Multi-granulation rough filters and rough fuzzy filters in Pseudo-BCI algebras

Abstract

This paper introduced the notions of rough filters, multi-granulation rough filters, and rough fuzzy filters in pseudo-BCI algebras and investigated some properties. First, a congruence relation was structured by a filter on pseudo-BCI algebra. Then rough filters and rough fuzzy filters were investigated. Next, the relationships between upper (lower) rough filters and upper (lower) approximations of their fuzzy homomorphic images were discussed. Furthermore, original rough filter model was extended to a multi-granulation rough filter model, where the set approximations were defined by using multi congruence relations on pseudo-BCI algebra.

Keywords

Pseudo-BCI algebra rough set fuzzy set multi-granulation rough set

1 Introduction

In 1965, Zadeh introduced fuzzy set theory [23]. Many problems which are essentially non-probabilistic in real life can be resolved by this theory. In response to uncertainty problems, Pawlak introduced rough set theory [12] in 1982. Rough set theory as a major approach to resolving uncertain problems. It played an important role in artificial intelligence, cognitive sciences, especially in the vague or imprecise knowledge, machine learning and knowledge discovery. In [10], the algebraic approach to rough ideals was researched. Biswas and Nanda put forward a notion of rough subgroups. In [7], Kuroki and Wang obtained some properties of upper (lower) approximations with respect to the fuzzy normal subgroups. And in [8], Kuroki presented the notion of rough ideals in semigroup as an extension of ideals in semigroups. In the study of modern fuzzy logic theory, algebraic systems played an important role (see [4 , 38]). In [3], W.A. Dudek and Y.B. Jun. introduced the notion of pseudo-BCI algebra. They investigated some properties of pseudo-BCI algebras. In [5], Y. B. Jun et al. presented the concept of pseudo-BCI ideal in pseudo-BCI algebras and researched its characterizations. Then, some pseudo-BCI algebras and pseudo-ideals (filters) are studied (see [1 , 31– 36]).

Furthermore Zadeh [24] proposed the granular computing, this concept described by a set was via the upper and lower approximations under a single granulation. It was induced from a single relation on the universe. The relation including tolerance relation, equivalence relation and so on. Thus, there were several extensions of the rough set model which were proposed before the multi-granulation rough sets. Such as the variable precision rough sets [39], the probabilistic rough sets [20], the fuzzy rough sets and rough fuzzy sets (see [2 , 19]), the rough set model was based on tolerance relation (see [6, 18]). Qian, Xu and Yao et al. investigated multi-granulation rough sets (see [14– 16 , 22]). The upper and lower approximations were induced by multiple binary relations. This theory developed rough set theory, especially in practical applications. In addition, the research on the combination of rough sets and soft sets has also attracted the attention of scholars (see [11 , 25– 27]).

The main object of this paper is to extend single-granulation rough filters to multi-granulation rough filters in pseudo-BCI algebras, where the set approximations are defined by using multi binary relations on the universe.

Some preliminary concepts in multi-granulation rough sets theory such as the lower approximation, upper approximation, filters in pseudo-BCI algebra are briefly reviewed in Section 2. In Section 3, the definitions of rough filters in pseudo-BCI algebra are introduced and some properties of such filters are investigated. Furthermore, the relationship between the upper (lower, respectively) rough filters and the upper (lower, respectively) approximations of their fuzzy homomorphic images are discussed. In Section 4, the definitions of multi-granulation rough filters are presented and some properties are investigated. Rough fuzzy filters are proposed, and its some useful properties are obtained in Section 5.

2 Preliminaries

Let is review of some fundamental notions of pseudo-BCI algebra and rough set in this section.

Definition 2.1. ([36]) A pseudo-BCI algebra is a structure (X; →, ↪, 1), where " → " and "↪" are binary operation on X and “1” is an element of X, verifying the axioms: ∀x, y, z ∈ X,

(y → z) → ((z → x) ↪ (y → x)) =1, (y ↪ z) ↪ ((z ↪ x) → (y ↪ x)) =1;

x → ((x → y) ↪ y) =1, x ↪ ((x ↪ y) → y) =1;

x → x = 1;

x → y = y → x = 1 ⇒ x = y;

x → y = 1 ⇔ x ↪ y = 1.

If (X; →, ↪, 1) is a pseudo-BCI algebra satisfying ∀x, y ∈ X, x → y = x ↪ y, then (X; →, 1) is a BCI algebra.

If (X; →, ↪, 1) is a pseudo-BCI algebra satisfying ∀x ∈ X, x → 1 =1, then (X; →, ↪, 1) ia a pseudo-BCK algebra.

Proposition 2.2. ([36]) Let (X ; →, ↪) be a pseudo-BCI algebra, then X satisfies the following properties, ∀x, y, z ∈ X,

1 ≤ x ⇒ x = 1;

x ≤ y ⇒ y → z ≤ x → z, y ↪ z ≤ x ↪ z;

x ≤ y, y ≤ z ⇒ x ≤ z;

x ↪ (y → z) = y → (x ↪ z);

x ≤ y → z ⇒ y ≤ x ↪ z;

x → y ≤ (z → x) → (z → y), x ↪ y ≤ (z ↪ x) ↪ (z ↪ y);

x ≤ y ⇒ z → x ≤ z → y, z ↪ x ≤ z ↪ y;

1 → x = x, 1 ↪ x = x;

((y → x) ↪ x) → x = y → x, ((y ↪ x) → x) ↪ x = y ↪ x;

x → y ≤ (y → x) ↪1, x ↪ y ≤ (y ↪ x) →1;

(x → y) →1 = (x → 1) ↪ (y ↪ 1), (x ↪ y) ↪1 = (x ↪ 1) → (y → 1);

x → 1 = x ↪ 1.

Definition 2.3. ([32]) A subset F of a pseudo-BCI algebra X is called a filter of X if it satisfies:

1 ∈ F;

x ∈ F, x → y ∈ F ⇒ y ∈ F;

x ∈ F, x ↪ y ∈ F ⇒ y ∈ F.

Definition 2.4. ([3]) By a pseudo-BCI subalgebra of a pseudo-BCI algebra X, we mean a subset S of X which satisfies ∀x, y ∈ S, x → y ∈ S, x ↪ y ∈ S.

Definition 2.5. ([34]) A filter F of a pseudo-BCI algebra X is said to be a normal filter of X if it satisfies ∀x, y ∈ F, x → y ∈ F ⇔ x ↪ y ∈ F.

Definition 2.6. Let $X = (X; \to, ↪, 1)$ be a pseudo-BCI algebra. An equivalence relation R on X is called a congruence relation on X if (x → u, y → v) ∈ R and (x ↪ u, y ↪ v) ∈ R for all (x, y), (u, v) ∈ R. We denote by [x] _R the R-congruence class containing the element x ∈ X. Let X/R be the set of all R-equivalence classes on X. ∀ [x] _R, [y] _R ∈ X/R, we define $\begin{matrix} [x]_{R} \to [y]_{R} \\ : = [x \to y]_{R} = {z \in X | (z, x \to y) \in R}, \\ [x]_{R} ↪ [y]_{R} \\ : = [x ↪ y]_{R} = {w \in X | (w, x ↪ y) \in R} . \end{matrix}$

Theorem 2.7. ([35]) Let (X ; →, ↪, 1) be a pseudo-BCI algebra and let F be a normal filter of X. R_F is a relation induced by filter F on X as follows, $\begin{matrix} \forall x, y \in X, {xR}_{F} y \Leftrightarrow (x \to y \in F, y \to x \in F) or \\ (x ↪ y \in F, y ↪ x \in F) . \end{matrix}$

Then R_F is a congruence relation on X, and (X/F ; →, ↪, C₁) is a pseudo-BCI algebra, where $X / F = {C_{x} | x \in F}, C_{x} = {y \in X | xRy},$ ∀x, y ∈ X, C_x → C_y = C_x→y, C_x ↪ C_y = C_x↪y, and C_x ≤ C_y if and only if C_x → C_y = C₁ or C_x ↪ C_y = C₁.

Definition 2.8. Let F be a filter of X and let R_F be a congruence relation induced by filter F on X.

Formally, 〈U, AT〉 is called an information system, where U is a non-empty finite set of objects, it is called the universe. AT is a non-empty finite set of attributes.

Definition 2.9. ([14]) Let 〈U, AT〉 be an information system in which A₁, A₂,…, A_m ⊆ AT, then ∀X ⊆ U, the pessimistic multi-granulation lower and upper approximations are defined by $\underline{\sum_{i = 1}^{m} A_{i}^{P}} (X)$ and $\bar{\sum_{i = 1}^{m} A_{i}^{P}} (X)$ , respectively, $\begin{matrix} \underline{\sum_{i = 1}^{m} A_{i}^{P}} (X) = {x \in U | [x]_{A_{1}} \subseteq X \\ \land [x]_{A_{2}} \subseteq X \land . . . \land [x]_{A_{m}} \subseteq X}; \\ \bar{\sum_{i = 1}^{m} A_{i}^{P}} (X) = {x \in U | [x]_{A_{1}} \cap X \neg = \emptyset \\ \lor [x]_{A_{2}} \cap X \neg = \emptyset \lor . . . \lor [x]_{A_{m}} \cap X \neg = \emptyset} . \end{matrix}$ where [x] _{A
_i} (1 ≤ i ≤ m) is the equivalence class of x in the terms of attributes A_i.

Definition 2.10. ([14]) Let 〈U, AT〉 be an information system in which A₁, A₂,…, A_m ⊆ AT, then ∀X ⊆ U, the optimistic multi-granulation lower and upper approximations are defined by $\underline{\sum_{i = 1}^{m} A_{i}^{O}} (X)$ and $\bar{\sum_{i = 1}^{m} A_{i}^{O}} (X)$ , respectively, $\begin{matrix} \underline{\sum_{i = 1}^{m} A_{i}^{O}} (X) = {x \in U | [x]_{A_{1}} \subseteq X \\ \lor [x]_{A_{2}} \subseteq X \lor . . . \lor [x]_{A_{m}} \subseteq X}; \\ \bar{\sum_{i = 1}^{m} A_{i}^{O}} (X) = {x \in U | [x]_{A_{1}} \cap X \neg = \emptyset \\ \land [x]_{A_{2}} \cap X \neg = \emptyset \land . . . \land [x]_{A_{m}} \cap X \neg = \emptyset} . \end{matrix}$ where [x] _{A
_i} (1 ≤ i ≤ m) is the equivalence class of x in the terms of attributes A_i.

Theorem 2.11. ([14]) Let I be an information system in which A₁, A₂,…, A_m ⊆ AT, then ∀X ⊆ U, we have following properties about the optimistic multi-granulation rough approximations: $\underline{\sum_{i = 1}^{m} A_{i}^{O}} (X) = \cup_{i = 1}^{m} \underline{A_{i}} (X), \bar{\sum_{i = 1}^{m} A_{i}^{O}} (X) = \cap_{i = 1}^{m} \bar{A_{i}} (X) .$

Theorem 2.12. ([14]) Let I be an information system in which A₁, A₂,…, A_m ⊆ AT, then ∀X ⊆ U, we have following properties about the pessimistic multi-granulation rough approximations: $\underline{\sum_{i = 1}^{m} A_{i}^{P}} (X) = \cap_{i = 1}^{m} \underline{A_{i}} (X), \bar{\sum_{i = 1}^{m} A_{i}^{P}} (X) = \cup_{i = 1}^{m} \bar{A_{i}} (X) .$

3 Rough filters in pseudo-BCI algebras

Definition 3.1. X is a pseudo-BCI algebra and let A be a non-empty subset of X. R is a congruence relation on X. Then A is called an upper (lower, respectively) rough filter of X if $\bar{R} (A)$ ( $\underline{R} (A)$ , respectively) is a filter of X.

Theorem 3.2. Let F be a filter of a pseudo-BCI algebra X and let R_F be a congruence relation induced by filter F. If A is a closed filter, then it is a upper rough filter of X.

Proof. (1) Since A is a filter of X, 1 ∈ A. So A⋂ ([1] _{R
_F}) ¬ = ∅. Therefore $1 \in \bar{R_{F}} (A)$ .

(2) Let x, y ∈ X with $x \to y, x \in \bar{R_{F}} (A)$ . Then [x→ y] _{R
_F} ⋂ A = ([x] _{R
_F} → [y] _{R
_F}) ⋂ A ¬ = ∅, [x] _{R
_F} ⋂ A ¬ = ∅. So we can find α, β ∈ A such that α ∈ [x] _{R
_F} → [y] _{R
_F}, β ∈ [x] _{R
_F}. Simultaneously, we have p ∈ [x] _{R
_F}, q ∈ [y] _{R
_F}. Since p, β ∈ [x] _{R
_F}, we have (p, x) ∈ R_F, (x, β) ∈ R_F. so (p, β) ∈ R_F, we get [p] _{R
_F} = [β] _{R
_F}. Since (p → p, β → p) = (1, β → p) ∈ R_F, so 1 → (β → p) = β → p ∈ A, β ∈ A, we can get p ∈ A. Since, p → q, p ∈ A, A is a filter of X, we obtain q ∈ A, q∈ [y] _{R
_F} ⋂ A ¬ = ∅, thus $y \in \bar{R_{F}} (A)$ .

(3) Let x, y ∈ X with $x ↪ y, x \in \bar{R_{F}} (A)$ . We can get $y \in \bar{R_{F}} (A)$ . The process of proof is similar to (2).

So A is a upper rough filter of X.

Theorem 3.3. Let F be a filter of a pseudo-BCI algebra X and let R_F be a congruence relation induced by filter F. A is a closed filter of a pseudo-BCI algebra X, it is a lower rough filter of X when $\underline{R_{F}} (A) \neg = \emptyset$ .

Proof. (1) Since A is a closed filter of X, so A is a subalgebra of X. Since $\underline{R_{F}} (A) \neg = \emptyset$ , $\underline{R_{F}} (A)$ is a subalgebra of X, so $1 \in \underline{R_{F}} (A)$ .

(2) Let x, y ∈ X with $x \to y, x \in \underline{R_{F}} (A)$ . Then [x → y] _{R
_F} = [x] _{R
_F} → [y] _{R
_F} ⊆ A, [x] _{R
_F} ⊆ A. If α ∈ [y] _{R
_F}, then (α, y) ∈ R_F. Since R_F is a congruence relation on X, we have (x → α, x → y) ∈ R_F, so x → α ∈ [x] _{R
_F} → [y] _{R
_F} = [x → y] _{R
_F} ⊆ A. Since A is a filter of X and x ∈ A, we can get α ∈ A, [y] _{R
_F} ⊆ A, $y \in \underline{R_{F}} (A)$ .

(3) Let x, y ∈ X with $x ↪ y, x \in \underline{R_{F}} (A)$ . We can get $y \in \underline{R_{F}} (A)$ . The process of proof is similar to (2).

So A is a lower rough filter of X.

Next we investigate some properties of fuzzy homomorphism about rough filters.

Definition 3.4. let f : X → Y be a fuzzy mapping, when X, Y are pseudo-BCI algebras, ∀x₁, x₂ ∈ X, y ∈ Y, $\begin{matrix} f (x_{1} \to x_{2}, y) \\ \geq \sup {f (x_{1}, y_{1}) \land f (x_{2}, y_{2}) | y = y_{1} \to y_{2}}, \\ f (x_{1} ↪ x_{2}, y) \\ \geq \sup {f (x_{1}, y_{1}) \land f (x_{2}, y_{2}) | y = y_{1} ↪ y_{2}}, \end{matrix}$

f is defined a fuzzy homomorphism. If f is surjective, f is defined a fuzzy surjective homomorphism.

F is a filter of pseudo-BCI algebra Y. R₂ is a congruence relation induced by filter F.

Theorem 3.5. f is a fuzzy surjective homomorphism of a pseudo-BCI algebra X to a pseudo-BCI algebra Y. Let F be a filter of a pseudo-BCI algebra Y and let A be a subset of X, R₁ can be defined as $\begin{matrix} R_{1} = {(x_{1}, x_{2}) \in X \times X | (y_{1}, y_{2}) \\ \in R_{2}, f (x_{1} \to x_{2}, y_{1} \to y_{2}) \\ \geq \sup (f (x_{1}, y_{1}) \land f (x_{2}, y_{2})), y_{1} \to y_{2} \in Y} \\ ⋃ {(x_{1}, x_{2}) \in X \times X | (y_{1}, y_{2}) \\ \in R_{2}, f (x_{1} ↪ x_{2}, y_{1} ↪ y_{2}) \\ \geq \sup (f (x_{1}, y_{1}) \land f (x_{2}, y_{2})), y_{1} ↪ y_{2} \in Y}, \end{matrix}$

Then, we can get

R₁ is a congruence relation on X;

If R₂ is complete and f is a fuzzy bijection, then R₁ is complete;

${t \in Y | f (k, t) \geq θ, k \in \bar{R_{1}} (A)} = \bar{R_{2}} (B)$ , B = {n ∈ Y|f (m, n) ≥ θ, m ∈ A};

${t \in Y | f (k, t) \geq θ, k \in \underline{R_{1}} (A)} \subseteq \underline{R_{2}} (B)$ . If f is a fuzzy bijection, then ${t \in Y | f (k, t) \geq θ, k \in \underline{R_{1}} (A)} = \underline{R_{2}} (B)$ . B = {n ∈ Y|f (m, n) ≥ θ, m ∈ A}.

Proof. (1) It is easy to prove that (1) hold.

(2) Let x ∈ [x₁ → x₂] _{R
₁}. Since R₂ is complete, R₂ is a congruence relation, and the definition of R₁, we can obtain y ∈ [y₁ → y₂] _{R
₂} = [y₁] _{R
₂} → [y₂] _{R
₂} ⊆ Y. Since f is a fuzzy surjective homomorphism, we have f (x₁ → x₂, y) ≥ sup {f (x₁, y₁) ∧ f (x₂, y₂) |y = y₁ → y₂}. Since f is a fuzzy bijection, so we can find $x_{1}^{'} \in [x_{1}]_{R_{1}}, x_{2}^{'} \in [x_{2}]_{R_{1}}$ , satisfying $f (x_{1}^{'} \to x_{2}^{'}, y) \geq \sup {f (x_{1}, y_{1}) \land f (x_{2}, y_{2}) | y = y_{1} \to y_{2}}$ . Thus x ∈ [x₁] _{R
₁} → [x₂] _{R
₁} and [x₁ → x₂] _{R
₁} ⊆ [x₁] _{R
₁} → [x₂] _{R
₁}. Simultaneously, we have [x₁] _{R
₁} → [x₂] _{R
₁} ⊆ [x₁ → x₂] _{R
₁}. Similarly, Let x ∈ [x₁ ↪ x₂] _{R
₁}, we can get [x₁] _{R
₁} ↪ [x₂] _{R
₁} = [x₁ ↪ x₂] _{R
₁}.

Thus R₁ is complete.

(3) $\forall y \in {t \in Y | f (k, t) \geq θ, k \in \bar{R_{1}} (A)}$ , there is a $x \in \bar{R_{1}} (A)$ , such that f (x₁ → x₂, y) ≥ sup {f (x₁, y₁) ∧ f (x₂, y₂) |y = y₁ → y₂, (y₁, y₂) ∈ R₂}. R₁ is a congruence relation. Hence [x] _{R
₁}∩ A = [x₁] _{R
₁} → [x₂] _{R
₁} ∩ A ¬ = ∅. Then we can find $x_{1}^{'} \in [x]_{R_{1}}, x_{2}^{'} \in [x]_{R_{1}}, x^{'} = x_{1}^{'} \to x_{2}^{'} \in [x_{1}]_{R_{1}} \to [x_{2}]_{R_{1}} \cap A$ . By the definition R₁, we have $f (x_{1}^{'} \to x_{2}^{'}, y^{'}) \geq \sup {f (x_{1}^{'}, y_{1}^{'}) \land f (x_{2}^{'}, y_{2}^{'}) | y^{'} = y_{1}^{'} \to y_{2}^{'}, (y_{1}^{'}, y_{2}^{'}) \in R_{2}}$ . So we can obtain y′ ∈ [y] _{R
₂}. So [y] _{R
₂}∩ B ≠ ∅. Then ${t \in Y | f (k, t) > θ, k \in \bar{R_{1}} (A)} \subseteq \bar{R_{2}} (B)$ .

Conversely, let $y \in \bar{R_{2}} (B)$ , we can get a x ∈ X such that f (x, y) ≥ θ. Hence [y] _{R
₂}∩ B ¬ = ∅, we can find x′ ∈ A such that y′ ∈ [y] _{R
₂} ∩ B. By the definition of R₁, we have x′ ∈ [x] _{R
₁}. So [x] _{R
₁}∩ A ¬ = ∅. Then $x \in \bar{R_{1}} A$ . So $y \in {t | f (k, t) > θ, k \in \bar{R_{1}} (A)}$ , $\bar{R_{2}} (B) \subseteq {t \in Y | f (k, t) > θ, k \in \bar{R_{1}} (A)}$ .

From the above, we obtain ${t \in Y | f (k, t) \geq θ, k \in \bar{R_{1}} (A)} = \bar{R_{2}} (B)$ .

(4) Let $y \in {t \in Y | f (k, t) \geq θ, k \in \underline{R_{1}} (A)}$ . Then we can find a $x \in \underline{R_{1}} (A)$ such that f (x, y) ≥ θ, [x] _{R
₁} ⊆ A. Let y′ ∈ [y] _{R
₂}, then we can find x′ ∈ X such that f (x′, y′) ≥ θ. We have y′ ∈ B, [y] _{R
₂} ⊆ B. So ${t \in Y | f (k, t) \geq θ, k \in \underline{R_{1}} (A)} \subseteq \underline{R_{2}} (B)$ .

Let $y \in \underline{R_{2}} (B)$ . Then we can find a x ∈ X such that f (x, y) ≥ θ, [y] _{R
₂} ⊆ B. Since f is a fuzzy bijection. Let x′ ∈ [x] _{R
₁}, then there is a y′ and f (x′, y′) ≥ θ and y′ ∈ [y] _{R
₂} ⊆ B and [x] _{R
₁} ⊆ A which $x \in \underline{R_{1}} (A)$ . Thus $\underline{R_{2}} (B) \subseteq {t \in Y | f (k, t) \geq θ, k \in \underline{R_{1}} (A)}$ . From the above, we have ${t \in Y | f (k, t) \geq θ, k \in \underline{R_{1}} (A)} = \underline{R_{2}} (B)$ .

Theorem 3.6. f is a fuzzy surjective homomorphism of X to Y. A is a subset of a pseudo-BCI algebra X. B = {y ∈ Y|f (x, y) ≥ θ, x ∈ A}. R₁ can be defined as, ∀x₁, x₂ ∈ X, $\begin{matrix} R_{1} & = & {(x_{1}, x_{2}) \in X \times X | (y_{1}, y_{2}) \in R_{2}, \\ f (x_{1} \to x_{2}, y_{1} \to y_{2}) \\ \geq & \sup (f (x_{1}, y_{1}) \land f (x_{2}, y_{2})), y_{1} \to y_{2} \in Y} \\ ⋃ {(x_{1}, x_{2}) \in X \times X | (y_{1}, y_{2}) \in R_{2}, \\ f (x_{1} ↪ x_{2}, y_{1} ↪ y_{2}) \\ \geq \sup (f (x_{1}, y_{1}) \land f (x_{2}, y_{2})), y_{1} ↪ y_{2} \in Y}, \end{matrix}$ then $\bar{R_{1}} (A)$ is a filter of X if and only if $\bar{R_{2}} (B)$ is a filter of Y.

Proof. Necessity. (1) $\bar{R_{1}} (A)$ is a filter of X. We have $1 \in \bar{R_{1}} (A)$ , [1] _{R
₁}⋂ A ¬ = ∅. Hence, ∃x ∈ [1] _{R
₁} ⋂ A, then we can find y ∈ B, f (x, y) ≥ θ. By the definition of R₁, we can get y ∈ [1′] _{R
₂}, 1′ ∈ Y is the image of the fuzzy mapping of $1 \in \bar{R_{1}} (A)$ so $[1^{'}]_{R_{2}} ⋂ B \neg = \emptyset, 1^{'} \in \bar{R_{2}} (B)$ .

(2) Let x′, y′ ∈ Y with $x^{'}, x^{'} \to y^{'} \in \bar{R_{2}} (B)$ . We have [x′] _{R
₂}⋂ B ¬ = ∅, [x′ → y′] _{R
₂} ⋂ B ¬ = ∅. Then we have x, z ∈ X such that f (x, x′) ≥ θ, f (z, x′ → y′) ≥ θ. When b′ ∈ [x′] _{R
₂} ⋂ B, we can find b ∈ A such that f (b, b′) ≥ θ. By the definition of R₁, we have b ∈ [x] _{R
₁}, b∈ [x] _{R
₁} ⋂ A ¬ = ∅. So $x \in \bar{R_{1}} (A)$ .

Since f is a fuzzy surjective homomorphism, so we can find y ∈ X such that f (y, y′) ≥ θ, y′ ∈ Y. Assume μ′ = (z′ → (x′ → y′)) → y′, then μ′ ∈ Y. Since $\begin{matrix} f (z \to (x \to y), w) \geq f (z, z^{'}) \land f (x \to y, x^{'} \to y^{'}) \\ \geq \sup {f (z, x^{'} \to y^{'}) \land f (x, x^{'}) \land f (y, y^{'})} \\ \geq \sup {f (x, x^{'}) \land f (y, y^{'}) \land f (x, x^{'}) \land f (y, y^{'})} . \end{matrix}$

Thus, we have w = (x′ → y′) → (x′ → y′) =1′ ∈ Y. And we can get $\begin{matrix} f (u, u^{'}) \geq \sup {f (z \to (x \to y), w) \\ \land f (y, y^{'})] | u^{'} = w \to y^{'} = y^{'}} . \end{matrix}$

Since [x′→ y′] _{R
₂} ⋂ B ¬ = ∅, so $\begin{matrix} [x^{'} \to y^{'}]_{R_{2}} ⋂ B & = & [x^{'}]_{R_{2}} \to [y^{'}]_{R_{2}} ⋂ B \\ = & [x^{'}]_{R_{2}} \to [μ^{'}]_{R_{2}} ⋂ B \\ = & [x^{'} \to μ^{'}]_{R_{2}} ⋂ B \neg = \emptyset . \end{matrix}$

Then we can find a ∈ A such that f (a, a′) ≥ θ, a′ ∈ [x′ → μ′] _{R
₂}. By the definition of R₁, we have a ∈ [x → μ] _{R
₁}, so $[x \to μ]_{R_{1}} ⋂ A \neg = \emptyset, x \to μ \in \bar{R_{1}} (A)$ . Since $\bar{R_{1}} (A)$ is a filter of X, $x \in \bar{R_{1}} (A)$ , so $μ \in \bar{R_{1}} (A)$ . Then by the Theorem 3.5.(3), $μ^{'} = y^{'} \in {t \in Y | f (k, t) \geq θ, k \in \bar{R_{1}} (A)} = \bar{R_{2}} (B)$ .

(3) Let x′, y′ ∈ B with $x^{'}, x^{'} ↪ y^{'} \in \bar{H_{2}} (B)$ . We can get $y^{'} \in {t \in Y | f (k, t) \geq θ, k \in \bar{R_{1}} (A)} = \bar{R_{2}} (B)$ . The process of proof is similar to (2).

Thus $\bar{R_{2}} (B)$ is a filter of Y.

Sufficiency. (1^′) If $\bar{R_{2}} (B)$ is a filter of Y. Since $f (1, 1^{'}) \geq θ, 1^{'} \in \bar{R_{2}} (B)$ , hence [1′] _{R
₂}⋂ B ¬ = ∅. f is a fuzzy surjective homomorphism, we can find x′ ∈ [1′] _{R
₂} ⋂ B such that f (x, x′) ≥ θ, x ∈ A. By the definition of R₁, we can obtain x ∈ [1] _{R
₁}. So $[1]_{R_{1}} ⋂ A \neg = \emptyset, 1 \in \bar{R_{1}} (A)$ .

(2^′) Let x, y ∈ X which $x, x \to y \in \bar{R_{1}} (A)$ . We have f (x, x′) ≥ θ, f (x → y, (x → y) ′) ≥ f (x, x′) ∧ f (y, y′) ≥ θ, so $x^{'} \in \bar{R_{2}} (B), (x \to y)^{'} = x^{'} \to y^{'} \in \bar{R_{2}} (B)$ . Since $\bar{R_{2}} (B)$ is a filter of Y, so $y^{'} \in \bar{R_{2}} (B), [y^{'}]_{R_{2}} ⋂ B \neg = \emptyset$ . f is a fuzzy surjective homomorphism, so we can get z ∈ X such that f (z, z′) ≥ θ, z′ ∈ [y′] _{R
₂} ⋂ B. By the definition of R₁, we can get z ∈ [y] _{R
₁}, z ∈ A, so $[y]_{R_{1}} ⋂ A \neg = \emptyset, y \in \bar{R_{1}} (A)$ .

(3^′) Let x, y ∈ X which $x, x ↪ y \in \bar{R_{1}} (A)$ . we can get $y \in \bar{R_{1}} (A)$ . The process of proof is similar to (2^′).

Thus $\bar{R_{1}} (A)$ is a filter of X.

Theorem 3.7. f is a fuzzy surjective isomorphism of X to Y. A is a subset of pseudo-BCI algebra X. B = {y ∈ Y|f (x, y) ≥ θ, x ∈ A}. R₂ is a congruence relation on Y. R₁ can be defined as $\begin{matrix} R_{1} & = & {(x_{1}, x_{2}) \in X \times X | (y_{1}, y_{2}) \\ \in R_{2}, f (x_{1} \to x_{2}, y_{1} \to y_{2}) \\ \geq & \sup (f (x_{1}, y_{1}) \land f (x_{2}, y_{2})), y_{1} \to y_{2} \in Y} \\ ⋃ {(x_{1}, x_{2}) \in X \times X | (y_{1}, y_{2}) \\ \in R_{2}, f (x_{1} ↪ x_{2}, y_{1} ↪ y_{2}) \\ \geq & \sup (f (x_{1}, y_{1}) \land f (x_{2}, y_{2})), y_{1} ↪ y_{2} \in Y} . \end{matrix}$ then $\underline{R_{1}} (A)$ is a filter of X if and only if $\underline{R_{2}} (B)$ is a filter of Y.

Proof. By the Theorem 3.5.(4), we can get ${t \in Y | f (k, t) \geq θ, k \in \underline{R_{1}} (A)} = \underline{R_{2}} (B)$ . The proof process is similar to Theorem 3.6.

4 multi-granulation rough filters in pseudo-BCI algebras

Definition 4.1. X is a pseudo-BCI algebra and A is a non-empty subset of X. R₁, R₂, ⋯, R_m are congruence relations on X. Then A is called a pessimistic multi-granulation upper (lower, respectively) rough filter of X if $\bar{\sum_{i = 1}^{m} R_{i}^{P}} (A)$ ( $\underline{\sum_{i = 1}^{m} R_{i}^{P}} (A)$ , respectively) is a filter of X. $\begin{matrix} \underline{\sum_{i = 1}^{m} R_{i}^{P}} (A) & = & {x \in X | [x]_{R_{F_{1}}} \subseteq X \land [x]_{R_{F_{2}}} \\ \subseteq X \land . . . \land [x]_{R_{F_{m}}} \subseteq X}; \\ \bar{\sum_{i = 1}^{m} R_{i}^{P}} (A) & = & {x \in X | [x]_{R_{F_{1}}} \cap X \neg = \emptyset \lor [x]_{R_{F_{2}}} \\ \cap X \neg = \emptyset \lor . . . \lor [x]_{R_{F_{m}}} \cap X \neg = \emptyset} . \end{matrix}$

Theorem 4.2. Let F₁, F₂, ⋯, F_m be filters of pseudo-BCI algebra X and let R_{F
₁}, R_{F
₂}, ⋯, R_{F
_m} be congruence relations induced by filter F₁, F₂, ⋯, F_m, respectively. A is a closed filter of a pseudo-BCI algebra X, it is a pessimistic multi-granulation lower rough filter of X when $\underline{\sum_{i = 1}^{m} R_{F_{i}}^{P}} (A) \neg = \emptyset$ .

Proof. (1) Since $\underline{\sum_{i = 1}^{m} R_{F_{i}}^{P}} (A) = ⋂ \underline{R_{F_{i}}} (A) \neg = \emptyset$ , By Theorem 3.2, we can get $\underline{R_{F_{i}}} (A)$ is a filter of pseudo-BCI algebra X. Thus ∀R_{F
_i} (A), we have $1 \in \underline{R_{F_{i}}} (A)$ . That means 1 ∈ [1] _{∩R_{F
_i}} ⊆ A. Therefore $1 \in ⋂ \underline{R_{F_{i}}} (A) = \underline{\sum_{i = 1}^{m} R_{F_{i}}^{P}} (A)$ .

(2) Let x, y ∈ X with $x, x \to y \in \underline{\sum_{i = 1}^{m} R_{F_{i}}^{P}} (A)$ . Then $[x \to y]_{\sum_{i = 1}^{m} R_{F_{i}}^{P}} = ⋂_{i = 1}^{m} [x \to y]_{R_{F_{i}}} \subseteq A$ , $[x]_{\sum_{i = 1}^{m} R_{F_{i}}^{P}} = ⋂_{i = 1}^{m} [x]_{R_{F_{i}}} \subseteq A$ . If $α \in [y]_{\sum_{i = 1}^{m} R_{F_{i}}^{P}} = ⋂_{i = 1}^{m} [y]_{R_{F_{i}}}$ , then $(α, y) \in ⋂_{i = 1}^{m} \underline{R_{F_{i}}}$ . Since $\underline{R_{F_{i}}}$ is a congruence relation, we have $(x \to α, x \to y) \in ⋂_{i = 1}^{m} \underline{R_{F_{i}}}$ , $x \to α \in ⋂_{i = 1}^{m} [x \to y]_{R_{F_{i}}} \subseteq A$ , since A is a filter of X, x ∈ A, so α ∈ A, $[y]_{⋂_{i = 1}^{m} R_{F_{i}}} \subseteq A$ , $y \in \underline{\sum_{i = 1}^{m} R_{F_{i}}^{P}} (A)$ .

(3) Let x, y ∈ X with $x, x ↪ y \in \underline{\sum_{i = 1}^{m} R_{F_{i}}^{P}} (A)$ . We can get $y \in \underline{\sum_{i = 1}^{m} R_{F_{i}}^{P}} (A)$ . The process of proof is similar to (2).

So A is a pessimistic multi-granulation lower rough filter.

However, Let F₁, F₂, ⋯, F_m be filters of pseudo-BCI algebra X and let R_{F
₁}, R_{F
₂}, ⋯, R_{F
_m} be congruence relations induced by filter F₁, F₂, ⋯, F_m, respectively. A is a filter of a pseudo-BCI algebra X, it may be not a pessimistic multi-granulation upper rough filter of X.

Example 4.3. Let X = {a, b, c, d, e, 1} with two binary operations in Tables 1 and 2. Then (X ; ≤, →, ↪) is a pseudo-BCI algebra. So F₁ = {a, b, 1} and F₂ = {c, 1} are filters. Let R_{F
₁} = {(a, b), (a, a), (b, b), (1, 1)}, R_{F
₂} = {(c, 1)}. So we can get [a] _{R
_{F
₁}} = {a, b}, [1] _{R
_{F
₁}} = {1}, and [c] _{R
_{F
₂}} = [1] _{R
_{F
₂}} = {c, 1}. Put A = {b, 1}, then $\bar{\sum_{i = 1}^{2} R_{F_{i}}^{p}} (A) = {a, b, c, 1}$ . By verification, $\bar{\sum_{i = 1}^{2} R_{F_{i}}^{p}} (A) = {a, b, c, 1}$ is not a filter. Since b → d = c, but we can not find d in $\bar{\sum_{i = 1}^{2} R_{F_{i}}^{p}} (A) = {a, b, c, 1}$ .

Table 1
Cayley table of operation " → "

→ a b c d e 1

a 1 a d e c b

b b 1 e c d a

c d e 1 a b c

d e c b 1 a d

e c d a b 1 c

1 a b c d e 1

→	a	b	c	d	e	1
a	1	a	d	e	c	b
b	b	1	e	c	d	a
c	d	e	1	a	b	c
d	e	c	b	1	a	d
e	c	d	a	b	1	c
1	a	b	c	d	e	1

Table 2

Cayley table of operation "↪"

↪	a	b	c	d	e	1
a	1	a	e	c	d	b
b	b	1	d	e	c	a
c	e	d	1	b	a	c
d	c	e	a	1	b	d
e	d	c	b	a	1	e
1	a	b	c	d	e	1

Definition 4.4. Let R₁ and R₂ be congruence relations on X, a partial relation can be defined, i.e. X/R₁ ⪯ X/R₂, if and only if, ∀ [x_i] _{R
₁} ∈ X/R₁, there exist [x_j] _{R
₂} ∈ X/R₂, such that [x_i] _{R
₁} ⊆ [x_j] _{R
₂} where X/R₁, X/R₂ are partitions induced by congruence relations R₁, R₂, respectively. We say that R₁ is finer than R₂, R₂ is coarser than R₁.

Remark 4.5. Let F₁ and F₂ be filters of pseudo-BCI algebra X and let R_{F
₁}, R_{F
₂} be congruence relations induced by filters F₁ and F₂, respectively. The binary relation ⪯ satisfies the following property, $F_{1} \subseteq F_{2} \Leftrightarrow R_{F_{1}} ⪯ R_{F_{2}} .$

Proof. Necessity. Let x ∈ F₁ ⊆ F₂, by Definition 2.6. we have, for each [x] _{R
_{F
₁}} ⊆ F₁, there exists [x] _{R
_{F
₂}} ⊆ F₂, such that [x] _{R
_{F
₁}} ⊆ [x] _{R
_{F
₂}}.

Sufficiency. Since [x] _{R
_{F
₁}} ⊆ [x] _{R
_{F
₂}}. Obviously, F₁ ⊆ F₂.

Theorem 4.6. Let F₁, F₂, ⋯, F_m be filters of pseudo-BCI algebra X and let R_{F
₁}, R_{F
₂}, ⋯, R_{F
_m} be congruence relations induced by filters F₁, F₂, ⋯, F_m, respectively. Satisfying, R_{F
₁} ⪯ R_{F
₂} ⪯… ⪯ R_{F
_m}. A is a closed filter of a pseudo-BCI algebra X, it is a pessimistic multi-granulation upper rough filter of X, then $\bar{\sum_{i = 1}^{m} R_{F_{i}}^{P}} (A) = \bar{R_{F_{m}}}$ .

Proof. $\forall x \in \bar{\sum_{i = 1}^{m} R_{F_{i}}^{P}} (A)$ , there exist i ∈ 1, 2, ⋯, m such that [x] _{R
_{F
_i}}∩ A ¬ = ∅. Since R_{F
₁} ⪯ R_{F
₂} ⪯… ⪯ R_{F
_m}, we can get [x] _{R
_{F
_i}} ⊆ [x] _{R
_{F
_m}}, so x∈ [x] _{R
_{F
_m}} ∩ A ¬ = ∅. It follows that $x \in \bar{R_{F_{m}}} (A)$ . Thus $\bar{\sum_{i = 1}^{m} R_{F_{i}}^{P}} (A) \subseteq \bar{R_{F_{m}}}$ . When $x \in \bar{R_{F_{m}}} (A)$ , by the Definition 3.4, $\bar{R_{F_{m}}} (A) \subseteq \bar{\sum_{i = 1}^{m} R_{F_{i}}^{P}} (A)$ , obviously. So from discussion above, we have $\bar{\sum_{i = 1}^{m} R_{F_{i}}^{P}} (A) = \bar{R_{F_{m}}} (A)$ .

By the Theorem 3.2, we have $\bar{R_{F_{m}}} (A)$ is a filter on X. Thus A is a pessimistic multi-granulation upper rough filter.

Definition 4.7. X is a pseudo-BCI algebra and A is a non-empty subset of X. R₁, R₂, ⋯, R_m are congruence relations on X. Then A is called a optimistic multi-granulation upper (lower, respectively) rough filter of X if $\bar{\sum_{i = 1}^{m} R_{i}^{O}} (A)$ ( $\underline{\sum_{i = 1}^{m} R_{i}^{O}} (A)$ , respectively) is a filter of X. $\begin{matrix} \underline{\sum_{i = 1}^{m} R_{i}^{O}} (A) & = & {x \in X | [x]_{R_{F_{1}}} \subseteq X \lor [x]_{R_{F_{2}}} \\ \subseteq X \lor \dots \lor [x]_{R_{F_{m}}} \subseteq X}; \\ \bar{\sum_{i = 1}^{m} R_{i}^{O}} (A) & = & {x \in X | [x]_{R_{F_{1}}} \cap X \neg = \emptyset \land [x]_{R_{F_{2}}} \\ \cap X \neg = \emptyset \land \dots \land [x]_{R_{F_{m}}} \cap X \neg = \emptyset} . \end{matrix}$

Theorem 4.8. Let F₁, F₂, ⋯, F_m be filters of pseudo-BCI algebra X and let R_{F
₁}, R_{F
₂}, ⋯, R_{F
_m} be congruence relations induced by filter F₁, F₂, ⋯, F_m, respectively. A is a closed filter of a pseudo-BCI algebra X, it is an optimistic multi-granulation upper rough filter of X.

Proof. (1) By the Theorem 3.2, we have $\bar{R_{F_{i}}} (A)$ is a filter of X. Thus we have 1∈ [1] _{R
_{F
_i}} ⋂ A ¬ = ∅. By the Definition 4.1., we can get $1 \in \bar{\sum_{i = 1}^{m} R_{F_{i}}^{O}} (A)$ .

(2) Let x, y ∈ X with $x, x \to y \in \bar{\sum_{i = 1}^{m} R_{F_{i}}^{O}} (A)$ . Then ∀i ∈ 1, 2, ⋯, m, we have [x] _{R
_{F
_i}}⋂ A ¬ = ∅, [x→ y] _{R
_{F
_i}} ⋂ A ¬ = ∅. Thus we can find α, β ∈ A such that α∈ [x → y] _{R
_{F
_i}} ⋂ A ¬ = ∅, β ∈ [x] _{R
_{F
_i}} ⋂ A ¬ = ∅. Since R_{F
_i} is a congruence relation. Thus we have α = p→ q ∈ [x → y] _{R
_{F
_i}} ⋂ A = [x] _{R
_{F
_i}} → [y] _{R
_{F
_i}} ⋂ A ¬ = ∅, p ∈ [x] _{R
_{F
_i}}, q ∈ [y] _{R
_{F
_i}}. We have (p, β) ∈ R_{F
_i}. Since (p → p, β → p) = (1, β → p) ∈ R_{F
_i}, so 1 → (β → p) = β → p ∈ A. Since A is a closed filter, β ∈ A, we have q ∈ A. Since p → q ∈ A, we have q ∈ A. Thus q ∈ [y] _{R
_{F
_i}} ⋂ A. We can get $y \in \bar{\sum_{i = 1}^{m} R_{F_{i}}^{O}} (A)$ .

(3) let x, y ∈ X with $x, x ↪ y \in \bar{\sum_{i = 1}^{m} R_{F_{i}}^{O}} (A)$ . We can get $y \in \bar{\sum_{i = 1}^{m} R_{F_{i}}^{O}}$ . The process of proof is similar to (2).

Thus $\bar{\sum_{i = 1}^{m} R_{F_{i}}^{O}} (A)$ is a filter of X. A is an optimistic multi-granulation upper rough filter.

However, let F₁, F₂, ⋯, F_m be filters of pseudo-BCI algebra, let R_{F
₁}, R_{F
₂}, ⋯, R_{F
_m} be congruence relations induced by filter F₁, F₂, ⋯, F_m, respectively. A is a filter of a pseudo-BCI algebra X, it may be not an optimistic multi-granulation lower rough filter of X.

Example 4.9. Let X = {a, b, c, d, e, 1} with two binary operations in Tables 1 and 2. Then (X ; ≤, →, ↪) is a pseudo-BCI algebra. So F₁ = {a, b, 1} and F₂ = {c, 1} are filters. Let R_{F
₁} = {(a, b), (a, a), (b, b), (1, 1)}, R_{F
₂} = {(c, 1)}. So we can get [a] _{R
_{F
₁}} = [b] _{R
_{F
₁}} = {a, b}, [1] _{R
_{F
₁}} = {1}, and [c] _{R
_{F
₂}} = [1] _{R
_{F
₂}} = {c, 1}. Put A = {a, b, c, d, e, 1}, then $\underline{\sum_{i = 1}^{2} R_{F_{i}}^{O}} (A) = {a, b, c, 1}$ . By verification, $\underline{\sum_{i = 1}^{2} R_{F_{i}}^{O}} (A) = {a, b, c, 1}$ is not a filter. Since b → d = c, but we can not find d in $\underline{\sum_{i = 1}^{2} R_{F_{i}}^{O}} (A) = {a, b, c, 1}$ .

Theorem 4.10. Let F₁, F₂, ⋯, F_m be filters of pseudo-BCI algebra X and let R_{F
₁}, R_{F
₂}, ⋯, R_{F
_m} be congruence relations induced by filter F₁, F₂, ⋯, F_m, respectively. Satisfying, R_{F
₁} ⪯ R_{F
₂} ⪯… ⪯ R_{F
_m}. When A is a closed filter of a pseudo-BCI algebra X, it is an optimistic multi-granulation lower rough filter of X if $\underline{R_{F_{n}}} \neg = \emptyset$ . Then $\underline{\sum_{i = 1}^{m} R_{F_{i}}^{O}} (A) = \underline{R_{F_{1}}} (A)$ .

Proof. $\forall x \in \underline{\sum_{i = 1}^{m} R_{F_{i}}^{O}} (A)$ , there exist i ∈ 1, 2, ⋯, m such that [x] _{R
_{F
_i}} ⊆ A. Since R_{F
₁} ⪯ R_{F
₂} ⪯… ⪯ R_{F
_m}, we can get [x] _{R
_{F
₁}} ⊆ [x] _{R
_{F
_i}}, so x ∈ [x] _{R
_{F
₁}} ⊆ A. It follows that $x \in \underline{R_{F_{1}}} (A)$ . Thus $\underline{\sum_{i = 1}^{m} R_{F_{i}}^{O}} (A) \subseteq \underline{R_{F_{1}}}$ . When $x \in \underline{R_{F_{1}}} (A)$ , by the Definition 4.1, $\underline{R_{F_{1}}} (A) \subseteq \underline{\sum_{i = 1}^{m} R_{F_{i}}^{O}} (A)$ , obviously. So from discussion above, we have $\underline{\sum_{i = 1}^{m} R_{F_{i}}^{O}} (A) = \underline{R_{F_{1}}} (A)$ .

By the Theorem 3.3, we have $\underline{R_{F_{1}}}$ is a filter of X. Thus A is a optimistic multi-granulation lower rough filter.

5 Rough fuzzy filters in pseudo-BCI algebras

Definition 5.1. A fuzzy set μ : X → [0, 1] is called a fuzzy pseudo-filter (briefly, fuzzy filter) of pseudo-BCI algebra X if it satisfies: ∀x, y ∈ X [19],

μ (1) ≥ μ (x);

μ (y) ≥ μ (x → y) ∧ μ (x);

μ (y) ≥ μ (x ↪ y) ∧ μ (x).

Proposition 5.2. ([36]) Let μ be a fuzzy filter of a pseudo-BCI algebra X. If x ≤ y, then μ (x) ≤ μ (y) where x, y ∈ X.

Definition 5.3. R is a congruence relation on a pseudo-BCI algebra X. μ is a fuzzy subset of X. Then we define the fuzzy upper approximation $\bar{R} (μ) (x)$ and fuzzy lower approximation $\underline{R} (μ) (x)$ as follows: $\begin{matrix} \bar{R} (μ) (x) = \lor_{x^{'} \in [x]_{R}} μ (x^{'}), \\ \underline{R} (μ) (x) = \land_{x^{'} \in [x]_{R}} μ (x^{'}) . \end{matrix}$ when μ satisfying: $\bar{R} (μ) (x) \neg = \underline{R} (μ) (x)$ . μ is defined a rough fuzzy set.

Definition 5.4. A fuzzy set μ of a pseudo-BCI algebra X is called an upper (a lower, respectively) rough fuzzy filter when $\bar{R} (μ)$ ( $\underline{R} (μ)$ , respectively) is a fuzzy filter of X.

Theorem 5.5. Let F be a filter of a pseudo-BCI algebra X and let R_F be a congruence relation induced by filter F. μ is a fuzzy filter of X, then $\bar{R_{F}} (μ)$ is a fuzzy filter of X.

Proof. (1) μ is a fuzzy filter of X, then ∀x ∈ X, μ (1) ≥ μ (x). Hence we have: $\begin{matrix} \bar{R_{F}} (μ) (1) \geq \lor_{y \in [1]_{R_{F}}} μ (y) \\ = μ (1) \geq \lor_{x^{'} \in [x]_{R_{F}}} μ (x^{'}) = \bar{R_{F}} (μ) (x) . \end{matrix}$

(2) ∀x, y ∈ X, Since μ is a fuzzy filter of X, R_F is a congruence relation, so we have $\begin{matrix} \bar{R_{F}} (μ) (y) = \lor_{y^{'} \in [y]_{R_{F}}} μ (y^{'}) \\ \geq \lor_{x^{'} \to y^{'} \in [x]_{R_{F}} \to [y]_{R_{F}}, x^{'} \in [x]_{R_{F}}} \\ (μ (x^{'} \to y^{'}) \land μ (x^{'})) \\ \geq \lor_{x^{'} \to y^{'} \in [x \to y]_{R_{F}}, x^{'} \in [x]_{R_{F}}} (μ (x^{'} \to y^{'}) \land μ (x^{'})) \\ \geq (\lor_{x^{'} \to y^{'} \in [x \to y]_{R_{F}}, x^{'} \in [x]_{R_{F}}} μ (x^{'} \to y^{'})) \\ \land (\lor_{x^{'} \in [x]_{R_{F}}} μ (x^{'})) \\ = \bar{R_{F}} (μ) (x \to y) \land \bar{R_{F}} (μ) (x) . \end{matrix}$

(3) Similarly, we can obtain $\bar{R_{F}} (μ) (y) \geq \bar{R_{F}} (μ) (x ↪ y) \land \bar{R_{F}} (μ) (x)$ .

Thus, $\bar{R_{F}} (μ)$ is a fuzzy filter of X.

However, Let F be a filter of a pseudo-BCI algebra X and R_F a congruence relation induced by filter F. μ is a fuzzy filter of X, then $\underline{R_{F}} (μ)$ may be not a fuzzy filter of X.

Example 5.6. Let X = {a, b, c, d, 1} with two binary operations in Tables 3 and 4. Then (X ; ≤, →, ↪) is a pseudo-BCI algebra. So F = {a, b, c, d, 1} is a filter. Let $\begin{matrix} R_{F} & = & {(a, c), (c, a), (a, a), (c, c), (b, b), \\ (d, 1), (1, d), (d, d) (1, 1)} . \end{matrix}$

Table 3
Cayley table of operation " → "

→ a b c d 1

a 1 1 1 1 1

b d 1 1 1 1

c d c 1 1 1

d c c c 1 1

1 a b c d 1

→	a	b	c	d	1
a	1	1	1	1	1
b	d	1	1	1	1
c	d	c	1	1	1
d	c	c	c	1	1
1	a	b	c	d	1

Table 4

Cayley table of operation "↪"

↪	a	b	c	d	1
a	1	d	1	1	1
b	d	1	1	1	1
c	d	d	1	1	1
d	c	b	c	1	1
1	a	b	c	d	1

So we can get [a] _{R
_F} = {a, c}, [1] _{R
_F} = {d, 1}. Define fuzzy filter μ : X → [0, 1] as following: $μ (1) = 1, μ (a) = μ (d) = μ (c) = 0.5, μ (b) = 0.7 .$

Then μ is a fuzzy filter of X, but $\begin{matrix} \underline{R_{F}} (μ) (1) = 0.5, \underline{R_{F}} (μ) (b) = 0.7, \\ \underline{R_{F}} (μ) (1) \underline{R_{F}} (μ) (b) . \end{matrix}$

That is, μ is not a lower rough fuzzy filter.

6 Conclusions

In this paper, we investigate rough filters, rough fuzzy filters, multi-granulation rough filters and discussed their properties. When a congruence relation was structured by a filter of a pseudo-BCI algebra, rough filter and rough fuzzy filter can be defined. In addition, by this congruence, we investigate the relations between upper (lower) rough filters and upper (lower) approximations of fuzzy homomorphic images. Next, we proposed multi-granulation rough filters by establishing multi congruence relations. Those congruence relations were structured by multi filters of pseudo-BCI algebra.

Author contributions

All authors have contributed equally to this paper.

Conflicts of interest

The authors declare no conflicts of interest.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61573240, 61473239).

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Multi-granulation rough filters and rough fuzzy filters in Pseudo-BCI algebras

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Rough filters in pseudo-BCI algebras

4 multi-granulation rough filters in pseudo-BCI algebras

Table 1 Cayley table of operation " → " → a b c d e 1 a 1 a d e c b b b 1 e c d a c d e 1 a b c d e c b 1 a d e c d a b 1 c 1 a b c d e 1

Table 3 Cayley table of operation " → " → a b c d 1 a 1 1 1 1 1 b d 1 1 1 1 c d c 1 1 1 d c c c 1 1 1 a b c d 1

Author contributions

Conflicts of interest

Footnotes

Acknowledgments

References

Table 1
Cayley table of operation " → "

→ a b c d e 1

a 1 a d e c b

b b 1 e c d a

c d e 1 a b c

d e c b 1 a d

e c d a b 1 c

1 a b c d e 1

Table 3
Cayley table of operation " → "

→ a b c d 1

a 1 1 1 1 1

b d 1 1 1 1

c d c 1 1 1

d c c c 1 1

1 a b c d 1