(∈,∈ ∨ q )-intuitionistic fuzzy BCI -subalgebras of a BCI -algebra

Abstract

In this paper, we first introduce the concept of quasi-coincidence of an intuitionistic fuzzy point within an intuitionistic fuzzy set. By using this new idea, we further introduce the notions of (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebras of BCI-algebras and investigate some of their related properties. Some characterization theorems of these generalized intuitionistic fuzzy BCI-subalgebras are derived. Then we study the cartesian product of (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebras. Finally, we introduce the homomorphism of intuitionistic fuzzy BCI-algebras, and prove that image and inverse image of an intuitionistic fuzzy set on a BCI-algebra should be an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra on the basis of homomorphism of BCI-algebra.

Keywords

fuzzy set cartesian product homomorphism

1 Introduction

The notion of intuitionistic fuzzy sets (IFSs) was introduced by Atanassov [6] in 1983 as a generalization of traditional fuzzy sets [40]. Fuzzy sets give a degree of membership of an element in a given set, IFS gives both a membership degree and a non-membership degree. Both degrees belong to the interval [0, 1] and their sum should not exceed 1. The membership and non-membership values induce an indeterminacy index, which models the hesitancy of deciding the degree to which an object satisfies a particular property. As the basis for the study of IFS theory, many operations and relations over IFSs were introduced [7]. The theory of IFSs becomes a vigorous area of research in differentdomains such as engineering, medical science, social science, physics, statics, graph theory, artificial intelligent, signal processing, multi-agent systems, pattern recognition, computer networks, expert systems, robotics, automata theory, decision making and so on.

On the other hand, Murali [24] proposed a definition of a fuzzy point belonging to a fuzzy subset under a natural equivalence on a fuzzy set. The idea of quasi-coincidence of a fuzzy point with a fuzzy set, which is mentioned in [26] played a vital role to generate some different types of fuzzy subgroups. Yuan et al. [39] redefined (α, β)-intuitionistic fuzzy subgroups. Subsequently, Abdullah et al. [1-5], Dudek et al. [14], Larimi et al. [20] and Narayanan et al. [25] extended these results to semigroups, near-rings and hemirings.

The study of BCK/BCI-algebra was initiated by Imai and Iśeki [15, 16] in 1966 as a generalization of the concept of set-theoretic difference and propositional calculus. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. The study of structures of fuzzy sets in BCK/BCI-algebraic structures had been carried out by many researchers [17 –23]. Hong et al. [17, 23] introduced the notion of intuitionistic fuzzy BCK/BCI-subalgebras. Bhowmik et al. [8], Jana et al. [9–13] and Senapati et al. [28 –38] has done lot of works on BCK/BCI-algebra and B/BG/G-algebras which is related to these algebras. To the best of our knowledge no works are available on (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebras. For this reason we are motivated to develop these theories for BCI-algebras.

Inspired by the previous works, a natural problem is whether of a BCK/BCI-algebras can be characterized by the corresponding properties of IFSs. In this paper, we combine the theories of IFSs with BCI-subalgebras to broaden application fields of theory of fuzzy sets and provide more ways to study fuzzy algebras. We introduce the notion of (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebras and find the cartesian product of (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebras on BCI-algebra. Finally, we show that image and inverse image of an IFS on BCI-algebras will be an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebras on the basis of homomorphism of BCI-algebras.

2 Preliminaries

In this section, some elementary aspects that are necessary for this paper are included.

By a BCI-algebra [15] we mean an algebra (X, ∗ , 0) of type (2, 0) satisfying the following axioms for all x, y, z ∈ X: (i) ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) =0 (ii) (x ∗ (x ∗ y)) ∗ y = 0 (iii) x ∗ x = 0 (iv) x ∗ y = 0 and y ∗ x = 0 imply x = y.

If a BCI-algebra X satisfies 0 ∗ x = 0 for all x ∈ X, then we say that X is a BCK-algebra [21].

We can define a partial ordering “≤” by x ≤ y if and only if x ∗ y = 0.

Any BCK-algebra X satisfies the following axioms for all x, y, z ∈ X: (1) (x ∗ y) ∗ z = (x ∗ z) ∗ y (2) ((x ∗ z) ∗ (y ∗ z)) ∗ (x ∗ y) =0 (3) x ∗ 0 = x (4) x ∗ y = 0 ⇒ (x ∗ z) ∗ (y ∗ z) =0, (z ∗ y) ∗ (z∗x) =0 .

A non-empty subset S of X is called a BCI-subalgebra of X if x ∗ y ∈ S for any x, y ∈ S.

Let X be a non-empty set. A fuzzy set μ of X is defined as a mapping μ : X → [0, 1], where [0, 1] is the usual interval of real numbers. we take $𝔽 (X)$ as the set of all fuzzy subsets of X.

A fuzzy set A = {〈x, μ_A (x) 〉 : x ∈ X} in a BCK/BCI-algebra X is called a fuzzy BCI-subalgebra of X if it satisfies for all x, y ∈ X $μ_{A} (x * y) \geq min {μ_{A} (x), μ_{A} (y)} .$

A fuzzy set A in a set X is of the form $μ_{A} (y) = {\begin{matrix} t \in (0, 1] & if y = x \\ 0 & if y \neq x . \end{matrix}$ is said to be a fuzzy point with support x and value t and is denoted by x_t . For a fuzzy point x_t and a fuzzy set μ of a set X, Pu and Liu [26] gave meaning to the symbol x_tΦμ, where Φ ∈ {∈, q, ∈ ∨ q, ∧ q}. To say that x_t ∈ μ (respectively, x_tqμ) means that μ (x) ≥ t (respectively μ (x) + t > 1), and in this case, x_t is said to belong to (respectively, be quasi-coincident with) a fuzzy set μ. To say that x_t ∈ ∨ qμ (respectively, x_t ∈ ∧ qμ) means that x_t ∈ μ or x_tqμ (respectively, x_t ∈ μ and x_tqμ). To say that $x_{t} \bar{φ} μ$ means that x_tΦμ does not hold, where Φ ∈ {∈, q, ∈ ∨ q, ∈ ∧ q}.

Atanassov introduced the notion of an IFS defined on a non-empty set X, an object having the the form A = (μ_A, ν_A) = {〈x, μ_A (x) , ν_A (x) 〉 : x ∈ X}, where the functions μ_A : X → [0, 1] and ν_A : X → [0, 1] denote the degree of membership (namely, μ_A (x)) and the degree of nonmembership (namely, ν_A (x)) of each element x ∈ X to the set A respectively, and 0 ≤ μ_A (x) + ν_A (x) ≤1 for all x ∈ X.

An IFS A = {〈x, μ_A (x) , ν_A (x) 〉 : x ∈ X} in X is called an intuitionistic fuzzy BCI-subalgebra [19] of X if it satisfies the following two conditions (1) μ_A (x ∗ y) ≥ min {μ_A (x) , μ_A (y)} and (2) ν_A (x ∗ y) ≤ max {ν_A (x) , ν_A (y)} for all x, y ∈ X.

Definition 2.1. [7] For every two IFSs A = (μ_A, ν_A) and B = (μ_B, ν_B) in X, we define (1) A ⊆ B iff μ_A (x) ≤ μ_B (x) and ν_A (x) ≥ ν_B (x) for all x ∈ X (2) A = B iff μ_A (x) = μ_B (x) and ν_A (x) = ν_B (x) for all x ∈ X (3) (A ∪ B) (x) = {max {μ_A (x) , μ_B (x)} , min {ν_A (x) , ν_B (x)} | x ∈ X} (4) (A ∩ B) (x) = {min {μ_A (x) , μ_B (x)} , max {ν_A (x) , ν_B (x)} | x ∈ X} .

3 (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebras

In this section, (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebras of BCI-algebras are firstly defined and introduced. Some Properties of (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebras are investigated and given in this section. In what follows, we simply use X to denote a BCI-algebra unless otherwise specified.

Definition 3.1. Let A = (μ_A, ν_A) be an IFS in a set X is of the form $μ_{A} (y) = {\begin{matrix} t \in (0, 1] & if y = x \\ 0 & if y \neq x, \end{matrix}$ and $ν_{A} (y) = {\begin{matrix} m \in [0, 1) & if y = x \\ 0 & if y \neq x . \end{matrix}$

Then 〈x, x_t, x_m〉 is said to be intuitionistic fuzzy point in X with support x and values x_t and x_m such that 0 ≤ t + m ≤ 1, where t (respectively, m) is the degree of membership (respectively, nonmembership) of 〈x, x_t, x_m〉. For an intuitionistic fuzzy point 〈x, x_t, x_m〉 and an IFS A = 〈x, μ_A, ν_A〉 in X. We give the meaning of the symbol (x_tΦμ_A, x_mΦν_A), where Φ ∈ {∈, q, ∈ ∨ q, ∈ ∧ q} . To say that x_t ∈ μ_A (respectively, x_tqμ_A) and x_m ∈ ν_A (respectively, x_mqν_A) means that μ_A (x) ≥ t (respectively, μ_A (x) + t > 1) and ν_A (x) ≤ m (respectively, ν_A (x) + m < 1), and in this case we say that, x_t is said to belong to (respectively, be quasi-coincident with) and x_m is said to belong to (respectively, be quasi-coincident with) an IFS A = 〈x, μ_A, ν_A〉. To say that x_t ∈ ∨ q (respectively, x_t ∈ ∧ q) and x_m ∈ ∨ q (respectively, x_m ∈ ∧ q) means that x_t ∈ μ_A or x_tqμ_A (respectively, x_t ∈ μ_A and x_tqμ_A) and x_m ∈ ν_A or x_mqν_A (respectively, x_m ∈ ν_A and x_mqν_A). To say that $(x_{t} \bar{Φ} μ_{A}, x_{m} \bar{Φ} ν_{A})$ means that x_tΦμ_A does not hold and x_mΦν_A does not hold, where Φ ∈ {∈, q, ∈ ∨ q, ∈ ∧ q}.

Definition 3.2. Let A = (μ_A, ν_A) be a IFS of X and (t, m) ∈ (0.5, 1] × [0.5, 1), we define U (μ_A, ν_A ; t, m) = {x ∈ X|μ_A (x) ≥ t and ν_A (x) ≤ m} is called a t-level cut of μ_A and m-level cut of ν_A of the IFS A = (μ_A, ν_A).

Definition 3.3. Let A = (μ_A, ν_A) be IFS in X. Then A = (μ_A, ν_A) is called an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X if (1) x_t ∈ μ_A, y_s ∈ μ_A ⇒ (x ∗ y) _min(t,s) ∈ ∨ qμ_A, for all t, s ∈ (0, 1] and for all x, y ∈ X (2) x_m ∈ ν_A, y_n ∈ ν_A ⇒ (x ∗ y) _max(m,n) ∈ ∨ qν_A, for all m, n ∈ [0, 1) and for all x, y ∈ X.

We now illustrate the above definitions by using some examples.

Example 3.4. Consider a BCI-algebra X = {0, a, b, c} with the following Caley table: $\begin{matrix} * & 0 & a & b & c \\ 0 & 0 & a & b & c \\ a & a & 0 & c & b \\ b & b & c & 0 & a \\ c & c & b & a & 0 \end{matrix}$

We define IFS A = (μ_A, ν_A) : X→ (0, 1] ×[0, 1) by $μ_{A} (x) = {\begin{matrix} 1 & if x = 0, b \\ 0.6 & if x = a, c, \end{matrix}$ and $ν_{A} (x) = {\begin{matrix} 0 & if x = 0, b \\ 0.4 & if x = a, c . \end{matrix}$

It is routine to check that A = (μ_A, ν_A) is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X.

Theorem 3.5. Let A = (μ_A, ν_A) be an IFS of X. Then A = (μ_A, ν_A) is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X if and only if for all x, y ∈ X, the following conditions hold, (1) μ_A (x ∗ y) ≥ min {μ_A (x) , μ_A (y) , 0.5}, and (2) ν_A (x ∗ y) ≤ max {ν_A (x) , ν_A (y) , 0.5}.

Proof: Let A = (μ_A, ν_A) be an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X. Suppose that the conditions (1) and (2) does not hold. Then there exists x, y ∈ X such that μ_A (x ∗ y) < min{μ_A (x) , μ_A (y) , 0.5} and ν_A (x ∗ y) > max {ν_A (x) , ν_A (y) , 0.5}. Now, for some t ∈ (0, 1] and m ∈ [0, 1), we have μ_A (x ∗ y) < t ≤ min {μ_A (x) , μ_A (y) , 0.5} and ν_A (x ∗ y) > m ≥ max {ν_A (x) , ν_A (y) , 0.5}. Now, if min {μ_A (x) , μ_A (y)} <0.5 and max {ν_A (x) , ν_A (y)}>0.5, then μ_A (x ∗ y) < t ≤ min {μ_A (x) , μ_A (y)} and ν_A (x ∗ y) > m ≥ max {ν_A (x) , ν_A (y)}, i.e. μ_A (x)≥t, μ_A (y) ≥ t but, $(x * y)_{t} \bar{\in} μ_{A}$ , and ν_A (x) ≤ m, ν_A (y) ≤ m, but $(x * y)_{m} \bar{\in} ν_{A}$ , i.e. $x * y \bar{\in} A$ . Also,μ_A (x ∗ y) + t < 0.5 + 0.5 = 1 and ν_A (x ∗ y) + m>0.5 + 0.5 = 1. Thus, $(x * y)_{t} \bar{q} μ_{A}$ and (x ∗ y) _m $\bar{q} ν_{A}$ , which imply that $x * y \bar{\in \lor q} A$ , is a contradiction. Therefore, (1) and (2) are valid. Again, ifmin {μ_A (x) , μ_A (y)} ≥0.5 and max {ν_A (x) , ν_A (y)} ≤0.5, then μ_A (x ∗ y) <0.5 and ν_A (x ∗ y) >0.5, which implies that μ_A (x) ≥0.5, μ_A (y) ≥0.5 but, $(x * y)_{min (0.5, 0.5)} = (x * y)_{0.5} \bar{\in} μ_{A}$ , and ν_A (x) ≤0.5, ν_A (y) ≤0.5 but, $(x * y)_{max (0.5, 0.5)} = (x * y)_{0.5} \bar{\in} ν_{A}$ , i.e. $(x * y)_{0.5} \bar{\in} A$ . Also, μ_A (x∗ y) +0.5 < 0.5 +0.5 = 1 and ν_A (x ∗ y) +0.5 > 0.5 + 0.5 = 1, in the same way it can be shown that $(x * y)_{0.5} \bar{q} A$ . Thus, $(x * y)_{0.5} \bar{\in \lor q} A$ , which is a contradiction. Therefore, (1) and (2) are valid.

Conversely, we assume that an IFS A = (μ_A, ν_A) satisfies the conditions (1) and (2). Let x, y ∈ X, t, s ∈ (0, 1] and m, n ∈ [0, 1) be such that μ_A (x) ≥ t, μ_A (y) ≥ s, ν_A (x) ≤ m and ν_A (y) ≤ n. So, by hypothesis μ_A (x ∗ y) ≥ min {μ_A (x) , μ_A (y) , 0.5} and ν_A (x∗y) ≤ min {ν_A (x) , ν_A (y) , 0.5}, which implies thatμ_A (x ∗ y) ≥ min {t, s, 0.5} and μ_A (x ∗ y) ≤ min {m, n, 0.5} . Now, if {t, s} ≤0.5 and {m, n} ≥0.5 then, μ_A (x) ≥ t, μ_A (y) ≥ s which imply (x ∗ y) _min(t,s) ∈ μ_A and ν_A (x) ≤ m, ν_A (y) ≤ n which imply (x ∗ y) _max(m,n) ∈ ν_A, i.e. (x ∗ y) _{{min(t,s),max(m,n)}} ∈ A.Again, if {t, s} <0.5 and {m, n} >0.5, then similarly we can show that (x ∗ y) _{{min(t,s),max(m,n)}}qA. Therefore, (x ∗ y) _{{min(t,s),max(m,n)}} ∈ ∨ qA. Hence, A = (μ_A, ν_A) is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X.□

Lemma 3.6. For an IFS A = (μ_A, ν_A) of X, the following conditions are equivalent (1) x_t ∈ μ_A, y_s ∈ μ_A ⇒ (x ∗ y) _max(t,s) ∈ ∨ qμ_A, and x_m ∈ ν_A, y_n ∈ ν_A ⇒ (x ∗ y) _min(m,n) ∈ ∨ qν_A, for all t, s ∈ (0, 1], m, n ∈ [0, 1) and x, y ∈ X. (2) μ_A (x ∗ y) ≥ min {μ_A (x) , μ_A (y) , 0.5}, and ν_A (x ∗ y) ≤ max {ν_A (x) , ν_A (y) , 0.5}.

Proof: (1) ⇒ (2). Let (1) be valid. Suppose that there exist x, y ∈ X such that μ_A (x ∗ y) < min {μ_A (x) , μ_A (y) , 0.5} and ν_A (x ∗ y) > max {ν_A (x) , ν_A (y) , 0.5}. If min {μ_A (x) , μ_A (y)} <0.5 and max {ν_A (x) , ν_A (y)} >0.5, then μ_A (x ∗ y) < min {μ_A (x) , μ_A (y)} and ν_A (x∗y) > max {ν_A (x) , ν_A (y)}. Let us assume t ∈ (0, 1]and m ∈ [0, 1) such that μ_A (x ∗ y) < t < min {μ_A(x) , μ_A (y)} and ν_A (x ∗ y) > m > max {ν_A (x) , ν_A (y)}. This implies that μ_A (x) ≥ t and μ_A (y) ≥ t, and ν_A (x) ≤ m, ν_A (y) ≤ m but $(x * y)_{(t, m)} \bar{\in \lor q} A$ , which is a contradiction. Again, consider the case, if i min {μ_A (x) , μ_A (y)} ≥0.5 and max {ν_A (x) , ν_A (y)}≤0.5, then μ_A (x ∗ y) <0.5 and ν_A (x ∗ y) >0.5. Thus, μ_A (x) ≥0.5, μ_A (y) ≥0.5 and ν_A (x) ≤0.5, ν_A (y) ≤0.5 but $(x * y)_{(0.5, 0.5)} \bar{\in \lor q} A$ , which is acontradiction. Hence, μ_A (x ∗ y) ≥ min {μ_A (x) , μ_A (y) , 0.5} and ν_A (x ∗ y) ≤ max {ν_A (x) , ν_A (y) , 0.5}, for all x, y ∈ X.

(2) ⇒ (1). For x, y ∈ X, let μ_A (x) ≥ t, μ_A (y)≥s, ν_A (x) ≤ m and ν_A (y) ≤ n. Then μ_A (x∗ y) ≥min {μ_A (x) , μ_A (y) , 0.5} and ν_A (x ∗ y) ≤ max{ν_A (x) , ν_A (y) , 0.5}, so μ_A (x ∗ y) ≥ min {t, s, 0.5} and ν_A (x ∗ y) ≤ max {m, n, 0.5}. Hence, we obtain μ_A (x ∗ y) ≥ min {t, s} and ν_A (x ∗ y) ≤ max {m, n}, for min {t, s} ≤0.5 and max {m, n} ≥0.5. Also, we obtain μ_A (x ∗ y) ≥0.5 and ν_A (x ∗ y) ≤0.5, for min {t, s} >0.5 and max {m, n} <0.5. Therefore, (x ∗ y) _{min(t,s),max(m,n)} ∈ ∨ qA. □

Theorem 3.7. An IFS A = (μ_A, ν_A) is an intuitionistic (∈, ∈ ∨ q)-fuzzy BCI-subalgebras of X if and only if the set U (μ_A, ν_A ; t, m) = {x ∈ X | μ_A (x) ≥ t and ν_A (x) ≤ m} is an intuitionistic fuzzy BCI-subalgebras of X for all m ∈ [0.5, 1) and t ∈ (0.5, 1].

Proof: Assume that an IFS A = (μ_A, ν_A) is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X. Let x, y ∈ U (μ_A, ν_A ; t, m) with m ∈ [0.5, 1) and t ∈ (0.5, 1]. Then μ_A (x) ≥ t, μ_A (y) ≥ t, ν_A (x) ≤ m and ν_A (y) ≤ m. Then from Theorem 3 that μ_A (x ∗ y) ≥ min {μ_A (x) , μ_A (y) , 0.5} ≥ min {t, 0.5} = t and ν_A (x ∗ y) ≤ max {ν_A (x) , ν_A (y) , 0.5} ≤ {m, 0.5} = m, so that x ∗ y ∈ U (μ_A, ν_A ; t, m). Therefore, U (μ_A, ν_A ; t, m) is a BCI-subalgebra of X.

Conversely, let A be an IFS of X such that the set U (μ_A, ν_A ; t, m) = {x ∈ X|μ_A (x) ≥ t and ν_A (x) ≤ m} is a BCI-subalgebra of X for all m ∈ [0.5, 1) and t ∈ (0.5, 1]. If there exist x, y ∈ X such that μ_A (x ∗ y) < min {μ_A (x) , μ_A (y) , 0.5} and ν_A (x ∗ y)> max {ν_A (x) , ν_A (y) , 0.5}, then we take m ∈ (0.5, 1) and t ∈ (0.5, 1) such that μ_A (x ∗ y) < t < min{μ_A (x) , μ_A (y) , 0.5} and ν_A (x ∗ y) > m > max{ν_A (x) , ν_A (y) , 0.5}. Thus, x, y ∈ U (μ_A, ν_A ; t, m) with t < 0.5 and m > 0.5, and so x ∗ y ∈ U (μ_A, ν_A ; t, m) i.e. μ_A (x ∗ y) ≥ t and ν_A (x ∗ y) ≤ mwhich is a contradiction. Therefore, μ_A (x ∗ y)≥ min {μ_A (x) , μ_A (y) , 0.5} and ν_A (x ∗ y) ≤ max{ν_A (x) , ν_A (y) , 0.5} , for all x, y ∈ X. Using Theorem 3, we conclude that A is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X.□

Corrollary 3.8. Every intuitionistic fuzzy BCI-subalg- ebra of X is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X.

The converse of the Corollary 3.8 is not true in general as seen in the following example.

Example 3.9. Consider a BCI-algebra X = {0, a, b, c} given in the Caley table 3.4. We define IFS A = (μ_A, ν_A) : X → (0, 1] × [0, 1) by $μ_{A} (x) = {\begin{matrix} 0.8 & if x = 0 \\ 0.6 & if x = a \\ 0.7 & if x = b \\ 0.6 & if x = c, \end{matrix}$ and $ν_{A} (x) = {\begin{matrix} 0.2 & if x = 0 \\ 0.3 & if x = a \\ 0.2 & if x = b \\ 0.4 & if x = c . \end{matrix}$

It is justified that A = (μ_A, ν_A) is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X, but not an intuitionistic fuzzy BCI-subalgebras of X, because $ν_{A} (a * b) = 0.4 ≰ 0.3 = max {ν_{A} (a), ν_{A} (b)} .$

Proposition 3.10. Let A and B be two (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebras of X. Then A ∩ B = (μ_A∩B, ν_A∩B) is also an (∈, ∈ ∨ q)-intuition fuzzy BCI-subalgebra of X.

Proof: Proof of the Proposition is straight forward.□

Theorem 3.11. Let {A_i| i ∈ ∧ is an index set} be a family of (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebras of X. Then A = ∩ _i∈∧A_i is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X, where ∩_i∈∧A_i = 〈x, ⋀ _i∈∧μ_{A
_i}, ⋁ _i∈∧ν_{A
_i}〉 and ⋀_i∈∧μ_{A
_i} (x) = inf {μ_Ai (x) | i ∈ ∧ and x ∈ X} ⋁_i∈∧ν_{A
_i} (x) = sup {ν_{A
_i} (x) | i ∈ ∧ and x ∈ X}.

Proof: Let us take x, y ∈ X and t₁, t₂ ∈ (0, 1], and m₁, m₂ ∈ [0, 1) be such that x_{t
₁} ∈ μ_A and y_{t
₂} ∈ μ_A, and x_{m
₁} ∈ ν_A and y_{m
₂} ∈ ν_A. Assume that $(x * y)_{min (t_{1}, t_{2})} \bar{\in \lor q} μ$ and $(x * y)_{max (m_{1}, m_{2})} \bar{\in \lor q} ν_{A}$ . Then μ_A (x ∗ y) < min {t₁, t₂} and μ_A (x ∗ y) + min {t₁, t₂} ≤1, and ν_A (x ∗ y) > max {m₁, m₂} and ν_A (x ∗ y) + max {m₁, m₂} ≥1, which imply that

$μ_{A} (x * y) < 0.5 and ν_{A} (x * y) > 0.5$ (1)

Now, we define Δ₁ = {i ∈ ∧ | (x ∗ y) _{min{t₁,t₂}} ∈ μ_{A
_i} and (x ∗ y) _{max{m₁,m₂}} ∈ ν_{A
_i}} and Δ₂ = {[i∈ ∧ $| (x * y)_{min {t_{1}, t_{2}}} q μ_{A_{i}}} \cap {j \in \land | (x * y)_{min {t_{1}, t_{2}}} \bar{\in}$ μ_j] and [{i ∈ ∧ | (x ∗ y) _{max{m₁,m₂}}qμ_{A
_i}} ∩ {j ∈ ∧ | $(x * y)_{max {m_{1}, m_{2}}} \bar{\in} ν_{j}]} .$ Then ∧ = Δ₁ ∪ Δ₂ and Δ₁ ∩ Δ₂ = φ. If Δ₂ = φ, then (x ∗ y) _{min{t₁,t₂}} ∈ μ_{A
_i} and (x ∗ y) _{max{m₁,m₂}} ∈ ν_{A
_i} for all i∈ ∧, i.e. μ_{A
_i} (x ∗ y) ≥ min {t₁, t₂} and ν_{A
_i} (x ∗ y) ≤ max {m₁, m₂} for all i∈ ∧, which indicate μ_A (x ∗ y) ≥ min {t₁, t₂} and ν_A (x ∗ y) ≤ max {m₁, m₂}. This is a contradiction. Hence, Δ₂ ≠ φ, and so for every i ∈ Δ₂ we have μ_{A
_i} (x ∗ y) < min {t₁, t₂} and μ_{A
_i} (x ∗ y) + min {t₁, t₂} >1, and ν_{A
_i} (x ∗ y) > max {m₁, m₂} and ν_{A
_i} (x ∗ y) + max {m₁, m₂} <1. It follows that min {t₁, t₂} >0.5 and max {m₁, m₂} <0.5. Now, x_{t
₁} ∈ μ_A and x_{m
₁} ∈ ν_A implies that μ_A (x) ≥ t₁ and ν_A (x) ≤ m₁, and thus μ_{A
_i} (x)≥μ_A (x) ≥ t₁ ≥ min {t₁, t₂} >0.5 and ν_{A
_i} (x) ≤ ν_A (x) ≤ m₁ ≤ max {m₁, m₂} <0.5 for all i∈ ∧. Similarly, we get μ_{A
_i} (y) >0.5 and ν_{A
_i} (y) <0.5 for all i∈ ∧. We suppose that t = μ_{A
_i} (x ∗ y) <0.5 and m = ν_{A
_i} (x ∗ y) >0.5. Taking that t < r < 0.5 and m > n > 0.5, we get x_r ∈ μ_{A
_i} and y_r ∈ μ_{A
_i}, but $(x * y)_{min (r, r)} = (x * y)_{r} \bar{\in \lor q} μ_{A_{i}}$ and x_n ∈ ν_{A
_i} and y_n ∈ ν_{A
_i}, but $(x * y)_{max (n, n)} = (x * y)_{n} \bar{\in \lor q} ν_{A_{i}}$ . This contradicts that A = (μ_{A
_i}, ν_{A
_i}) is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X. Hence, μ_{A
_i} (x ∗ y) ≥0.5 and ν_{A
_i} (x ∗ y) ≤0.5 for all i∈ ∧, so μ_A (x ∗ y) ≥0.5 and ν_A (x ∗ y) ≤0.5 which contradicts (1). Therefore, (x ∗ y) _{min{t₁,t₂}} ∈ ∨ qμ_A and (x ∗ y) _{max{m₁,m₂}} ∈ ∨ qν_A and consequently, A = (μ_{A
_i}, ν_{A
_i}) is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X. □

Theorem 3.12. Let A = (μ_A, ν_A) be an IFS of X. Then non-empty set A_(t,m), where t ∈ (0.5, 1] and m ∈ [0.5, 1) is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X if and only if max {μ_A(x ∗ y) , 0.5} ≥ min {μ_A (x) , μ_A (y)} and min {ν_A (x ∗ y) , 0.5} ≤ max {ν_A (x) , ν_A (y)}, for all x, y ∈ X.

Proof: Suppose that A_(t,m) is a BCI-subalgebra of X. If there exist a, b ∈ X such that max {μ_A(a ∗ b) , 0.5} < t = min {μ_A (a) , μ_A (b)} and min {ν_A(a ∗ b) , 0.5} > m = max {ν_A (a) , ν_A (b)}, then t∈(0.5, 1], m ∈ [0.5, 1), so a_t ∈ μ_A, b_t ∈ μ_A anda_m ∈ ν_A, b_m ∈ ν_A but $(a * b)_{t} \bar{\in} μ_{A}$ and $(a * b)_{m} \bar{\in} ν_{A}$ . Since a, b ∈ A_(t,m) and A_(t,m) is a BCI-subalge-bra of X, but a ∗ b ∉ A_(t,m). This is a contradiction, and so for all x, y ∈ X, we have max{μ_A (x ∗ y) , 0.5} ≥ min {μ_A (x) , μ_A (y)} and min {ν_A(x ∗ y) , 0.5} ≤ max {ν_A (x) , ν_A (y)}. Conversely, sup-pose that the condition is valid. Let t ∈ (0.5, 1], m ∈ [0.5, 1) and x, y ∈ A_(t,m). Then, we have x_t ∈ μ_A, y_t ∈ μ_A and x_m ∈ ν_A, y_m ∈ ν_A, which equivalently, μ_A (x) ≥ t, μ_A (y) ≥ t and ν_A (x) ≤ m, ν_A (y) ≤ m. Hence, max {μ_A (x ∗ y) , 0.5} ≥ min {μ_A (x) , μ_A (y)} ≥ t > 0.5 and min {ν_A (x ∗ y) , 0.5} ≤ max {ν_A (x) , ν_A (y)} ≤ m < 0.5. and so, μ_A (x ∗ y) ≥ t and ν_A (x ∗ y) ≤ m, i.e. (x ∗ y) _t ∈ μ_A and (x ∗ y) _m ∈ ν_A. Therefore, x ∗ y ∈ A_(t,m). This shows that A_(t,m) is a BCI-subalgebra of X. □

For any t, m ∈ [0, 1] and fuzzy subset μ inX, denote $μ_{\hat{t}} = {x \in X | μ (x) \geq t}$ , 〈μ〉_t = {x ∈ X|x_tqμ}, [μ] _t = {x ∈ X|x_t ∈ qμ}, $\hat{ν_{m}} = {x \in X | μ (x) \leq m}$ and $\hat{[ν]_{m}} = {x \in X | x_{m} \bar{\in} \lor \bar{q} ν}$ . Clearly, $A^{(t, m)} = μ_{A_{\hat{t}}} \cap \hat{μ_{A_{m}}}$ for all t, m ∈ [0, 1].

Theorem 3.13. Let A be an IFS of a BCI-algebra X. Then (1) A is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X if and only if non-empty subset $μ_{A_{\hat{t}}}$ and $\hat{ν_{A_{m}}}$ are BCI-subalgebras of X for all t ∈ [0, 0.5) and m ∈ (0.5, 1] (2) A is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X if and only if non-empty subsets $μ_{A_{\hat{t}}}$ and $\hat{[ν]_{m}}$ are BCI-subalgebras of X for all r ∈ [0, 0.5) and s ∈ (0, 1] (3) A is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X if and only if non-empty subsets 〈μ_A〉_t and $\hat{ν_{A_{m}}}$ are BCI-subalgebras of X for all t, m ∈ (0.5, 1] (4) A is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X if and only if non-empty subsets 〈μ_A〉_tand $\hat{[ν_{A}]_{m}}$ are BCI-subalgebras of X for all t ∈ (0.5, 1] and m ∈ (0, 1] (5) A is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X if and only if non-empty subsets [μ_A] _t and $\hat{ν_{A_{m}}}$ are BCI-subalgebras of X for all t ∈ (0, 1] and m ∈ (0.5, 1] (6) A is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X if and only if non-empty subsets [μ_A] _t and $[\hat{ν}]_{m}$ are BCI-subalgebras of X for all t, m ∈ (0, 1].

Proof: We have to prove only (1) and (2). The prove of the others are similar. (1) Let A be an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X and assume that $μ_{A_{\hat{t}}} \neq φ$ and $\hat{ν_{A_{m}}} \neq φ$ for some t ∈ [0, 0.5) and m ∈ (0.5, 1]. Let $x, y \in μ_{A_{\hat{t}}}, \hat{ν_{A_{m}}}$ . Then μ_A (x) ≥ t, μ_A (y) ≥ t and ν_A (x) ≤ m, ν_A (y) ≤ m. Since A is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X, so we have μ_A (x ∗ y) ≥ min {μ_A (x) , μ_A (y) , 0.5} ≥ t and ν_A (x ∗ y) ≤ max {ν_A (x) , ν_A (y) , 0.5} ≤ m. Hence, $x * y μ_{A_{\hat{t}}}$ and $x * y \in \hat{ν_{A_{m}}}$ . Therefore, $μ_{A_{\hat{t}}}$ and $\hat{ν_{A_{m}}}$ are BCI-subalgebras of X.

Conversely, assume that the given condition hold. Let x, y ∈ X and choose t = min {μ_A (x) , μ_A (y) , 0.5} +ɛ and m = max {ν_A (x) , ν_A (y) , 0.5} + ɛ, where ɛ > 0. Then t ∈ [0, 0.5), m ∈ (0.5, 1] and $x, y \in μ_{A_{\hat{t}}}, \hat{ν_{A_{m}}}$ . Since $μ_{A_{\hat{t}}}$ and $\hat{ν_{A_{m}}}$ are BCI-subalgebras of X, we have $x * y \in μ_{A_{\hat{t}}}$ and $x * y \in \hat{ν_{A_{m}}}$ , and so μ_A (x ∗ y) > t = min {μ_A (x) , μ_A (y) , 0.5} + ɛ and ν_A (x ∗ y) < m = max {ν_A (x) , ν_A (y) , 0.5} + ɛ. Hence, μ_A (x ∗ y) ≥ min {μ_A (x) , μ_A (y) , 0.5} and ν_A (x ∗ y) ≤ max {ν_A (x) , ν_A (y) , 0.5}. Since ɛ is arbitrary. Thus, A is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X.

(2) Let A be an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X and assume that $μ_{A_{\hat{t}}} \neq φ$ and $\hat{[ν_{A}]_{m}} \neq φ$ for some t ∈ [0, 0.5] and m ∈ (0, 1]. We have to only show that $\hat{[ν_{A}]_{m}}$ is a BCI-subalgebra of X, let x, y ∈ X. Then $x_{m} \bar{\in} \lor \bar{q} ν_{A}$ and $y_{m} \bar{\in} \lor \bar{q} ν_{A}$ , which indicate that ν_A (x) < m or ν_A (x) + m ≤ 1, and ν_A (y) < m or ν_A (y) + m ≤ 1.

We follow the two cases:

Case 1: m ∈ (0, 0.5]. Then 1 - m ≥ 0.5 ≥ m. (1a) If ν_A (x) < m and ν_A (y) < m, then ν_A (x ∗ y) $\leq max {ν_{A} (x), ν_{A} (y), 0.5} = 0.5 \leq 1 - m \Rightarrow (x * y)_{m} \bar{q} ν_{A}$ . (1b) If ν_A (x) + m ≤ 1 and ν_A (y) + m ≤ 1, then $ν_{A} (x * y) \leq max {ν_{A} (x), ν_{A} (y), 0.5} \leq 1 - m \Rightarrow (x * y)_{m} \bar{q} ν_{A}$ .

Case 2: m ∈ (0.5, 1]. Then m > 0.5 > 1 - m. (2a) If ν_A (x) < m and ν_A (y) < m, then $ν_{A} (x * y) \leq max {ν_{A} (x), ν_{A} (y), 0.5} < m \Rightarrow (x * y)_{m} \bar{\in} ν_{A}$ . (2b) If ν_A (x) + m ≤ 1 and ν_A (y) + m ≤ 1, then ν_A (x ∗ y) ≤ max {ν_A (x) , ν_A (y) , 0.5} ≤ max {1 - m, $1 - m, 0.5} < m \Rightarrow (x * y)_{m} \bar{q} ν_{A}$ .

Hence, in any case $(x * y)_{m} \bar{\in} \lor \bar{q} ν_{A}$ , i.e. $x * y \in \hat{[ν_{A}]_{m}}$ . Thus, $\hat{[ν_{A}]_{m}}$ is a BCI-subalgebra of X. Converse part is obvious. Thus, A is an (∈, ∈ ∨ q)-intuitionistic BCI-subalgebra of X. □

4 Cartesian product of (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-algebras

In this section, we consider the Cartesian product of (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-algebras of BCI-algebra. Before we study the product of (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-algebras of the BCI-algebras, we first define product of intuitionistic fuzzy subsets of X.

Definition 4.1. Let A = (μ_A, ν_A) and (μ_B, ν_B) be two IFSs in a set X. The cartesian product of A and B is also a BCI-algebra defined by A × B = {(μ_A × μ_B (x, y) , ν_A × ν_B (x, y)) | x, y ∈ X} under the binary operation ∗ defined in A × B by (x₁, y₁) ∗ (x₂, y₂) = (x₁ ∗ x₂, y₁ ∗ y₂) where μ_A × μ_B : X × X → [0, 1] and ν_A × ν_B : X × X → [0, 1] defined by (μ_A × μ_B) (x, y) = min {μ_A (x) , μ_B (y)}, (ν_A × ν_B) (x, y) = max {ν_A (x) , ν_B (y)}.

Theorem 4.2. Let A and B be (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebras of X. Then A × B is also an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X.

Proof: Let A and B be two IFSs in X and (x₁, y₁) , (x₂, y₂) ∈ X × X. Then (x₁, y₁) ∗ (x₂, y₂) = (x₁ ∗ x₂, y₁ ∗ y₂). Now, ((μ_A × μ_B) ((x₁, y₁) ∗ (x₂, y₂)))= ((μ_A × μ_B) (x₁ ∗ x₂, y₁ ∗ y₂) = min {μ_A (x₁ ∗ x₂) , μ_B (y₁ ∗ y₂)} ≥ min {μ_A (x₁ ∗ x₂) , μ_B (y₁ ∗ y₂) , 0.5}≥ min {{μ_A (x₁∗ x₂) , 0.5} , {μ_B (y₁ ∗ y₂) , 0.5}} ≥min {min {μ_A (x₁) , μ_A (x₂) , 0.5} , min {μ_B (y₁) , μ_B (y₂) , 0.5}} = min {min {μ_A (x₁) , μ_B (y₁)} , min {μ_A (x₂) , μ_B(y₂)} , 0.5} = min {(μ_A × μ_B) (x₁, y₁) , (μ_A × μ_B)(x₂, y₂) , 0.5} and ((ν_A × ν_B) ((x₁, y₁) ∗ (x₂, y₂))) = ((ν_A × ν_B) (x₁ ∗ x₂, y₁ ∗ y₂) = max {ν_A (x₁ ∗ x₂) , ν_B (y₁ ∗ y₂)} ≤ max {ν_A (x₁ ∗ x₂) , ν_B (y₁ ∗ y₂) , 0.5}≤ max {{ν_A (x₁ ∗ x₂) , 0.5} , {ν_B (y₁ ∗ y₂) , 0.5}} ≤ max{max {ν_A (x₁) , ν_A (x₂) , 0.5} , max {ν_B (y₁) , ν_B (y₂) , 0.5}} = max {max {ν_A (x₁) , ν_B (y₁)} , max {ν_A (x₂) , ν_B(y₂)} , 0.5} = max {(ν_A × ν_B) (x₁, y₁) , (ν_A × ν_B) (x₂, y₂) , 0.5}. Hence, A × B is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X × X.□

The following example shows that the product of two (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebras of X is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X.

Example 4.3. Consider a BCI-algebra X = {0, a, b, c} with the Caley table given in Example 3.4. We define IFS A of X such that $\begin{matrix} X & 0 & a & b & c \\ A & 〈 0.5, 0.4 〉 & 〈 0.2, 0.2 〉 & 〈 0.5, 0.4 〉 & 〈 0.2, 0.5 〉 \end{matrix}$ and IFS B of X such that $\begin{matrix} X & 0 & a & b & c \\ B & 〈 0.3, 0.2 〉 & 〈 0.2, 0.3 〉 & 〈 0.3, 0.5 〉 & 〈 0.2, 0.0 〉 \end{matrix}$

Then, A and B are (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebras of X.

Now, for (a, b) , (b, c) ∈ X × X, we have (μ_A×μ_B) ((a, b) ∗ (b, c)) = (μ_A × μ_B) (a ∗ b, b ∗ c) = (μ_A × μ_B) (c, a) = min {μ_A (c) , μ_B (a)} =0.2, (μ_A×μ_B) (a, b) = min {μ_A (a) , μ_B (b)} =0.2, and (μ_A×μ_B) (b, c) = min {μ_A (b) , μ_B (c)} =0.2 . Therefore,(μ_A × μ_B) ((a, b) ∗ (b, c)) ≥ min {(μ_A × μ_B) (a, b) , (μ_A × μ_B) (b, c) , 0.5} hold.

Again, (ν_A × ν_B) ((a, b) ∗ (b, c)) = (ν_A × ν_B)(c, a) = max {ν_A (c) , ν_B (a)} =0.5, (ν_A × ν_B) (a, b)= max {ν_A (a) , ν_B (b)} =0.2 and (ν_A × ν_B) (b, c)= max {ν_A (b) , ν_B (c)} =0.4. Therefore, (ν_A × ν_B)((a, b) ∗ (b, c)) ≤ max {(ν_A × ν_B) (a, b) , (ν_A × ν_B)(b, c) , 0.5} hold. Hence, A × B is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X × X.

5 Images and preimages of intuitionistic fuzzy BCI-algebras

A mapping from f : X → Y of BCI-algebras is called a homomorphism if f (x ∗ y) = f (x) ∗ f (y) for all x, y ∈ X. If A = (μ_A, ν_A) and B = (μ_B, ν_B) are IFSs of X and Y respectively, then pre-image of B = (μ_B, ν_B) under f is an IFS f^-1 (B) = (μ_f^-1(B), ν_f^-1(B)) where μ_f^-1(B) (x) = μ_B (f (x)) and ν_f^-1(B) (x) = ν_B (f (x)) for any x ∈ X and image of A = (μ_A, ν_A) under f is an IFS f (A) = (μ_f(A), ν_f(A)) is defined by $μ_{f (A)} (y) = {\begin{matrix} sup_{x \in f^{- 1} (y)} {μ_{A} (x)} & if y \in f (X) \\ 0 & if y \notin f (X), \end{matrix}$ and $ν_{f (A)} (y) = {\begin{matrix} inf_{x \in f^{- 1} (y)} {ν_{A} (x)} & if y \in f (X) \\ 1 & if y \notin f (X) . \end{matrix}$

Theorem 5.1. Let f : X → Y be a homomorphism of BCI-algebras. If A = (μ_A, ν_A) is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of Y, then inverse-image f^-1 (A) of A is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X.

Proof: Since f is a homomorphism, for any x, y∈X, we have f^-1 (μ_A) (x ∗ y) = μ_A (f (x ∗ y))= μ_A (f (x) ∗ f (y)) ≥ min {μ_A (f (x)) , μ_A (f (y)) , 0.5} = min {f^-1 (μ_A (x)) , f^-1 (μ_A (y)) , 0.5} and f^-1(ν_A) (x∗ y) = ν_A (f (x ∗ y)) = ν_A (f (x) ∗ f (y)) ≤max {ν_A (f (x)) , ν_A (f (y)) , 0.5} = max {f^-1 (ν_A (x)) , f^-1 (ν_A (y)) , 0.5} . Therefore, f^-1 (A) is an (∈, ∈∨q)-intuitionistic fuzzy BCI-subalgebra of X. □

For the homomorphic image of an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebras, we have the following theorem.

Theorem 5.2. Let f be a homomorphism between BCI-algebras X and Y. For every (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra A = (μ_A, ν_A) of X, the image f (A) is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of Y.

Proof: We have to prove

$f^{- 1} (y_{1}) * f^{- 1} (y_{2}) \subseteq f^{- 1} (y_{1} * y_{2})$ (2) for all y₁, y₂ ∈ Y. If x ∈ f^-1 (y₁) ∗ f^-1 (y₂), then x = x₁ ∗ x₂ for some x₁ ∈ f^-1 (y₁) and x₂ ∈ f^-1 (y₂). Since f is a homomorphism, then f (x) = f (x₁ ∗ x₂) = f (x₁) ∗ f (x₂) = y₁ ∗ y₂, which implies that x ∈ f^-1 (y₁ ∗ y₂). Hence, (2) holds. Now, let y₁, y₂ ∈ Y be arbitrarily given. Assume that y₁ ∗ y₂ ∉ Im (f). Then for an IFS A = (μ_A, ν_A), we get f (μ_A) (y₁ ∗ y₂) =0 and f (ν_A) (y₁ ∗ y₂) =0. But, if y₁ ∗ y₂ ∉ Im (f), i.e. f^-1 (y₁ ∗ y₂) = φ, then f^-1 (y₁) = φ or f^-1 (y₂) = φ by (2). Thus f (μ_A) (y₁) =0 or f (μ_A) (y₂) =0 and f (ν_A) (y₁) =0 or f (ν_A) (y₂) =0, and so, f (μ_A) (y₁ ∗ y₂) =0 = min {f (μ_A) (y₁) , f (μ_A) (y₂) , 0.5}, f (ν_A) (y₁ ∗ y₂) =0 = max {f (ν_A) (y₁) , f (ν_A) (y₂) , 0.5}. Suppose that f^-1 (y₁ ∗ y₂) ∉ φ. Then we have considered the following cases: $f^{- 1} (y_{1}) = φ or f^{- 1} (y_{2}) = φ$ (3) $f^{- 1} (y_{1}) \notin φ and f^{- 1} (y_{2}) \notin φ$ (4)

From (1) it implies that f (μ_A) (y₁ ∗ y₂) ≥0 = min {f (μ_A) (y₁) , f (μ_A) (y₂) , 0.5} and f (ν_A) (y₁ ∗ y₂) ≤0 = max {f (ν_A) (y₁) , f (ν_A) (y₂) , 0.5} .

From case (4), we get from (2) that $\begin{matrix} f (μ_{A}) (y_{1} * y_{2}) & = & sup_{x \in f^{- 1} (y_{1} * y_{2})} μ_{A} (x) \\ \geq & sup_{x \in f^{- 1} (y_{1}) * f^{- 1} (y_{2})} μ_{A} (x) \\ = & sup_{x_{1} \in f^{- 1} (y_{1}), x_{2} \in f^{- 1} (y_{2})} μ_{A} (x_{1} * x_{2}) \end{matrix}$

It follows from definition of an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra that $\begin{matrix} f (μ_{A}) (y_{1} * y_{2}) \\ \geq & sup_{x_{1} \in f^{- 1} (y_{1}), x_{2} \in f^{- 1} (y_{2})} min {μ_{A} (x_{1}), μ_{A} (x_{2}), 0.5} \\ = & sup_{x_{1} \in f^{- 1} (y_{1})} min {sup_{x_{2} \in f^{- 1} (y_{2})} μ_{A} (x_{1}), μ_{A} (x_{2}), 0.5} \\ = & sup_{x_{1} \in f^{- 1} (y_{1})} min {μ_{A} (x_{1}), sup_{x_{2} \in f^{- 1} (y_{2})} μ_{A} (x_{2}), 0.5} \\ = & min {sup_{x_{1} \in f^{- 1} (y_{1})} μ_{A} (x_{1}), f (μ_{A}) (y_{2}), 0.5} \\ = & min {f (μ_{A}) (y_{1}), f (μ_{A}) (y_{2}), 0.5}, \end{matrix}$ and $\begin{matrix} f (ν_{A}) (y_{1} * y_{2}) \\ \leq & inf_{x_{1} \in f^{- 1} (y_{1}), x_{2} \in f^{- 1} (y_{2})} max {ν_{A} (x_{1}), ν_{A} (x_{2}), 0.5} \\ = & inf_{x_{1} \in f^{- 1} (y_{1})} max {inf_{x_{2} \in f^{- 1} (y_{2})} ν_{A} (x_{1}), ν_{A} (x_{2}), 0.5} \\ = & inf_{x_{1} \in f^{- 1} (y_{1})} max {ν_{A} (x_{1}), inf_{x_{2} \in f^{- 1} (y_{2})} ν_{A} (x_{2}), 0.5} \\ = & max {inf_{x_{1} \in f^{- 1} (y_{1})} ν_{A} (x_{1}), f (ν_{A}) (y_{2}), 0.5} \\ = & max {f (ν_{A}) (y_{1}), f (ν_{A}) (y_{2}), 0.5} . □ \end{matrix}$

6 Conclusions and future work

To investigate the structure of an algebraic system, it is clear that (intutionistic fuzzy) subalgebras with special properties play an important role. By using the combined notions of “belongingness” and “quasicoincidence” of intuitionistic fuzzy points and IFSs we introduced the notions of (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebras of a BCI-algebra. We established the Cartesian products (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebras. Finally, we investigated images and preimages of these algebras.

We believe that our results presented in this paper will give a foundation for further study the algebraic theory of BCI-algebras. In our future study of structure of BCI-algebras, the following topics will be considered and discussed. (i) to find (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-ideals of BCI-algebras, (ii) to find (∈, ∈ ∨ q)-interval-valued intuitionistic fuzzy BCI-subalgebras of BCI-algebras, (iii) to find (∈, ∈ ∨ q)-interval-valued intuitionistic fuzzy BCI-ideals of BCI-algebras, (iv) to find the relationships between these BCI-subalgebras and BCI-ideals of BCI-algebras.

Footnotes

Acknowledgments

We are very grateful to the anonymous referees for their valuable comments and suggestions that helped us to improve the paper.

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