Interval-valued intuitionistic fuzzy ideals of B-algebras

Abstract

In this paper, the concepts of interval-valued fuzzy ideals in B-algebras are introduced and investigated some of their properties. The homomorphic inverse image of interval-valued intuitionistic fuzzy ideals are studied. The intersection and Cartesian product of interval-valued fuzzy ideals in B-algebras are discussed.

Keywords

B-algebra interval-valued fuzzy ideals homomorphic image intersection

1 Introduction

After the introduction of the concept of fuzzy sets by Zadeh [9], several researches were conducted on the generalizations of the notion of fuzzy sets. Interval-valued fuzzy sets provides with a flexible mathematical framework to cope with imperfect and imprecise information. Atanassov [6, 8] introduced the concept of intuitionistic fuzzy sets and the interval-valued intuitionistic fuzzy, as a generation of the notion of fuzzy sets. Interval-valued intuitionistic fuzzy subgroups are discussed [1]. Fuzzy sets and intuitionistic fuzzy sets are widely used in various algebraic systems and other other fields [3 , 12]. Cho and Kim [4] discussed further relations between B-algebras and other topics especially quasigroups. Using the notion of interval-valued fuzzy set, Saeid [2] introduced the concept of interval-valued fuzzy B-subalgebras of a B-algebra, and studied some of their properties. Senapati et al. [10] introduced fuzzy closed ideals of B-algebras and fuzzy subalgebras of B-algebras with respect to t-norm. For the general development of B-algebras, the ideal theory and subalgebras play important role. To the best of our knowledge no works are available on interval-valued intuitionistic fuzzy ideals of B-algebras. For this reason we motivated to develop these theories for B-algebras.

In this paper, interval-valued intuitionistic fuzzy ideals of B-algebras are defined and lot of properties are investigated. The remainder of this article is structured as follows: Section 2 proceeds with a recapitulation of all required definitions and properties. In Section 3, the concepts of interval-valued intuitionistic fuzzy ideals are introduced and discussed their properties in details. In Section 4, the homomorphic image and inverse image of interval-valued intuitionistic fuzzy ideals are studied. In Section 5, conclusion and scope for future research are given.

2 Preliminaries

In this section, some elementary aspects that are necessary for this paper are included. Firstly, we review some basic facts for B-algebra (see, [2 , 6, 10]).

Definition 2.1. A nonempty set X with a constant 0 and a binary operation * is said to be B-algebra if it satisfies the following axioms.

x * x = 0 for all x ∈ X,

x * 0 = x for all x ∈ X,

(x * y) * z = x * (z * (0 * y)), for all x, y, z ∈ X.

A relation ≤ on a B-algebra X by x ≤ y if and only if x * y = 0.

Let (X, * , 0) be a B-algebra. Then for all x, y ∈ X, (x * y) * (0 * y) = x, 0 * (0 * x) = x.

A non-empty subset I of a B-algebra X is said to be an ideal of X if (1) 0 ∈ I, (2)x * y, y ∈ I ⇒ x ∈ I.

We now review some fuzzy logic concepts as follows:

A fuzzy set in set X is a function μ : X → [0, 1] and the complement of μ, denoted by $\bar{μ}$ , is the fuzzy set in X given by $\bar{μ} (x) = 1 - μ (x)$ . For t ∈ [0, 1], the set U (μ ; t) = {x ∈ X|μ (x) ≥ t} is called an upper level cut set and L (μ ; t) = {x ∈ X|μ (x) ≤ t} is called a lower level cut set.

An Intuitionistic fuzzy set (briefly, IFS) A in a nonempty set X is an object having the form A = {(x, α_A (x) , β_A (x)) |x ∈ X}. where the functions α_A and β_A denote the degree of membership and the degree of non-membership, respectively, and 0 ≤ α_A (x) + β_A (x) ≤1 for all x ∈ X. For the sake of simplicity, we shall use A = (α_A, β_A) for IFSA = {(x, α_A (x) , β_A (x)) |x ∈ X}.

Zadeh defined another type of fuzzy set called interval-valued fuzzy sets [?]. The membership value of an element of this is not a single number, it is an interval and this interval is a subinterval of the interval [0, 1]. An interval-valued fuzzy set (briefly, IVFS) A in a nonempty set X is an object having the form A = {< x, M_A (x) > |x ∈ X}, where M_A : X → D [0, 1] and D [0, 1] = {[a, b] | [a, b] ⊂ [0, 1]}. Combining the idea of intuitionistic fuzzy set and interval-valued fuzzy set, Atanassov and Gargov [6, 8] defined a new class of fuzzy set called interval-valued intuitionistic fuzzy sets (briefly, IVIFS).

Definition 2.2. An interval-valued intuitionistic fuzzy set A in a non-empty set X is an object having the form A = {< x, M_A (x) , N_A (x) > |x ∈ X}, where M_A : X → D [0, 1] and N_A : X → D [0, 1]. The intervals $M_{A} (x) = [M_{A}^{L} (x), M_{A}^{U} (x)]$ and $N_{A} (x) = [N_{A}^{L} (x), N_{A}^{U} (x)]$ denote the intervals of the degree of membership and degree of non-membership of the element x, and $0 \leq M_{A}^{L} (x) + M_{A}^{U} (x) \leq 1$ . For the sake of simplicity, we shall use symbol A = (M_A, N_A).

Definition 2.3. Let D₁, D₂ ∈ D [0, 1]. If D₁ = [a₁, b₁] and D₂ = [a₂, b₂], then $\begin{matrix} r max (D_{1}, D_{2}) = D_{1} \lor^{r} D_{2}, \\ r min (D_{1}, D_{2}) = D_{1} \land^{r} D_{2} . \end{matrix}$

Where D₁ ∨ ^rD₂ = [max(a₁, a₂) , max(b₁, b₂)], D₁ ∧ ^rD₂ = [min(a₁, a₂) , min(b₁, b₂)].

Now we call D₁ ≥ D₂ if and only if a₁ ≥ a₂ and b₁ ≥ b₂. Similarly, the relations D₁ ≤ D₂ and D₁ = D₂ are defined.

Definition 2.4. A fuzzy set μ : X → [0, 1] is called a fuzzy ideal of B-algebra X if it satisfies: for all x, y, z ∈ X,

μ (0) ≥ μ (x), (2) μ (x) ≥ μ (x * y) ∧ μ (y).

Definition 2.5. A fuzzy set μ : X → [0, 1] is called a anti-fuzzy ideal of B-algebra X if it satisfies: for all x, y, z ∈ X,

μ (0) ≤ μ (x), (2) μ (x) ≤ μ (x * y) ∧ μ (y).

Definition 2.6. An IFSA = (α_A, β_A) of B-algebra X is called the intuitionistic fuzzy ideals of X if it satisfies: for all x, y, z ∈ X,

α_A (0) ≥ α_A (x), β_A (0) ≤ β_A (x).

α_A (x) ≥ α_A (x * y) ∧ α_A (y), $β_{A} (x) \leq β_{A} (x * y) \lor β_{A} (y) .$

3 Interval-valued intuitionistic fuzzy ideals

In what follows, let X denote a B-algebra unless otherwise specified.

Definition 3.1. Let A = (M_A, N_A) be an IVIFS of X. Then A is called an interval-valued intuitionistic fuzzy ideal (briefly, IVIFI) if it satisfies the following conditions:

M_A (0) ≥ M_A (x), N_A (0) ≤ N_A (x),

M_A (x) ≥ M_A (x * y) ∧ ^rM_A (y),

N_A (x) ≤ N_A (x * y) ∨ ^rN_A (y), for all x, y ∈ X.

Example 3.1. Let X = {0, a, b, c, d, e} be a set with the following Cayley table:

*	0	a	b	c	d	e
0	0	b	a	c	d	e
a	a	0	b	d	e	c
b	b	c	e	0	b	a
c	c	d	e	0	b	a
d	d	e	c	a	0	b
e	e	c	d	b	a	0

Then (X, * , 0) be a B-algebra. Define an IVIFSA in X by $\begin{matrix} 0 & a & b \\ M_{A} & [0.61, 0.64] & [0.52, 0.56] & [0.43, 0.45] \\ N_{A} & [0.22, 0.25] & [0.33, 0.38] & [0.47, 0.49] \end{matrix}$ $\begin{matrix} c & d & e \\ M_{A} & [0.43, 0.45] & [0.43, 0.45] & [0.43, 0.45] \\ N_{A} & [0.47, 0.49] & [0.47, 0.49] & [0.47, 0.49] \end{matrix}$

Then A = (M_A, N_A) is an IVIFI of X.

Example 3.2. Let X = {0, a, b, c} be a set with the following Cayley table:

*	0	a	b	c
0	0	a	b	c
a	a	0	c	b
b	b	c	0	a
c	c	d	a	0

Then (X, * , 0) be a B-algebra. Define an IVIFSA in X by $\begin{matrix} 0 & a & b & c \\ M_{A} & [0.64, 0.74] & [0.42, 0.49] & [0.42, 0.58] & [0.42, 0.58] \\ N_{A} & [0.22, 0.26] & [0.40, 0.47] & [0.35, 0.39] & [0.35, 0.39] \end{matrix}$

Then A = (M_A, N_A) is not an IVIFI of X since $\begin{matrix} M_{A} (a) & = & [0.42, 0.49] < [0.42, 0.58] \\ = & [0.42, 0.58] \land^{r} [0.42, 0.58] \\ = & M_{A} (a * b) \land^{r} M_{A} (b) . \end{matrix}$

Theorem 3.1. Let A = (M_A, N_A) is an IVIFI of X. If x ≤ y, then M_A (x) ≥ M_A (y) , N_A (x) ≤ N_A (y).

Proof. Let x, y ∈ X such that x ≤ y. Then x * y = 0, and so $\begin{matrix} M_{A} (x) & \geq & M_{A} (x * y) \land^{r} M_{A} (y) \\ = & M_{A} (0) \land^{r} M_{A} (y) = M_{A} (y), \end{matrix}$ $\begin{matrix} N_{A} (x) & \leq & N_{A} (x * y) \lor^{r} N_{A} (y) \\ = & N_{A} (0) \lor^{r} N_{A} (y) = N_{A} (y) . \end{matrix}$ by Definition 3.1.

Theorem 3.2. Let $A = ([M_{A}^{L}, M_{A}^{U}], [N_{A}^{L}, N_{A}^{U}])$ is an IVIFS of X. Then A = (M_A, N_A) is an IVIFI of X if and only if the fuzzy sets $M_{A}^{L}$ and $M_{A}^{U}$ are both fuzzy ideals of X, $N_{A}^{L}$ , $N_{A}^{U}$ are both anti-fuzzy ideals of X.

Proof. Assume that A = (M_A, N_A) is an IVIFI of X. Then, for all x, y ∈ X, $[M_{A}^{L} (0), M_{A}^{U} (0)] = M_{A} (0) \geq M_{A} (x) = [M_{A}^{L} (x), M_{A}^{U} (x)]$ , it follows that $M_{A}^{L} (0) \geq M_{A}^{L} (x)$ and $M_{A}^{U} (0) \geq M_{A}^{U} (x)$ . $\begin{matrix} M_{A} (x) & = & [M_{A}^{L} (x), M_{A}^{U} (x)] \\ \geq & M_{A} (x * y) \land^{r} M_{A} (y) \\ = & [M_{A}^{L} (x * y), M_{A}^{U} (x * y)] \\ \land^{r} [M_{A}^{L} (y), M_{A}^{U} (y)] \\ = & [M_{A}^{L} (x * y) \land M_{A}^{L} (y), M_{A}^{U} (x * y) \\ \land M_{A}^{U} (y)] \end{matrix}$

Hence $M_{A}^{L} (x) \geq M_{A}^{L} (x * y) \land M_{A}^{L} (y)$ and $M_{A}^{U} (x) \geq M_{A}^{U} (x * y) \land M_{A}^{U} (y)$ . It follows from Definition 2.4 that $M_{A}^{L}, M_{A}^{U}$ are fuzzy ideals of X.

Since A is an IVIFI of X. Then, for all x, y ∈ X, $\begin{matrix} [N_{A}^{L} (0), N_{A}^{U} (0)] & = & N_{A} (0) \leq N_{A} (x) \\ = & [N_{A}^{L} (x), N_{A}^{U} (x)], \end{matrix}$ it follows that

$N_{A}^{L} (0) \leq N_{A}^{L} (x)$ and $N_{A}^{U} (0) \leq N_{A}^{U} (x)$ . $\begin{matrix} N_{A} (x) & = & [N_{A}^{L} (x), N_{A}^{U} (x)] \\ \leq & N_{A} (x * y) \lor^{r} N_{A} (y) \\ = & [N_{A}^{L} (x * y), N_{A}^{U} (x * y)] \\ \lor^{r} [N_{A}^{L} (y), N_{A}^{U} (y)] \\ = & [N_{A}^{L} (x * y) \lor N_{A}^{L} (y), N_{A}^{U} (x * y) \\ \lor N_{A}^{U} (y)] \end{matrix}$

Hence $N_{A}^{L} (x) \leq N_{A}^{L} (x * y) \lor N_{A}^{L} (y)$ and $N_{A}^{U} (x) \leq N_{A}^{U} (x * y) \lor N_{A}^{U} (y)$ . It follows from Definition 2.5 that $N_{A}^{L}, N_{A}^{U}$ are anti-fuzzy ideals of X.

Conversely, assume that $M_{A}^{L}$ and $M_{A}^{U}$ are both fuzzy ideals of X, then for all x, y ∈ X, $\begin{matrix} M_{A}^{L} (0) & \geq & M_{A}^{L} (x), M_{A}^{U} (0) \geq M_{A}^{U} (x), \\ M_{A}^{L} (x) & \geq & M_{A}^{L} (x * y) \land M_{A}^{L} (y) \\ M_{A}^{U} (x) & \geq & M_{A}^{U} (x * y) \land M_{A}^{U} (y) . \end{matrix}$

Hence $\begin{matrix} M_{A} (0) & = & [M_{A}^{L} (0), M_{A}^{U} (0)] \\ \geq & [M_{A}^{L} (x), M_{A}^{U} (x)] = M_{A} (x), \end{matrix}$ $\begin{matrix} M_{A} (x) & = & [M_{A}^{L} (x), M_{A}^{U} (x)] \\ \geq & [M_{A}^{L} (x * y) \land M_{A}^{L} (y), M_{A}^{U} (x * y) \land M_{A}^{U} (y)] \\ = & [M_{A}^{L} (x * y), M_{A}^{U} (x * y)] \land^{r} [M_{A}^{L}, M_{A}^{U} (y)] \\ = & M_{A} (x * y) \land^{r} M_{A} (y) \end{matrix}$

Assume that $N_{A}^{L}$ and $N_{A}^{U}$ are both anti-fuzzy ideals of X, then for all x, y ∈ X, $N_{A}^{L} (0) \leq N_{A}^{L} (x)$ , $N_{AL} (x) \leq N_{A}^{L} (x * y) \lor N_{A}^{L} (y)$ and $N_{A}^{U} (0) \leq N_{A}^{U} (x)$ , $N_{A}^{U} (x) \leq N_{A}^{U} (x * y) \lor N_{A}^{U} (y)$ . Hence $N_{A} (0) = [N_{A}^{L} (0), N_{A}^{U} (0)] \leq [N_{A}^{L} (x), N_{A}^{U} (x)] = N_{A} (x)$ , and $\begin{matrix} N_{A} (x) & = & [N_{A}^{L} (x), N_{A}^{U} (x)] \\ \leq & [N_{A}^{L} (x * y) \lor N_{A}^{L} (y), N_{A}^{U} (x * y) \lor N_{A}^{U} (y)] \\ = & [N_{A}^{L} (x * y), N_{A}^{U} (x * y)] \lor^{r} [N_{A}^{L} (y), N_{A}^{U} (y)] \\ = & N_{A} (x * y) \lor^{r} N_{A} (y) \end{matrix}$

Hence A = (M_A, N_A) is an IVIFI of X.

Theorem 3.3. Let $A = ([M_{A}^{L}, M_{A}^{U}], [N_{A}^{L}, N_{A}^{U}])$ is an IVIFS of X. Then A = (M_A, N_A) is an IVIFI of X if and only if the intuitionistic fuzzy sets $(M_{A}^{L}, N_{A}^{L})$ and $(M_{A}^{U}, N_{A}^{U})$ are both intuitionistic fuzzy ideals of X.

Proof. Assume that A = (M_A, N_A) is an IVIFI of X, then, for all x ∈ X, $\begin{matrix} [M_{A}^{L} (0), M_{A}^{U} (0)] & = & M_{A} (0) \geq M_{A} (x) \\ = & [M_{A}^{L} (x), M_{A}^{U} (x)], \\ [N_{A}^{L} (0), N_{A}^{U} (0)] & = & N_{A} (0) \leq N_{A} (x) \\ = & [N_{A}^{L} (x), N_{A}^{U} (x)], \end{matrix}$ that is $M_{A}^{L} (0) \geq M_{A}^{L} (x)$ , $M_{A}^{U} (0) \geq M_{A}^{U} (x)$ and $N_{A}^{L} (0) \leq N_{A}^{L} (x)$ , $N_{A}^{U} (0) \leq N_{A}^{U} (x)$ .

For all x, y ∈ X, we have $\begin{matrix} M_{A} (x) & \geq & M_{A} (x * y) \land^{r} M_{A} (y), \\ N_{A} (x) & \leq & N_{A} (x * y) \lor^{r} N_{A} (y), \end{matrix}$

This shows that $\begin{matrix} M_{A}^{L} (x) & \geq & M_{A}^{L} (x * y) \land M_{A}^{L} (y), \\ M_{A}^{U} (x) & \geq & M_{A}^{U} (x * y) \land M_{A}^{U} (y), \\ N_{A}^{L} (x) & \leq & N_{A}^{L} (x * y) \lor N_{A}^{L} (y), \\ N_{A}^{U} (x) & \leq & N_{A}^{U} (x * y) \lor N_{A}^{U} (y) . \end{matrix}$

It follows from Definition 2.6 that $(M_{A}^{L}, N_{A}^{L})$ and $(M_{A}^{U}, N_{A}^{U})$ are both intuitionistic fuzzy idealsof X.

Conversely, assume that $(M_{A}^{L}, N_{A}^{L})$ and $(M_{A}^{U}, N_{A}^{U})$ are both intuitionistic fuzzy ideals of X. Then $M_{A}^{L} (0) \geq M_{A}^{L} (x)$ , $M_{A}^{U} (0) \geq M_{A}^{U} (x)$ and $N_{A}^{L} (0) \leq N_{A}^{L} (x)$ , $N_{A}^{U} (0) \leq N_{A}^{U} (x)$ . $\begin{matrix} M_{A}^{L} (x) & \geq & M_{A}^{L} (x * y) \land M_{A}^{L} (y), \\ M_{A}^{U} (x) & \geq & M_{A}^{U} (x * y) \land M_{A}^{U} (y), \\ N_{A}^{L} (x) & \leq & N_{A}^{L} (x * y) \lor N_{A}^{L} (y), \\ N_{A}^{U} (x) & \leq & N_{A}^{U} (x * y) \lor N_{A}^{U} (y) . \end{matrix}$

Therefore M_A (0) ≥ M_A (x), M_A (x) ≥ M_A (x * y) ∧ ^rM_A (y), N_A (0) ≤ N_A (x), N_A (x) ≤ N_A (x * y) ∨ ^rN_A (y). Hence A = (M_A, N_A) is an IVIFI of X.

Definition 3.2. Let A = (M_A, N_A) and B = (M_B, N_B) be two IVIFS of X. Then the intersection of A and B is denoted by A ∩ B and is given by $A \cap B = (M_{A \cap B}, N_{A \cap B}) = (M_{A} \land^{r} M_{B}, N_{A} \lor^{r} N_{B}) .$

Theorem 3.4. Let A = (M_A, N_A) and B = (M_B, N_B) are IVIFI of X. Then A ∩ B is also an IVIFI of X.

Proof. Suppose A = (M_A, N_A) and B = (M_B, N_B) be two IVIFS of X, then for all x ∈ X, we have M_A (0) ≥ M_A (x), N_A (0) ≤ N_A (x), M_B (0) ≥ M_B (x), N_B (0) ≤ N_B (x). Hence $\begin{matrix} M_{A \cap B} (0) & = & M_{A} (0) \land^{r} M_{B} (0) \\ \geq & M_{A} (x) \land^{r} M_{B} (x) \\ = & M_{A \cap B} (x), \end{matrix}$ $\begin{matrix} N_{A \cap B} (0) & = & N_{A} (0) \lor^{r} N_{B} (0) \\ \leq & N_{A} (x) \lor^{r} N_{B} (x) \\ = & N_{A \cap B} (x) . \end{matrix}$

By Definition 3.1, we have, for all x, y ∈ X, $\begin{matrix} M_{A} (x) & \geq & M_{A} (x * y) \land^{r} M_{A} (y), \\ M_{B} (x) & \geq & M_{B} (x * y) \land^{r} M_{B} (y) . \\ N_{A} (x) & \leq & N_{A} (x * y) \lor^{r} N_{A} (y), \\ N_{B} (x) & \leq & N_{B} (x * y) \lor^{r} N_{B} (y) . \end{matrix}$ $\begin{matrix} M_{A \cap B} (x) & = & M_{A} (x) \land^{r} M_{B} (x) \\ \geq & (M_{A} (x * y) \land^{r} M_{A} (y)) \\ \land^{r} & (M_{B} (x * y) \land^{r} M_{B} (y)) \\ = & (M_{A} (x * y) \land^{r} M_{B} (x * y)) \\ \land^{r} & (M_{A} (y) \land^{r} M_{B} (y)) \\ = & M_{A \cap B} (x * y) \land^{r} M_{A \cap B} (y) . \end{matrix}$ $\begin{matrix} N_{A \cap B} (x) & = & N_{A} (x) \lor^{r} N_{B} (x) \\ \leq & (N_{A} (x * y) \lor^{r} N_{A} (y)) \\ \lor^{r} & (N_{B} (x * y) \lor^{r} N_{B} (y)) \\ = & (N_{A} (x * y) \lor^{r} N_{B} (x * y)) \\ \lor^{r} & (N_{A} (y) \lor^{r} N_{B} (y)) \\ = & N_{A \cap B} (x * y) \lor^{r} N_{A \cap B} (y) . \end{matrix}$

Hence A ∩ B is an IVIFI of X.

Theorem 3.5. Let {A_i|i = 1, 2, ⋯} be a family of IVIFI of X. Then ∩A_i is also an IVIFI of X.

Let $A = (M_{A}, N_{A}) = ([M_{A}^{L}, M_{A}^{U}], [N_{A}^{L}, N_{A}^{U}])$ is an IVIFS of X. Define IVIFS ⊗A and ⊕A by $\otimes A = (M_{A}, {\bar{M}}_{A}), \oplus A = ({\bar{N}}_{A}, N_{A}) .$

Where ${\bar{M}}_{A} = [1 - M_{A}^{U}, 1 - M_{A}^{L}]$ and ${\bar{N}}_{A} = [1 - N_{A}^{U}, 1 - N_{A}^{L}]$ .

Theorem 3.6. Let A = (M_A, N_A) is an IVIFS of X. Then A = (M_A, N_A) is an IVIFI of X if and only if ⊗A and ⊕A are both IVIFI of X.

Proof. Note that $\begin{matrix} M_{A} (0) & \geq & M_{A} (x) \Leftrightarrow {\bar{M}}_{A} (0) \leq {\bar{M}}_{A} (x), \\ N_{A} (0) & \leq & N_{A} (x) \Leftrightarrow {\bar{N}}_{A} (0) \geq {\bar{N}}_{A} (x) \\ M_{A} (x) & \geq & M_{A} (x * y) \land^{r} M_{A} (y) \\ \Leftrightarrow & {\bar{M}}_{A} (x) \leq [1, 1] - M_{A} (x * y) \land^{r} M_{A} (y) \\ \Leftrightarrow & {\bar{M}}_{A} (x) \leq {\bar{M}}_{A} (x * y) \lor^{r} {\bar{M}}_{A} (y) . \\ N_{A} (x) & \leq & N_{A} (x * y) \lor^{r} N_{A} (y) \\ \Leftrightarrow & {\bar{N}}_{A} (x) \geq [1, 1] - N_{A} (x * y) \lor^{r} N_{A} (y) \\ \Leftrightarrow & {\bar{N}}_{A} (x) \geq {\bar{N}}_{A} (x * y) \land^{r} {\bar{N}}_{A} (y) . \end{matrix}$

Hence A = (M_A, N_A) is an IVIFI of X if and only if ⊗A and ⊕A are both IVIFI of X.

Theorem 3.7. Let A = (M_A, N_A) is an IVIFI of X. Then the sets I_M = {x ∈ X|M_A (x) = M_A (0)} and I_N = {x ∈ X|N_A (x) = N_A (0)} are ideals of X.

Proof. Clearly 0 ∈ I_M. If x * y ∈ I_M and y ∈ I_M, then M_A (x * y) = M_A (y) = M_A (0). Hence $\begin{matrix} M_{A} (0) \geq M_{A} (x) & \geq & M_{A} (x * y) \land^{r} M_{A} (y) \\ = & M_{A} (0) \land^{r} M_{A} (0) = M_{A} (0) . \end{matrix}$

So M_A (x) = M_A (0) and x ∈ I_M. Therefore I_M = {x ∈ X|M_A (x) = M_A (0)} is an ideal of X.

Clearly 0 ∈ I_N. If x * y ∈ I_N and y ∈ I_N, then N_A (x * y) = N_A (y) = N_A (0). Hence $\begin{matrix} N_{A} (0) \leq N_{A} (x) & \leq & N_{A} (x * y) \lor^{r} N_{A} (y) \\ = & N_{A} (0) \lor^{r} N_{A} (0) = N_{A} (0) . \end{matrix}$

So N_A (x) = N_A (0) and x ∈ I_N. Therefore I_N = {x ∈ X|N_A (x) = N_A (0)} is an ideal of X.

Theorem 3.8.Let I be a nonempty subset of X and A = (M_A, N_A) is an IVIFS of X defined by

M_{A} (x) : = {\begin{array}{l} α, & if $ x \in I $ \\ β, & if $ x \notin I $ \end{array} N_{A} (x) : = {\begin{array}{l} γ, & if $ x \in I $ \\ δ, & if $ x \notin I $ \end{array}

Where $α = [α_{1}, α_{2}] \in D [0, 1]$ , $β = [β_{1}, β_{2}] \in D [0, 1]$ , $γ = [γ_{1}, γ_{2}] \in D [0, 1]$ and $δ = [δ_{1}, δ_{2}] \in D [0, 1]$ . And $[α_{1}, α_{2}] \geq [β_{1}, β_{2}]$ , $[γ_{1}, γ_{2}] \leq [δ_{1}, δ_{2}]$ . Then A ia an IVIFI of X if and only if I is an ideal of X.

Proof. Suppose that A ia an IVIFI of X. Then M_A (0) ≥ M_A (0), by define of A, we have M_A (0) = [α₁, α₂]. Thus 0 ∈ I. If x * y, y ∈ X, then M_A (x * y) = M_A (y) = [α₁, α₂]. Therefore M_A (x) ≥ M_A (x * y) ∧ ^rM_A (y) ≥ [α₁, α₂] ∧ ^r [α₁, α₂] = [α₁, α₂], that is to say M_A (x) = [α₁, α₂] and x ∈ I. Hence I ia an ideal of X.

Conversely, suppose that I is an ideal of X. Let x, y ∈ X. Consider two cases:

Case (1) If x * y, y ∈ I, then x ∈ I, thus M_A (x) = M_A (x * y) = M_A (y) = [α₁, α₂] and N_A (x) = N_A (x * y) = N_A (y) = [γ₁, γ₂]. Hence M_A (x) = M_A (x * y) ∧ ^rM_A (y) and N_A (x) = N_A (x * y) ∨ ^rN_A (y).

Case (2) If x * y ∉ I or y ∉ I, then M_A (x * y) = [β₁, β₂] or M_A (y) = [β₁, β₂], thus M_A (x) ≥ [β₁, β₂] = M_A (x * y) ∧ ^rM_A (y). In the same way we have N_A (x) ≤ N_A (x * y) ∨ ^rN_A (y).

Since I is an ideal of X, then 0 ∈ I. Therefore M_A (0) = [α₁, α₂] and N_A (0) = [γ₁, γ₂]. It follows that M_A (0) ≥ M_A (x) and N_A (0) ≤ N_A (x). Hence A ia an IVIFI of X.

Definition 3.3. Let A = (M_A, N_A) be an IVIFS of X. For [s₁, s₂] ∈ D [0, 1],[t₁, t₂] ∈ D [0, 1], the set $U (M_{A}, [s_{1}, s_{2}]) = {x \in X | M_{A} (x) \geq [s_{1}, s_{2}]}$ is called upper [s₁, s₂] lever set of A and $L (N_{A}, [t_{1}, t_{2}]) = {x \in X | N_{A} (x) \leq [t_{1}, t_{2}]}$ is called lower [t₁, t₂] lever set of A.

Theorem 3.9. If A = (M_A, N_A) be an IVIFI of X, then the nonempty sets U (M_A, [s₁, s₂]) and L (N_A, [t₁, t₂]) are ideals of X.

Proof. Since U (M_A, [s₁, s₂]) is nonempty, there exists x₀ ∈ U (M_A, [s₁, s₂]) and A = (M_A, N_A) be an IVIFI of X, then M_A (0) ≥ M_A (x₀) ≥ [s₁, s₂]. That is 0 ∈ U (M_A, [s₁, s₂]). Let x * y, y ∈ U (M_A, [s₁, s₂]). Then M_A (x * y) ≥ [s₁, s₂] and M_A (y) ≥ [s₁, s₂]. It follows that M_A (x) ≥ M_A (x * y) ∧ ^rM_A (y) ≥ [s₁, s₂], so that x ∈ U (M_A, [s₁, s₂]). Hence U (M_A, [s₁, s₂]) is ideal of X.

Since L (N_A, [t₁, t₂]) is nonempty, then there exists y₀ ∈ L (N_A, [t₁, t₂]) and A = (M_A, N_A) be an IVIFI of X, then N_A (0) ≤ N_A (y₀) ≤ [t₁, t₂]. That is 0 ∈ L (N_A, [t₁, t₂]). Let x * y, y ∈ L (N_A, [t₁, t₂]). Then N_A (x * y) ≤ [t₁, t₂] and N_A (y) ≤ [t₁, t₂]. It follows that N_A (x) ≤ N_A (x * y) ∨ ^rN_A (y) ≤ [t₁, t₂], so that x ∈ L (N_A, [t₁, t₂]). Hence L (N_A, [t₁, t₂]) is ideal of X.

Theorem 3.10. Let A = (M_A, N_A) be an IVIFS of X such that the sets U (M_A, [s₁, s₂]) and L (N_A, [t₁, t₂]) are ideals of X for every [s₁, s₂] ∈ D [0, 1],[t₁, t₂] ∈ D [0, 1]. Then A = (M_A, N_A) be an IVIFIof X.

Proof. Let U (M_A, [s₁, s₂]) and L (N_A, [t₁, t₂]) are ideals of X for every [s₁, s₂] ∈ D [0, 1],[t₁, t₂] ∈ D [0, 1].

If there exists a ∈ X such that M_A (0) < M_A (a), then taking [m₁, m₂] = (M_A (0) + M_A (a))/2, we have M_A (0) < [m₁, m₂] < M_A (a). It follows that a ∈ U (M_A, [m₁, m₂]), but 0 ∉ U (M_A, [m₁, m₂]), that is, U (M_A, [m₁, m₂]) is not an ideal of X. This is a contradiction. If there exists b ∈ X such that N_A (0) > N_A (b), then taking [n₁, n₂] = (N_A (0) + N_A (b))/2, we have N_A (0) > [n₁, n₂] > N_A (b). It follows that b ∈ L (N_A, [n₁, n₂]), but 0 ∉ L (N_A, [n₁, n₂]), that is, L (N_A, [n₁, n₂]) is not an ideal of X. This is a contradiction. Suppose there exists x₀, y₀ ∈ X such that M_A (x₀) < M_A (x₀ * y₀) ∧ ^rM_A (y₀), then taking [p₁, p₂] = (M_A (x₀) + M_A (x₀ * y₀) ∧ ^rM_A (y₀))/2, we have M_A (x₀) < [p₁, p₂] < M_A (x₀ * y₀) ∧ ^rM_A (y₀), it follows that x₀ * y₀ ∈ U (M_A, [p₁, p₂]), y₀ ∈ U (M_A, [p₁, p₂]), but x₀ ∉ U (M_A, [p₁, p₂]), so that U (M_A, [p₁, p₂]) is not an ideal of X. This is a contradiction. Finally, Suppose there exists x₁, y₁ ∈ X such that N_A (x₁) > N_A (x₁ * y₁) ∨ ^rN_A (y₁), then taking [q₁, q₂] = (N_A (x₁) + N_A (x₁ * y₁) ∨ ^rN_A (y₁))/2, we have N_A (x₁) > [q₁, q₂] > N_A (x₁ * y₁) ∨ ^rN_A (y₁), it follows that x₁ * y₁ ∈ L (N_A, [q₁, q₂]), y₁ ∈ L (N_A, [q₁, q₂]), but x_q ∉ L (N_A, [q₁, q₂]), so that L (N_A, [q₁, q₂]) is not an ideal of X. This is a contradiction. Therefore A = (M_A, N_A) be an IVIFI of X.

4 Direct product and homomorphic imageof IVIFI

In this section, direct product and homomorphism of IVIFI of B-algebra is defined and some results are studied.

Definition 4.1. Let A = (M_A, N_A) and B = (M_B, N_B) be two IVIFS of X. The direct product of A and B is defined by for all x, y ∈ X, $(A \times B) (x, y) = (M_{A \times B} (x, y), N_{A \times B} (x, y)),$ where $\begin{matrix} M_{A \times B} (x, y) = M_{A} (x) \land^{r} M_{B} (y), \\ N_{A \times B} (x, y) = N_{A} (x) \lor^{r} N_{B} (y) . \end{matrix}$

Theorem 4.1. Let A = (M_A, N_A) and B = (M_B, N_B) be two IVIFI of B-algebras X and Y, respectively. Then the direct product A × B of A and B is IVITI of X × Y.

Proof. Suppose A = (M_A, N_A) and B = (M_B, N_B) be two IVIFI of B-algebras X and Y, respectively, then for all (x, y) , (a, b) ∈ X × Y, we have, for all (x, y) ∈ X × Y, $\begin{matrix} M_{A \times B} (0, 0) & = & M_{A} (0) \land^{r} M_{B} (0) \\ \geq & M_{A} (x) \land^{r} M_{B} (y) \\ = & M_{A \times B} (x, y) . \\ N_{A \times B} (0, 0) & = & N_{A} (0) \lor^{r} N_{B} (0) \\ \leq & N_{A} (x) \lor^{r} N_{B} (y) \\ = & N_{A \times B} (x, y) . \end{matrix}$

Let (x₁, y₁) , (x₂, y₂) ∈ X × Y. Then $\begin{matrix} M_{A \times B} (x_{1}, y_{1}) & = & M_{A} (x_{1}) \land^{r} M_{B} (y_{1}) \\ \geq & (M_{A} (x_{1} * x_{2}) \land^{r} M_{A} (x_{2})) \\ \land^{r} & (M_{B} (y_{1} * y_{2}) \land^{r} M_{B} (y_{2})) \\ = & (M_{A} (x_{1} * x_{2}) \land^{r} M_{B} (y_{1} * y_{2})) \\ \land^{r} & (M_{A} (x_{2}) \land^{r} M_{B} (y_{2})) \\ = & M_{A \times B} (x_{1} * x_{2}, y_{1} * y_{2})) \\ \land^{r} & M_{A \times B} (x_{2}, y_{2}) \\ = & M_{A \times B} ((x_{1}, y_{1}) * (x_{2}, y_{2})) \\ \land^{r} & M_{A \times B} (x_{2}, y_{2}) . \\ N_{A \times B} (x_{1}, y_{1}) & = & N_{A} (x_{1}) \lor^{r} N_{B} (y_{1}) \\ \leq & (N_{A} (x_{1} * x_{2}) \lor^{r} N_{A} (x_{2})) \\ \lor^{r} & (N_{B} (y_{1} * y_{2}) \lor^{r} N_{B} (y_{2})) \\ = & (N_{A} (x_{1} * x_{2}) \lor^{r} N_{B} (y_{1} * y_{2})) \\ \lor^{r} & (N_{A} (x_{2}) \lor^{r} N_{B} (y_{2})) \\ = & N_{A \times B} (x_{1} * x_{2}, y_{1} * y_{2})) \\ \lor^{r} & N_{A \times B} (x_{2}, y_{2}) \\ = & N_{A \times B} ((x_{1}, y_{1}) * (x_{2}, y_{2})) \\ \lor^{r} & N_{A \times B} (x_{2}, y_{2}) . \end{matrix}$

Hence the direct product A × B of A and B is IVITI of X × Y.

Let f be a mapping from a B-algebra X into a B-algebra Y. Let B is an IVIFS in Y. Then the inverse image of B, is defined as f^-1 (B) = (f^-1 (M_B) , f^-1 (N_B)) with membership function and non-membership function respectively are given by f^-1 (M_B) (x) = M_B (f (x)) and f^-1 (N_B) (x) = N_B (f (x)).

A mapping f : X → Y is called a homomorphism if f (x * y) = f (x) * f (y), for all x, y ∈ X. Let X and Y be two B-algebras. Then f (0) =0′.

Theorem 4.2. Let X and Y be two B-algebras and f : X → Y be a homomorphism. If B = (M_B, N_B) is an IVIFI in Y, then the preimage f^-1 (B) = (f^-1 (M_B) , f^-1 (N_B)) of B under f is an IVIFI of X.

Proof. Suppose B = (M_B, N_B) is an IVIFI in Y. Let x₁, x₂ ∈ X. Then there exists y₁, y₂ ∈ X such that f (x₁) = y₁, f (x₂) = y₂. Hence $\begin{matrix} f^{- 1} (M_{B}) (0) & = & M_{B} (f (0)) = M_{B} (0^{'}) \\ \geq & M_{B} (y) = M_{B} (f (x)) \\ = & f^{- 1} (M_{B}) (x) . \\ f^{- 1} (N_{B}) (0) & = & N_{B} (f (0)) = N_{B} (0^{'}) \\ \leq & N_{B} (y) = M_{B} (f (x)) \\ = & f^{- 1} (M_{B}) (x) . \\ f^{- 1} (M_{B}) (x_{1}) & = & M_{B} (f (x_{1})) = M_{B} (y_{1}) \\ \geq & M_{B} (y_{1} * y_{2}) \land^{r} M_{B} (y_{2}) \\ = & M_{B} (f (x_{1}) * f (x_{2})) \land^{r} M_{B} (f (x_{2})) \\ = & M_{B} (f (x_{1} * x_{2})) \land^{r} M_{B} (f (x_{2})) \\ = & f^{- 1} (M_{B}) (x_{1} * x_{2}) \land^{r} f^{- 1} (M_{B}) (x_{2}) . \\ f^{- 1} (N_{B}) (x_{1}) & = & N_{B} (f (x_{1})) = N_{B} (y_{1}) \\ \leq & N_{B} (y_{1} * y_{2}) \lor^{r} N_{B} (y_{2}) \\ = & N_{B} (f (x_{1}) * f (x_{2})) \lor^{r} N_{B} (f (x_{2})) \\ = & N_{B} (f (x_{1} * x_{2})) \lor^{r} N_{B} (f (x_{2})) \\ = & f^{- 1} (N_{B}) (x_{1} * x_{2}) \lor^{r} f^{- 1} (N_{B}) (x_{2}) . \end{matrix}$

Hence the preimage f^-1 (B) = (f^-1 (M_B) , f^-1 (N_B)) of B under f is an IVIFI of X.

5 Conclusion

In the present paper, the notions of interval value intuitionistic fuzzy ideals of B-algebras has been introduced and some important properties of it are also studied. We have shown that the Cartesian product of any two interval value intuitionistic fuzzy ideals is an interval value intuitionistic fuzzy ideals. In our opinion, these definitions and main results can be similarly extended to some other algebraic systems.

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