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.
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.
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 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
A fuzzy set A in a set X is of the form
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.
(∈, ∈ ∨ 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.
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 t Φμ A , x m Φν A ), where Φ ∈ {∈, q, ∈ ∨ q, ∈ ∧ q} . To say that x t ∈ μ A (respectively, x t qμ A ) and x m ∈ ν A (respectively, x m qν 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 t qμ A (respectively, x t ∈ μ A and x t qμ A ) and x m ∈ ν A or x m qν A (respectively, x m ∈ ν A and x m qν A ). To say that means that x t Φμ A does not hold and x m Φν A does not hold, where Φ ∈ {∈, q, ∈ ∨ q, ∈ ∧ q}.
We now illustrate the above definitions by using some examples.
We define IFS A = (μ A , ν A ) : X→ (0, 1] ×[0, 1) by and
It is routine to check that A = (μ A , ν A ) is an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra of X.
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.□
(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. □
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.□
The converse of the Corollary 3.8 is not true in general as seen in the following example.
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
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∈ ∧μ j ] and [{i ∈ ∧ | (x ∗ y) max{m1,m2}qμ A i } ∩ {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 and x n ∈ ν A i and y n ∈ ν A i , but . 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. □
For any t, m ∈ [0, 1] and fuzzy subset μ inX, denote , 〈μ〉 t = {x ∈ X|x t qμ}, [μ] t = {x ∈ X|x t ∈ qμ}, and . Clearly, for all t, m ∈ [0, 1].
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 . Since and are BCI-subalgebras of X, we have and , 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 and for some t ∈ [0, 0.5] and m ∈ (0, 1]. We have to only show that is a BCI-subalgebra of X, let x, y ∈ X. Then and , 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). (1b) If ν A (x) + m ≤ 1 and ν A (y) + m ≤ 1, then .
Case 2: m ∈ (0.5, 1]. Then m > 0.5 > 1 - m. (2a) If ν A (x) < m and ν A (y) < m, then . (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, .
Hence, in any case , i.e. . Thus, is a BCI-subalgebra of X. Converse part is obvious. Thus, A is an (∈, ∈ ∨ q)-intuitionistic BCI-subalgebra of X. □
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.
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.
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.
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) = (μf-1(B), νf-1(B)) where μf-1(B) (x) = μ B (f (x)) and νf-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) = (μf(A), νf(A)) is defined by and
For the homomorphic image of an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebras, we have the following theorem.
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
It follows from definition of an (∈, ∈ ∨ q)-intuitionistic fuzzy BCI-subalgebra that
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.
