_{2}, a_{4}, a_{8}}, D^{cc} (u_{25}, u_{15}) = {a_{1}, a_{2},$ $a_{4}, a_{7}, a_{8}}, D^{cc} (u_{25}, u_{16}) = {a_{1}, a_{2}, a_{3}, a_{4}, a_{7}, a_{8}},$ $D^{cc} (u_{25}, u_{17}) = {a_{1}, a_{2}, a_{4}, a_{8}}, D^{cc} (u_{25}, u_{18}) =$ ${a_{1}, a_{2}, a_{3}, a_{4}, a_{8}}, D^{cc} (u_{25}, u_{19}) = {a_{1}, a_{2}, a_{5}},$ $D^{cc} (u_{25}, u_{20}) = {a_{1}, a_{2}, a_{3}, a_{8}}, D^{cc} (u_{25}, u_{21}) =$ ${a_{1}, a_{2}, a_{7}, a_{8}}, D^{cc} (u_{25}, u_{22}) = {a_{1}, a_{2}, a_{4}}, D^{cc}$ $(u_{25}, u_{23}) = {a_{1}, a_{2}, a_{4}, a_{8}}, D^{cc} (u_{25}, u_{24}) = {a_{2},$ $a_{3}, a_{8}}, D^{cc} (u_{25}, u_{25}) = \emptyset$ .

Obviously, core^cc (A) = {a₈} .

It should be noted that

Δ^cc (A) = a₈ ∧ (a₁ ∨ a₄) ∧ (a₁ ∨ a₅) ∧ (a₁ ∨ a₇) ∧ (a₂ ∨ a₃) ∧ (a₃ ∨ a₄) ∧ (a₃ ∨ a₇) ∧ (a₂ ∨ a₄ ∨ a₅) ∧ (a₂ ∨ a₄ ∨ a₇) ∧ (a₂ ∨ a₅ ∨ a₇) . Then the number of cc-reductions of A is as much as 1728 (contains duplicates). Here, we do not make a detailed list. But it is easy to be obtained.

7 Conclusions

In this paper, we have given cc-reduction in a fully fuzzy information system and divided attributes in a full fuzzy information system into three categories (i.e., necessary cc-attributes, relatively necessary cc-attributes, unnecessary cc-attributes) according to their importance. Moreover, we have obtained some invariant characterizations of fully fuzzy information systems under homomorphisms, which sheds light on the relationship between a fully fuzzy information system and its image fully fuzzy information system. This may have potential applications in knowledge discovery, decision making and reasoning about data. In the future, we will consider concrete applications of the proposed results.

Footnotes

Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and suggestions which have helped immensely in improving the quality of the paper. This work is supported by High Level Innovation Team Program from Guangxi Higher Education Institutions of China (Document No. [2018] 35), Natural Science Foundation of Guangxi (2018GXNSFDA294003, 2018GXNSFDA281028, 2018GXNSFAA294134), Key Laboratory of Software Engineering in Guangxi University for Nationalities (2018-18XJSY-03) and Engineering Project of Undergraduate Teaching Reform of Higher Education in Guangxi (2017JGA179).

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cc -reduction in a fully fuzzy information system

Abstract

Keywords

1 Introduction

2.1 Rough sets

Table 1 An information system about cars Price Mileage Size Max-speed x 1 high low full low x 2 low very low full low x 3 low high compact low x 4 high high full high x 5 high very-high full high x 6 low high full very high

2.3 Homomorphisms between fully fuzzyinformation systems

3 cc-reduction in a fully fuzzy informationsystem

Table 2 An information system about cars a1 a2 a3 a4 x1 0.55 0.84 0.55 0.32 x2 0.58 0.82 0.58 0.43 x3 0.58 0.82 0.58 0.43 x4 0.52 0.85 0.52 0.61 x5 0.22 0.88 0.22 0.36 x6 0.27 0.77 0.27 0.47

5 Invariant characterizations of fully fuzzyinformation systems underhomomorphisms

6 A numerical experiment

Footnotes

Acknowledgments

References

Table 1
An information system about cars

Price Mileage Size Max-speed

x ₁ high low full low

x ₂ low very low full low

x ₃ low high compact low

x ₄ high high full high

x ₅ high very-high full high

x ₆ low high full very high

Table 2
An information system about cars

a₁ a₂ a₃ a₄

x₁ 0.55 0.84 0.55 0.32

x₂ 0.58 0.82 0.58 0.43

x₃ 0.58 0.82 0.58 0.43

x₄ 0.52 0.85 0.52 0.61

x₅ 0.22 0.88 0.22 0.36

x₆ 0.27 0.77 0.27 0.47