Atanassov’s intuitionistic fuzzy bi-normed KU -ideals of a KU -algebra

Abstract

In this paper, by using the t-norm T and t-conorm S, we introduce the intuitionistic fuzzy bi-normed KU-ideals of a KU-algebra. Some properties of intuitionistic fuzzy bi-normed KU-ideals of a KU-algebra under the homomorphism are discussed. The direct product and the (T, S)-product of intuitionistic fuzzy bi-normed KU-ideals are particularly investigated. Some results obtained in this paper can be regarded as extended and generalized results recently given by S.M. Mostafa et al. concerning the intuitionistic fuzzy KU-ideals in KU-algebras.

Keywords

homomorphism upper(lower)-level cut

1 Introduction

The concept of triangular norms was first initiated in the theory of probabilistic metric spaces by K. Menger [14]. It turns out that the t-norms and its relatedt-conorms are crucial operations in fuzzy sets and other fuzzy structures, for instance, in fuzzy logics, fuzzy sub-hypergroups and fuzzy sub-semihypergroups, see [5–8 , 20]. In recent years, a systematic study concerning the properties and the related aspects of t-norms has been extensively discussed by E. P. Klement et al. [12, 13].

We observe that C. Prabpayak and U. Leerawat [18] introduced a new algebraic structure which is called a KU-algebra in 2009. They considered the homomorphisms of KU-algebras and investigated some of their related properties in [19]. In addition, S. M. Mostafa et al. [15] introduced the fuzzy KU-ideals of a KU-algebra. M. Akram et al. [2] and N. Yaqoob et al. [32] also introduced the cubic KU-subalgebras and KU-ideals of a KU-algebra. They discussed the relationship between a cubic KU-subalgebra and a cubic KU-ideal. M. Gulistan et al. [10] applied the soft set theory to KU-algebra. Moreover, G. Muhiuddin[17] applied the bipolar-valued fuzzy set theory to KU-algebras, and introduced the bipolar fuzzy KU-subalgebr and bipolar fuzzy KU-ideals of a KU-algebra. He considered the specifications of a bipolar fuzzy KU-subalgebra, a bipolar fuzzy KU-ideal of KU-algebras and discussed the relationship between a bipolar fuzzy KU-subalgebra and a bipolar fuzzy KU-ideal and provided conditions for a bipolar fuzzy KU-subalgebra to be a bipolar fuzzy KU-ideal. G. Gulistan et al. [9] further studied the (α, β)-fuzzy KU-ideals in KU-algebras and discussed some special properties. M. Akram et al. [1] introduced the notion of interval-valued $(\tilde{θ}, \tilde{δ})$ -fuzzy KU-ideals of KU-algebras and obtained some related properties. Recently, S. M. Mostafa et al. [16] have introduced the intuitionistic fuzzy KU-ideals in KU-algebras. T. Senapati [29, 30] introduced the notion of fuzzy KU-subalgebras and KU-ideals of KU-algebras with respect to a givent-norm and obtained some of their properties. T. Senapati [31] also introduced the intuitionistic fuzzy bi-normed, that is, the intuitionistic fuzzy (T, S)-normed KU-subalgebras of KU-algebras. Furthermore, T. Senapati studied the BCK/BCI-algebras in [21, 22], G-algebras [23], B-algebras [24, 25] and BG-algebras [4 , 26–28] which are closely related to KU-algebras.

In this paper, we introduce the new concepts of (imaginable) triangular norm and triangular conorm of the KU-ideals of a KU-algebra. Thus, most of the results recently given by S. M. Mostafa et al. (see [16]) are extended and generalized to instituitionistic fuzzy (T, S)-normed KU-ideals. We also consider and discuss the direct product and the (T, S)-product of intuitionistic fuzzy bi-normed KU-ideals. These results are new and are differ from the corresponding results given in [16]. In Section 2, some basic definitions and properties are given. In Section 3, we give the concepts and operations of intuitionistic fuzzy bi-normed KU-ideals of a KU-algebra and discussed their properties. In Section 4, the properties of intuitionistic fuzzy bi-normed fuzzy KU-ideals under homomorphisms will be investigated. In Section 5, the direct product and (T, S)-product of intuitionistic fruzzy bi-normed ((T, S)-normed) KU-ideals of a KU-algebra are introduced. Finally, in Section 6, we propose some possible research of this topics. Throughout this paper, when we mention the intuitionistic fuzzy bi-normed KU-ideals, we always mean the intuitionistic fuzzy (T, S)-normedKU-ideals.

2 Preliminaries

In this section, some elementary aspects that are necessary for the main part of the paper are included.

Definition 2.1. [18] By a KU-algebra we mean an algebra (X, ∗ , 0) of type (2, 0) with a single binary operation ∗ that satisfies the following axioms: for any x, y, z ∈ X,

(x ∗ y) ∗ ((y ∗ z) ∗ (x ∗ z)) =0,

x ∗ 0 =0,

0 ∗ x = x,

x ∗ y = 0 = y ∗ x implies x = y.

In what follows, we use (X, ∗ , 0) to denote a KU-algebra unless otherwise specified. For the sake of brevity, we call X a KU-algebra. We now define a partial ordering on the KU-algebra “≤” by x ≤ y if and only if y ∗ x = 0.

Definition 2.2. [15] In a KU-algebra, the following axioms are true: for any x, y, z ∈ X,

z ∗ z = 0,

z ∗ (x ∗ z) =0,

x ≤ y imply y ∗ z ≤ x ∗ z,

z ∗ (y ∗ x) = y ∗ (z ∗ x),

y ∗ ((y ∗ x) ∗ x) =0 .

A non-empty subset S of a KU-algebra X is called a KU-subalgebra [18] of X if x ∗ y ∈ S, for all x, y ∈ S. From the above definition, we observe that if a subset S of a KU-algebra satisfies only the above property then S becomes a KU-subalgebra.

Let (X, ∗ , 0) and (Y, ∗ ′, 0′) be KU-algebras. A homomorphism is a mapping f : X → Y satisfying f (x ∗ y) = f (x) ∗ ′f (y), for all x, y ∈ X.

Let X be the collection of objects denoted by x then a fuzzy set [33] A in X is defined as A = {< x, α_A (x) > : x ∈ X} where α_A (x) is called the membership value of x in A and 0 ≤ α_A (x) ≤1. For any fuzzy sets A and B of a set X, we define A ∩ B = min {α_A (x) , α_B (x)}for all x ∈ X .

By a triangular norm (briefly t-norm) T [14], we mean a binary operation on the unit interval [0, 1] which is commutative, associative, monotone and has 1 as its neutral element, i.e., it is a function T : [0, 1] ² → [0, 1] such that for all x, y, z ∈ [0, 1]: (i) T (x, y) = T (y, x); (ii) T (x, T (y, z)) = T (T (x, y) , z); (iii) T (x, y) ≤ T (x, z) if y ≤ z; (iv) T (x, 1) = x.

Some example of the t-norms are the minimum T_M (x, y) = min(x, y), the product T_P (x, y) = x . y and the Lukasiewicz t-norm T_L (x, y) = max(x + y - 1, 0) for all x, y ∈ [0, 1].

By a triangular conorm (t-conorm for short) S [20], we mean a binary operation on the unit interval [0, 1] which is commutative, associative, monotone and has 0 as neutral element, i.e., it is a function S : [0, 1] ² → [0, 1] such that for all x, y, z ∈ [0, 1]: (i) S (x, y) = S (y, x); (ii) S (x, S (y, z)) = S (S (x, y) , z); (iii) S (x, y) ≤ S (x, z) if y ≤ z; (iv) S (x, 0) = x.

Some example of the t-conorms are the maximum S_M (x, y) = max(x, y), the probabilistic sum S_P (x, y) = x + y - x . y and the Lukasiewicz t-conorm or (bounded sum) S_L (x, y) = min(x + y, 1) for all x, y ∈ [0, 1]. Also, it is well known [11, 12] that if T is a t-norm and S is a t-conorm, then T (x, y) ≤ min {x, y} and S (x, y) ≥ max {x, y} for all x, y ∈ [0, 1] , respectively.

We now give the following definitions:

Definition 2.3. Let P be a t-norm. Denote by △_P the set of elements x ∈ [0, 1] such that P (x, x) = x, that is, △_P = {x | x ∈ [0, 1] & P (x, x) = x} .

A fuzzy set A in X is said to satisfy the imaginable property with respect to P if Im (α_A) ⊆ △ _P.

Definition 2.4. [29, 30] A fuzzy set A in X is said to be a T-fuzzy KU-subalgebra of X if it satisfies the following condition: (T1) α_A (x ∗ y) ≥ T {α_A (x) , α_A (y)} for all x, y ∈ X. A fuzzy set A in X is said to be a T-fuzzy KU-ideal of X if it satisfies (T2) α_A (0) ≥ α_A (x) and (T3) α_A (x ∗ z) ≥ T {α_A (x ∗ (y ∗ z)) , α_A (y)} forall x, y, z ∈ X.

Definition 2.5. [3] The intuitionistic fuzzy sets defined on a non-empty set X as objects having the form A = {〈x, α_A (x) , β_A (x) 〉 : x ∈ X}, where the functions α_A (x) : X → [0, 1] and β_A (x) : X → [0, 1]. Now, we denote the degree of membership and the degree of non-membership of each element x ∈ X to the set A respectively, and 0 ≤ α_A (x) + β_A (x) ≤1 for all x ∈ X. Obviously, when β_A (x) =1 - α_A (x) for every x ∈ X, the set A becomes a fuzzy set. For the sake of simplicity, we shall use the symbol A = (α_A, β_A) for the intuitionistic fuzzy subset A = {〈x, α_A (x) , β_A (x) 〉 : x ∈ X}.

Definition 2.6. [31] An intuitionistic fuzzy set A = (α_A, β_A) in X is called an intuitionistic bi-normed (thai is, an (T, S)-normed) fuzzy KU-subalgebra of X if it satisfies the following conditions for all x, y ∈ X:

α_A (x ∗ y) ≥ T {α_A (x) , α_A (y)},

β_A (x ∗ y) ≤ S {β_A (x) , β_A (y)}.

3 Intuitionistic fuzzy bi-normed KU-ideals

In this section, the inituitionistic fuzzy bi-normed, that is, (T, S)-normed, KU-ideals of KU-algebras are firstly defined and introduced. Some Properties of intuitionistic fuzzy bi-normed KU-ideals are investigated and given in this section. In what follows, we simply use X to denote a KU-algebra unless otherwisespecified.

Definition 3.1. [31] An intuitionistic fuzzy set A = (α_A, β_A) in X is called an intuitionistic fuzzy bi-normed KU-ideal of X if it satisfies the following conditions for alll x, y, z ∈ X:

α_A (0) ≥ α_A (x) and β_A (0) ≤ β_A (x),

α_A (x ∗ z) ≥ T {α_A (x ∗ (y ∗ z)) , α_A (y)},

β_A (x ∗ z) ≤ S {β_A (x ∗ (y ∗ z)) , β_A (y)}.

We now illustrate the above definitions by using some examples.

Example 3.2. Let X={0, a, b, c, d} be a KU-algebra with the following Cayley table: $\begin{matrix} * & 0 a b c d \\ 0 & 0 a b c d \\ a & 0 0 0 0 a \\ b & 0 c 0 c d \\ c & 0 a b 0 a \\ d & 0 0 0 0 0 \end{matrix}$

Let T_m, S_m : [0, 1] × [0, 1] → [0, 1] be functions defined by T_m (x, y) = max(x + y - 1, 0) and S_m (x, y) = min(x + y, 1) for all x, y ∈ [0, 1]. Then T_m is a t-norm and S_m is a s-norm. Define an IFS A = (α_A, β_A) in X by $α_{A} (x) = {\begin{matrix} 0.9, & if x = 0 \\ 0.7, & if x = a, c \\ 0.5, & if x = b, d \end{matrix}$ and $β_{A} (x) = {\begin{matrix} 0.1, & if x = 0 \\ 0.3, & if x = a, c \\ 0.4, & if x = b, d . \end{matrix}$

Then A is clearly a bi-normed KU-ideal. Hence, A is an an intuitionistic fuzzy (T_m, S_m)-normed KU-ideal of X.

Definition 3.3. An intuitionistic fuzzy bi-normed KU-ideal A = (α_A, β_A) is called an intuitionistic fuzzy imaginable bi-normed KU-ideal of X if α_A and β_A satisfy the imaginable property with respect to T and S respectively.

Example 3.4. Let T_m be a t-norm, S_m a s-norm and X={0, a, b, c} be a KU-algebra in Example 3.2. Define an IFS A = (α_A, β_A) in X by α_A (0) =1, β_A (0) =0, α_A (x) =0 and β_A (x) =1 if x ∈ {a, b, c, d}. Then, it is easy to verify that α_A (x ∗ y) ≥ T_m {α_A (x) , α_A (y)} and β_A (x ∗ y) ≤ S_m {β_A (x) , β_A (y)} for all x, y ∈ X. Also, we have Im (α_A) ⊆ △ _{T
_m} and Im (β_A) ⊆ △ _{S
_m}. Hence, A is an intuitionistic fuzzy imaginable (T_m, S_m)-normed KU-ideal of X.

The sets {x | x ∈ X & α_A (x) = α_A (0)} and {x | x ∈ X & β_A (x) = β_A (0)} are denoted by I_{α_A} and I_{β_A}, respectively. The above two sets are clearly the KU-ideal of the KU-algebra X.

Theorem 3.5. Let A = (α_A, β_A) be an intuitionistic fuzzy bi-normed KU-ideal of X. Then the sets I_{α_A} and I_{β_A} are KU-ideals of X.

Proof. Let A be an intuitionistic fuzzy bi-normed KU-ideal of X. Then it is obvious that 0 ∈ I_{α_A}, I_{β_A}. Let x, y, z ∈ X be such that x ∗ (y ∗ z) ∈ I_{α_A}, I_{β_A} and y ∈ I_{α_A}, I_{β_A}. If α_A (x ∗ (y ∗ z)) = α_A (0) = α_A (y) and β_A (x ∗ (y ∗ z)) = β_A (0) = β_A (y), then we have α_A (x ∗ z) ≥ T {α_A (x ∗ (y ∗ z)) , α_A (y)} ≥ α_A (0) and β_A (x ∗ z) ≤ S {β_A (x ∗ (y ∗ z)) , β_A (y)} ≤ β_A (0). Since A is an intuitionistic fuzzy bi-normed KU-ideal of X, α_A (x ∗ z) = α_A (0) and β_A (x ∗ z) = β_A (0) i.e., x ∗ z ∈ I_{α_A}, I_{β_A}. Hence, I_{α_A} and I_{β_A} are indeed KU-idealsof X. □

Definition 3.6. [3] Let A = (α_A, β_A) and B = (α_B, β_B) be two IFSs on X. Then the intersection of A and B is denoted by A ∩ B and is given by A ∩ B = {min(α_A, α_B) , max(β_A, β_B)}.

The intersection of any two intuitionistic fuzzy bi-normed KU-ideals is also an intuitionistic fuzzy bi-normed KU-ideal which is proved in the following theorem.

Theorem 3.7. Let A₁ and A₂ be two intuitionistic fuzzy bi-normed KU-ideals of X. Then A₁ ∩ A₂ is an intuitionistic fuzzy bi-normed KU-ideal of X .

Proof. In proving the above theorem, we first let A₁ and A₂ be any two intuitionistic fuzzy bi-normed KU-ideals of X. Let x, y, z ∈ A₁ ∩ A₂. Then x, y, z ∈ A₁ and A₂. Now, α_A₁∩A₂ (x ∗ z) = min {α_{A
₁} (x ∗ z) , α_{A
₂} (x ∗ z)} ≥ min {T {α_{A
₁} (x ∗ (y ∗ z)) , α_{A
₁} (y)} , T {α_{A
₂} (x ∗ (y ∗ z)),α_{A
₂} (y)}} ≥ T {min {α_{A
₁} (x ∗ (y ∗ z)) , α_{A
₂} (x ∗ (y ∗ z))},min {α_{A
₁} (y) , α_{A
₂} (y)}} = T {α_A₁∩A₂ (x ∗ (y ∗ z)) , α_A₁∩A₂(y)} and β_A₁∩A₂ (x ∗ z) = max {β_{A
₁} (x ∗ z) , β_{A
₂} (x ∗ z)} ≤ max {T {β_{A
₁} (x ∗ (y ∗ z)) , β_{A
₁} (y)} , T {β_{A
₂} (x ∗ (y ∗ z)) , β_{A
₂} (y)}} ≤ T {max {β_{A
₁} (x ∗ (y ∗ z)) , β_{A
₂} (x ∗ (y ∗ z))} , max {β_{A
₁} (y) , β_{A
₂} (y)}} = T {β_A₁∩A₂ (x ∗ (y ∗ z)) , β_A₁∩A₂ (y)} . This shows that A₁ ∩ A₂ is an intuitionistic fuzzy bi-normed KU-ideal of X.□

The above theorem can be further generalized as the following form:

Theorem 3.8. Let {A_i : i = 1, 2, 3, 4, …} be a family of intuitionistic fuzzy bi-normed KU-ideals of a (KU-algebra X. Then ⋂A_i is an intuitionistic fuzzzy bi-normed KU-ideal of X, where ⋂A_i = (min α_{A
_i} (x) , max β_{A
_i} (x)).

Proposition 3.9. Let A be an intuitionistic fuzzy bi-normed KU-ideal of X and x, y, z ∈ X. Then the following statments hold:

α_A (x ∗ (x ∗ y)) ≥ α_A (y) and β_A (x ∗ (x ∗ y)) ≤ β_A (y).

If x ≤ y then α_A (x) ≥ α_A (y) and β_A (x) ≤ β_A (y).

If x ∗ y ≤ z then α_A (y) ≥ T {α_A (x) , α_A (z)} and β_A (y) ≤ S {β_A (x) , β_A (z)}.

Proof. (i) Taking z = x ∗ y in (TS4), (TS5) andusing (TS3) and x ∗ 0 = x, we get α_A (x ∗ (x ∗ y))≥T {α_A (x ∗ (y ∗ (x ∗ y))) , α_A (y)} = T {α_A (x ∗ (x ∗ (y ∗ y))) , α_A (y)} = T {α_A (x ∗ (x ∗ 0)) , α_A (y)} = T {α_A (0) , α_A (y)} = α_A (y) and β_A (x ∗ (x ∗ y)) ≤ S {β_A (x ∗ (y ∗ (x ∗ y))) , β_A (y)} = S {β_A (x ∗ (x ∗ (y ∗ y))) , β_A (y)} = S {β_A (x ∗ (x ∗ 0)) , β_A (y)} = S {β_A (0) , β_A (y)} = β_A (y).

(ii) Let x, y ∈ X be such that x ≤ y. Then y ∗ x = 0. This implies α_A (x) = α_A (0 ∗ x) ≥ T {α_A (0 ∗ (y ∗ x)) , α_A (y)} = T {α_A (0 ∗0) , α_A (y)} = T {α_A (0) , α_A (y)} ≥ α_A (y) and β_A (x) = β_A (0 ∗ x) ≤ S {β_A (0 ∗ (y ∗ x)) , β_A (y)} = S {β_A (0 ∗0) , β_A (y)} = S {β_A (0) , β_A (y)} ≤ β_A (y).

(iii) Let x, y, z ∈ X be such that x ∗ y ≤ z. Then z ∗ (x ∗ y) =0.

From (TS4), we deduce the following inequalities: α_A (z ∗ y) ≥ T {α_A (z ∗ (x ∗ y)) , α_A (x)}, if we put z = 0 then α_A (0 ∗ y) = α_A (y) ≥ T {α_A (0 ∗ (x ∗ y)),α_A (x)} = T {α_A (x ∗ y) , α_A (x)} ≥ T {T {α_A (x ∗ (z ∗ y)),α_A (z)} , α_A (x)} = T {T {α_A (z ∗ (x ∗ y)) , α_A (z)} , α_A (x)}= T {T {α_A (0) , α_A (z)}, α_A (x)} = T {α_A (z) , α_A (x)} .

Again, from (TS5) , we have the following inequalities: β_A (z ∗ y) ≤ S {β_A (z ∗ (x ∗ y)) , β_A (x)}, if we put z = 0 then β_A (0 ∗ y) = β_A (y) ≤ S {β_A (0 ∗ (x ∗ y)) , β_A (x)} = S {β_A (x ∗ y) , β_A (x)} ≤ S {S {β_A (x ∗ (z ∗ y)) , β_A (z)} , β_A (x)} = S {S {β_A (z ∗ (x ∗ y)) , β_A (z)} , β_A (x)} = S {S {β_A (0) , β_A (z)} , β_A (x)} = S {β_A (z) , β_A (x)} .

Therefore, α_A (y) ≥ T {α_A (x) , α_A (z)} and β_A (y) ≤ {β_A (x) , β_A (z)}, for all x, y, z ∈ X. □

We now state the following theorem.

Theorem 3.10. An IFS A = (α_A, β_A) is an intuitionistic fuzzy bi-normed KU-ideal of X if and only if the fuzzy sets A₁ = {x | x ∈ A & α_A (x)} and $A_{2} = {x | x \in A & {\bar{β}}_{A} (x)}$ are either empty or T-fuzzy KU-ideals of X.

Proof. Let A be an intuitionistic fuzzy bi-normed KU-ideal of the KU-algebra X. Then, it is clear that A₁ is a T-fuzzy KU-ideal of X. For every x, y, z ∈ X, we have ${\bar{β}}_{A} (0) = 1 - β_{A} (0) \geq 1 - β_{A} (x) = {\bar{β}}_{A} (x)$ and ${\bar{β}}_{A} (x * z) = 1 - β_{A} (x * z) \geq 1 - S {β_{A} (x * (y * z)), β_{A} (y)} = T {1 - β_{A} (x * (y * z)), 1 - β_{A} (y)} = T {{\bar{β}}_{A} (x * (y * z)), {\bar{β}}_{A} (y)}$ . Hence, A₂ is a T-fuzzy KU-ideal of X.

Conversely, assume that A₁ and A₂ are T-fuzzyKU-ideals of X. Then α_A (0) ≥ α_A (x) and β_A (0)≤β_A (x) for all x ∈ X. For every x, y, z ∈ X,α_A (x ∗ z) ≥ T {α_A (x ∗ (y ∗ z)) , α_A (y)} and 1 - β_A $(x * z) = {\bar{β}}_{A} (x * z) \geq T {{\bar{β}}_{A} (x * (y * z)), {\bar{β}}_{A} (y)} = T {1 - β_{A} (x * (y * z)), 1 - β_{A} (y)} = 1 - S {β_{A} (x * (y * z)) β_{A} (y)}$ . That is, β_A (x ∗ z) ≤ S {β_A (x ∗ (y ∗ z)) , β_A (y)}. Hence, we deduce that A = (α_A, β_A) is an intuitionistic fuzzy bi-normed KU-ideal of X. □

The two operators oplus A and ⨂A on IFS as follows:

Definition 3.11. [3] Let A = (α_A, β_A) be an IFS defined on X. The operators oplus A and ⨂A are defined as $oplus A = (α_{A}, {\bar{α}}_{A})$ and $⨂ A = ({\bar{β}}_{A}, β_{A})$ respectively.

Theorem 3.12. If A = (α_A, β_A) is an intuitionistic fuzzy bi-normed fuzzy KU-ideal of X, then (i) oplus A, and (ii) ⨂A, both are intuitionistic fuzzy bi-normedKU-ideals.

Proof. Straightforward. □

Let A = (α_A, β_A) be an intuitionistic fuzzy KU-ideal of X. For $\tilde{s}, \tilde{t} \in [0, 1]$ , the set $U (α_{A} : \tilde{s}) = {x | x \in X & α_{A} (x) \geq \tilde{s}}$ is called the upper $\tilde{s}$ -level of A and $L (β_{A} : \tilde{t}) = {x | x \in X & β_{A} (x) \leq \tilde{t}}$ is called the lower $\tilde{t}$ -level of A.

Theorem 3.13. Let A = (α_A, β_A) be an IFS in X. Then A is an intuitionistic fuzzy bi-normed KU-ideal of X if and oly if for all $\tilde{s}, \tilde{t} \in [0, 1]$ , the upper level set $U (α_{A} : \tilde{s})$ and the lower level set $L (β_{A} : \tilde{t})$ are either empty or KU-ideals of X.

Proof. Assume that A is an intuitionistic fuzzy bi-normed KU-ideal of X and x, y, z ∈ X. Let $\tilde{s} \in [0, 1]$ be such that $U (α_{A} : \tilde{s}) \neq φ$ and $x \in U (α_{A} : \tilde{s})$ . Then $α_{A} (x) \geq \tilde{s}$ . By (TS4) we get $α_{A} (0) = α_{A} (y * 0) \geq T {α_{A} (y * (x * 0)), α_{A} (x)} = T {α_{A} (y * 0), α_{A} (x)} = T {α_{A} (0), α_{A} (x)} \geq \tilde{s} .$ Thus, $0 \in U (α_{A} : \tilde{s})$ . Now letting $x * (y * z), y \in U (α_{A} : \tilde{s})$ , This implies that $α_{A} (x * z) \geq T {α_{A} (x * (y * z)), α_{A} (y)} \geq \tilde{s} .$ Therefore, $x * z \in U (α_{A} : \tilde{s})$ . Hence, $U (α_{A} : \tilde{s})$ is a KU-ideal of X.

Again, let $\tilde{t} \in [0, 1]$ be such that $L (β_{A} : \tilde{t}) \neq φ$ and $x \in L (β_{A} : \tilde{t})$ , then $β_{A} (x) \leq \tilde{t}$ . Then, by (TS5) we get the following equalities: $β_{A} (0) = β_{A} (y * 0) \leq S {β_{A} (y * (x * 0)), β_{A} (x)} = S {β_{A} (y * 0), β_{A} (x)} = S {β_{A} (0), β_{A} (x)} \leq \tilde{t} .$ Thus, $0 \in L (β_{A} : \tilde{t})$ . Now letting $x * (y * z), y \in L (β_{A} : \tilde{t})$ , This implies that $β_{A} (x * z) \leq S {β_{A} (x * (y * z)), β_{A} (y)} \leq \tilde{t} .$ Therefore, $x * z \in L (β_{A} : \tilde{t})$ . Hence, $L (β_{A} : \tilde{t})$ is a KU-ideal of X.

Conversely, suppose that for every $\tilde{s}, \tilde{t} \in [0, 1]$ , $U (α_{A} : \tilde{s})$ and $L (β_{A} : \tilde{t})$ are KU-ideals of X. Then, in the contrary, we consider the followings:

Let x₀, y₀, z₀ ∈ X be such that α_A (x₀ ∗ z₀) < T {α_A (x₀ ∗ (y₀ ∗ z₀)) , α_A (y₀)} and β_A (x₀ ∗ z₀) > S {β_A (x₀ ∗ (y₀ ∗ z₀)) , β_A (y₀)} . Assume that ${\tilde{s}}_{0} = \frac{1}{2} [α_{A} (x_{0} * z_{0}) + T {α_{A} (x_{0} * (y_{0} * z_{0})), α_{A} (y_{0})}]$ and ${\tilde{t}}_{0} = \frac{1}{2} [β_{A} (x_{0} * z_{0}) + S {β_{A} (x_{0} * (y_{0} * z_{0})), β_{A} (y_{0})}] .$

Then, we have $α_{A} (x_{0} * z_{0}) < {\tilde{s}}_{0} \leq T {α_{A} (x_{0} * (y_{0} * z_{0})), α_{A} (y_{0})} \leq min {α_{A} (x_{0} * (y_{0} * z_{0})), α_{A} (y_{0})}$ and so $x_{0} * z_{0} \notin U (α_{A} : \tilde{s})$ but $x_{0} * (y_{0} * z_{0}), y_{0} \in U (α_{A} : \tilde{s})$ . Also, we have $β_{A} (x_{0} * z_{0}) > {\tilde{t}}_{0} \geq S {β_{A} (x_{0} * (y_{0} * z_{0})), β_{A} (y_{0})} \geq max {β_{A} (x_{0} * (y_{0} * z_{0})), β_{A} (y_{0})}$ and so $x_{0} * z_{0} \notin L (β_{A} : \tilde{t})$ but $x_{0} * (y_{0} * z_{0}), y_{0} \in L (β_{A} : \tilde{t})$ . This is a contradiction and hence we see that α_A and β_A satisfies (TS4) and (TS5) respectively. Therefore A forms an intuitionistic fuzzy bi-normed KU-ideal of X. □

We now state a characterization theorem of intuitionistic fuzzy bi-normed KU-ideals of a KU-algebara X.

Theorem 3.14. If every intuitionistic fuzzy bi-normed KU-ideal A of X has the finite image, then every descending chain of KU-ideals of X terminates at finite number of steps.

Proof. Suppose that there exists a strictly descending chain S₀⊋S₁⊋S₂⋯ of KU-ideals of X which does not terminates at finite number of steps. Define an IFS A in X by $α_{A} (x) = {\begin{matrix} \frac{n}{n + 1} & if x \in S_{n} \ S_{n + 1} \\ 1 & if x \in \cap_{n = 0}^{\infty} S_{n} \end{matrix}$ and $β_{A} (x) = {\begin{matrix} \frac{1}{n + 1} & if x \in S_{n} \ S_{n + 1} \\ 0 & if x \in \cap_{n = 0}^{\infty} S_{n} \end{matrix}$ where n = 0, 1, 2, … and S₀ stands for X. Since $0 \in \cap_{n = 0}^{\infty} S_{n}$ , α_A (0) =1 ≥ α_A (x) and β_A (0) =0 ≤ β_A (x) for all x ∈ X. Let x, y, z ∈ X. Assume that x ∗ (y ∗ z) ∈ S_n \ S_n+1 and y ∈ S_k \ S_k+1 for n = 0, 1, 2, …; k = 0, 1, 2, …. Without loss of generality, we may assume that n ≤ k. Then obviously, we have x ∗ (y ∗ z) and y ∈ S_n, so x ∗ z ∈ S_n because S_n is a KU-ideal of X. Hence, $\begin{matrix} α_{A} (x * z) & \geq & \frac{n}{n + 1} = min {α_{A} (x * (y * z)), α_{A} (y)} \\ \geq T {α_{A} (x * (y * z)), α_{A} (y)} \\ β_{A} (x * z) & \leq & \frac{1}{n + 1} = max {β_{A} (x * (y * z)), β_{A} (y)} \\ \leq S {β_{A} (x * (y * z)), β_{A} (y)} . \end{matrix}$

If $x * (y * z), y \in \cap_{n = 0}^{\infty} S_{n}$ , then $x * z \in \cap_{n = 0}^{\infty} S_{n}$ . Thus we deduce that $\begin{matrix} α_{A} (x * z) & = & 1 = min {α_{A} (x * (y * z)), α_{A} (y)} \\ \geq T {α_{A} (x * (y * z)), α_{A} (y)} \\ β_{A} (x * z) & = & 0 = max {β_{A} (x * (y * z)), β_{A} (y)} \\ \leq S {β_{A} (x * (y * z)), β_{A} (y)} . \end{matrix}$

If $x * (y * z) \notin \cap_{n = 0}^{\infty} S_{n}$ and $y \in \cap_{n = 0}^{\infty} S_{n}$ , then there exists a positive integer r such that x ∗ (y ∗ z) ∈ S_r \ S_r+1. It follows that x ∗ z ∈ S_r so that $\begin{matrix} α_{A} (x * z) & \geq & \frac{r}{r + 1} = min {α_{A} (x * (y * z)), α_{A} (y)} \\ \geq T {α_{A} (x * (y * z)), α_{A} (y)} \\ β_{A} (x * z) & \leq & \frac{1}{r + 1} = max {β_{A} (x * (y * z)), β_{A} (y)} \\ \leq S {β_{A} (x * (y * z)), β_{A} (y)} . \end{matrix}$

Finally suppose that $x * (y * z) \in \cap_{n = 0}^{\infty} S_{n}$ and $y \notin \cap_{n = 0}^{\infty} S_{n}$ . Then y ∈ S_s \ S_s+1 for some positive integer s. It follows that x ∗ z ∈ S_s, and hence $\begin{matrix} α_{A} (x * z) \\ \geq \frac{s}{s + 1} = min {α_{A} (x * (y * z)), α_{A} (y)} \\ \geq T {α_{A} (x * (y * z)), α_{A} (y)} \\ β_{A} (x * z) \\ \leq \frac{1}{s + 1} = max {β_{A} (x * (y * z)), β_{A} (y)} \\ \leq S {β_{A} (x * (y * z)), β_{A} (y)} . \end{matrix}$

Thus, we have proved that A is a T-fuzzy KU-ideal with an infinite number of different values, which is a contradiction. This completes the proof. □

4 Images and preimages of intuitionistic fuzzy bi-normed KU-ideals

Let f be a mapping from the set X into the set Y and B be an IFS in Y. Then the inverse image of B, is defined as f^-1 (B) = (f^-1 (α_B) , f^-1 (β_B)) in X with the membership function and non-membership function respectively are given by f^-1 (α_B) (x) = α_B (f (x)) and f^-1 (β_B) (x) = β_B (f (x)). It can be shown that f^-1 (B) is an IFS

We prove in below the following Proposition.

Theorem 4.1. An onto homomorphic preimage of an intuitionistic fuzzy bi-normed KU-ideal is also an intuitionistic fuzzy bi-normed KU-ideal.

Proof. Let f : X → Y be an onto homomorphism of KU-algebras, B be an intuitionistic fuzzy bi-normed KU-ideal of Y, and f^-1 (B) the preimage of B under f. Then for all x ∈ X, f^-1 (α_B) (0) = α_B (f (0)) ≥ α_B (f (x)) = f^-1 (α_B) (x) and f^-1 (β_B) (0) = β_B (f (0))≤β_B (f (x)) = f^-1 (β_B) (x). Now let x, y, z ∈ X, then f^-1 (α_B) (x ∗ z) = α_B (f (x ∗ z)) = α_B (f (x) ∗ f (z))≥T {α_B (f (x) ∗ (f (y) ∗ f (z))) , α_B (f (y))} = T {α_B(f (x ∗ (y ∗ z))) , α_B (f (y))} = T {f^-1 (α_B) (x ∗ (y ∗ z)),f^-1 (α_B) (y)} and f^-1 (β_B) (x ∗ z) = β_B (f (x ∗ z)) =β_B (f (x) ∗ f (z)) ≤ S {β_B (f (x) ∗ (f (y) ∗ f (z))),β_B (f (y))} = S {β_B (f (x ∗ (y ∗ z))) , β_B (f (y))} = S {f^-1 (β_B) (x ∗ (y ∗ z)) , f^-1 (β_B) (y)} . This completes the proof.□

Definition 4.2. Let f be a mapping from the set X to the set Y. If A = (α_A, β_A) is an IFS in X and B is the image of A, then B is given by $α_{B} (y) = {\begin{matrix} sup_{x \in f^{- 1} (y)} α_{A} (x), \\ if f^{- 1} (y) = {x \in X, f (x) = y} \neq φ \\ 0, otherwise \end{matrix}$ and $β_{B} (y) = {\begin{matrix} inf_{x \in f^{- 1} (y)} β_{A} (x), \\ if f^{- 1} (y) = {x \in X, f (x) = y} \neq φ \\ 0, otherwise . \end{matrix}$

Definition 4.3. An IFS A in the KU-algebra X is said to have the sup-property and inf-property if for any subset T ⊆ X there exist t₀ ∈ T such that $α_{A} (t_{0}) = sup_{t_{0} \in T} α_{A} (t)$ and $β_{A} (t_{0}) = inf_{t_{0} \in T} β_{A} (t)$ respectively.

For the homomorphic image of an intuitionistic fuzzy bi-normed KU-ideals, we have the following theorem.

Theorem 4.4. Let f : X → Y be an onto homomorphism of KU-algebras. If A is an intuitionistic fuzzy bi-normed KU-ideal of X, then the image B of A under f is an intuitionistic fuzzy bi-normed KU-ideal of Y.

Proof. Let f : X → Y be an onto homomorphism of KU-algebras, A be an intuitionistic fuzzy bi-normed KU-ideal X with supremum and infimum properties, and B be the image of A under f. Since A is an intuitionistic fuzzy bi-normed KU-ideal of X, we have α_A (0) ≥ α_A (x) and β_A (0) ≤ β_A (x), for all x ∈ X.

Note that 0 ∈ f^-1 (0′) where 0, 0′ are the zero of X and Y respectively. Thus, $α_{B} (0^{'}) = sup_{t \in f^{- 1} (0^{'})} α_{A} (t) = α_{A} (0) \geq α_{A} (x)$ and $β_{B} (0^{'}) = inf_{t \in f^{- 1} (0^{'})} β_{A} (t) = β_{A} (0) \leq β_{A} (x)$ which implies that $α_{B} (0^{'}) \geq sup_{t \in f^{- 1} (x^{'})}$ α_A (t) = α_B (x′) and $β_{B} (0^{'}) \leq inf_{t \in f^{- 1} (x^{'})} β_{A} (t) = β_{B} (x^{'})$ .

For any x′, y′, z′ ∈ Y, let x₀ ∈ f^-1 (x′), y₀ ∈ f^-1 (y′) and z₀ ∈ f^-1 (z′) be such that $\begin{matrix} α_{A} (x_{0} * y_{0}) & = & sup_{t \in f^{- 1} (x^{'} * z^{'})} α_{A} (t), \\ β_{A} (x_{0} * y_{0}) & = & inf_{t \in f^{- 1} (x^{'} * z^{'})} β_{A} (t), \\ α_{A} (y_{0}) & = & sup_{t \in f^{- 1} (y^{'})} α_{A} (t), \\ β_{A} (y_{0}) & = & sup_{t \in f^{- 1} (y^{'})} β_{A} (t), \end{matrix}$ $\begin{matrix} α_{A} (x_{0} * (y_{0} * z_{0})) = α_{B} {f (x_{0} * (y_{0} * z_{0}))} \\ = α_{B} (x^{'} * (y^{'} * z^{'})) \\ = sup_{(x_{0} * (y_{0} * z_{0})) \in f^{- 1} (x^{'} * (y^{'} * z^{'}))} α_{A} (x_{0} * (y_{0} * z_{0})) \\ = sup_{t \in f^{- 1} (x^{'} * (y^{'} * z^{'}))} α_{A} (t) \end{matrix}$ and $\begin{matrix} β_{A} (x_{0} * (y_{0} * z_{0})) = β_{B} {f (x_{0} * (y_{0} * z_{0}))} \\ = β_{B} (x^{'} * (y^{'} * z^{'})) \\ = inf_{(x_{0} * (y_{0} * z_{0})) \in f^{- 1} (x^{'} * (y^{'} * z^{'}))} β_{A} (x_{0} * (y_{0} * z_{0})) \\ = inf_{t \in f^{- 1} (x^{'} * (y^{'} * z^{'}))} β_{A} (t) . \end{matrix}$

Then $\begin{matrix} α_{A} (x^{'} * z^{'}) = sup_{t \in f^{- 1} (x^{'} * z^{'})} α_{A} (t) = α_{A} (x_{0} * z_{0}) \\ \geq T {α_{A} (x_{0} * (y_{0} * z_{0})), α_{A} (y_{0})} \\ = T {sup_{t \in f^{- 1} (x^{'} * (y^{'} * z^{'}))} α_{A} (t), sup_{t \in f^{- 1} (y^{'})} α_{A} (t)} \\ = T {α_{A} (x^{'} * (y^{'} * z^{'})), α_{A} (y^{'})} \end{matrix}$ and $\begin{matrix} β_{A} (x^{'} * z^{'}) = inf_{t \in f^{- 1} (x^{'} * z^{'})} β_{A} (t) = β_{A} (x_{0} * z_{0}) \\ \leq S {β_{A} (x_{0} * (y_{0} * z_{0})), β_{A} (y_{0})} \\ = S {inf_{t \in f^{- 1} (x^{'} * (y^{'} * z^{'}))} β_{A} (t), inf_{t \in f^{- 1} (y^{'})} β_{A} (t)} \\ = S {β_{A} (x^{'} * (y^{'} * z^{'})), β_{A} (y^{'})} . \end{matrix}$

Thus, B is is an intuitionistic fuzzy bi-normed KU-ideal of Y.□

5 Product of intuitionistic fuzzy bi-normed KU-ideals

In this section, we consider the direct product and the bi-normed product of intuitionistic fuzzy KU-ideals of the KU-algebra X with respect to the derived t-conorms. Several properties of such products are investigated and studied. Before we study the product of intuitionistic fuzzy KU-ideals of the KU-algebras, we first define some kind of product of intuitionistic fuzzy subsets of X. The results in this section are new results.

Definition 5.1. Let A₁ = (α_{A
₁}, β_{A
₁}) and A₂ = (α_{A
₂}, β_{A
₂}) be two intuitionistic fuzzy subsets of a KU-algebra X. Then the (T, S)-product of A₁ and A₂, denoted by $[A_{1} \cdot A_{2}]_{(T, S)} = {{[α_{A_{1}} \cdot α_{A_{2}}]}_{T}, {[β_{A_{1}} \cdot β_{A_{2}}]}_{S}}$ and is defined by $\begin{matrix} {[α_{A_{1}} \cdot α_{A_{2}}]}_{T} (x) & = & T (α_{A_{1}} (x), α_{A_{2}} (x)), \\ {[β_{A_{1}} \cdot β_{A_{2}}]}_{S} (x) & = & S (β_{A_{1}} (x), β_{A_{2}} (x)), \end{matrix}$ for all x ∈ X.

Definition 5.2. Let A₁ and A₂ be two IFSs of X. Then the (T, S)-product of A₁ and A₂, [A₁ . A₂] _T is called an intuitionistic fuzzy bi-normed KU-ideal of X if for all x, y, z ∈ X it satisfies

[α_A
₁ . α_A
₂] _T (0) ≥ [α_A
₁ . α_A
₂] _T (x) and [β_A
₁ . β_A
₂] _S (0) ≤ [β_A
₁ . β_A
₂] _S (x),

[α_A
₁ . α_A
₂] _T (x ∗ z) ≥ T {[α_A
₁ . α_A
₂] _T (x ∗ (y ∗ z)) , [α_A
₁ . α_A
₂] _T (y)},

[β_A
₁ . β_A
₂] _S (x ∗ z) ≤ S {[β_A
₁ . β_A
₂] _S (x ∗ (y ∗ z)) , [β_A
₁ . β_A
₂] _S (y)}.

Theorem 5.3. Let A₁ and A₂ be two intuitionistic fuzzy bi-normed KU-idealsl of a KU-algebra X. If T^* is a t-norm and S^* is a t-conorm which dominates T and S respectively, i.e., $\begin{matrix} T^{*} (T (a, b), T (c, d)) & \geq & T (T^{*} (a, c), T^{*} (b, d)) \\ and S^{*} (S (a, b), S (c, d)) & \leq & S (S^{*} (a, c), S^{*} (b, d)) \end{matrix}$ for all a, b, c and d ∈ [0, 1], Then the (T^*, S^*)-product of A₁ and A₂, [A₁ · A₂] _(T^*,S^*) is an intuitionistic fuzzy bi-normed KU-ideal of X.

Proof. For any x, y, z ∈ X, we have

[α_{A
₁} . α_{A
₂}] _T (0) = T {α_{A
₁} (0) , α_{A
₂} (0)} ≥ T {α_{A
₁} (x) , α_{A
₂} (x)} = [α_{A
₁} . α_{A
₂}] _T (x), [β_{A
₁} . β_{A
₂}] _S (0) = S {β_{A
₁} (0) , β_{A
₂} (0)} ≤ S {β_{A
₁} (x) , β_{A
₂} (x)} = [β_{A
₁} . β_{A
₂}] _S (x) and $\begin{matrix} {[α_{A_{1}} . α_{A_{2}}]}_{T^{*}} (x * z) = T^{*} {α_{A_{1}} (x * z), α_{A_{2}} (x * z)} \\ \geq T^{*} {T {α_{A_{1}} (x * (y * z)), α_{A_{1}} (y)}, \\ T {α_{A_{2}} (x * (y * z)), α_{A_{2}} (y)}} \\ \geq T (T^{*} (α_{A_{1}} (x * (y * z)), α_{A_{2}} (x * (y * z))), \\ T^{*} (α_{A_{1}} (y), α_{A_{2}} (y))) \\ = T {{[α_{A_{1}} . α_{A_{2}}]}_{T^{*}} (x * (y * z)), {[α_{A_{1}} . α_{A_{2}}]}_{T^{*}} (y)}, \end{matrix}$

$\begin{matrix} {[β_{A_{1}} . β_{A_{2}}]}_{S^{*}} (x * z) = S^{*} {β_{A_{1}} (x * z), β_{A_{2}} (x * z)} \\ \leq S^{*} {S {β_{A_{1}} (x * (y * z)), β_{A_{1}} (y)}, \\ S {β_{A_{2}} (x * (y * z)), β_{A_{2}} (y)}} \\ \leq S (S^{*} (β_{A_{1}} (x * (y * z)), β_{A_{2}} (x * (y * z))), \\ S^{*} (β_{A_{1}} (y), β_{A_{2}} (y))) \\ = S {{[β_{A_{1}} . β_{A_{2}}]}_{S^{*}} (x * (y * z)), {[β_{A_{1}} . β_{A_{2}}]}_{S^{*}} (y)} . \end{matrix}$

Hence, [A₁ · A₂] _(T^*,S^*) is an intuitionistic fuzzy bi-normed KU-ideal of X.□

Let f : X → Y be an epimorphism of KU-algebras. Let T, T^* be t-norms and S, S^* be t-conorms such that T^*, S^* dominates T and S respectively. If A₁ and A₂ be two intuitionistic fuzzy bi-normed KU-ideal of Y, then the (T^*, S^*)-product of A₁ and A₂, [A₁ · A₂] _(T^*,S^*) is an intuitionistic fuzzy bi-normed KU-ideal of Y. Since every epimorphic preimage of an intuitionistic fuzzy bi-normed KU-ideal is an intuitionistic fuzzy bi-normed KU-ideal, the preimages f^-1 (A₁), f^-1 (A₂) and f^-1 ([A₁ · A₂] _(T^*,S^*)) are the T-fuzzy KU-ideals of X. The next theorem state the relatioship between the f^-1 ([A₁ · A₂] _(T^*,S^*)) and the (T^*, S^*)-product [f^-1 (A₁) · f^-1 (A₂)] _(T^*,S^*) of f^-1 (A₁) and f^-1 (A₂).

Theorem 5.4. Let f : X → Y be an epimorphism of KU-algebras. Let T, T^* be the t-norms and S, S^* the t-conorms such that T^*, S^* dominates T and S, respectively. Let A₁ and A₂ be two intuitionistic fuzzy bi-normed KU-ideasl of Y. If [A₁ · A₂] _(T^*,S^*) is the (T^*, S^*)-product of A₁ and A₂ and [f^-1 (A₁) · f^-1 (A₂)] _(T^*,S^*) = {f^-1 ([α_{A
₁} · α_{A
₂}] _{T
^*}) , f^-1 ([β_{A
₁} · β_{A
₂}] _{S
^*})} is the (T^*, S^*)-product of f^-1 (A₁) and f^-1 (A₂). Then we have $f^{- 1} ([α_{A_{1}} \cdot α_{A_{2}}]_{T^{*}}) = [f^{- 1} (α_{A_{1}}) \cdot f^{- 1} (α_{A_{2}})]_{T^{*}}$ and $f^{- 1} ([β_{A_{1}} \cdot β_{A_{2}}]_{S^{*}}) = [f^{- 1} (β_{A_{1}}) \cdot f^{- 1} (β_{A_{2}})]_{S^{*}} .$

Proof. For any x ∈ X, we derive the following equalities: f^-1 ([α_{A
₁} · α_{A
₂}] _{T
^*}) (x) = [α_{A
₁} · α_{A
₂}] _{T
^*} (f (x)) = T^* (α_{A
₁} (f (x)) , α_{A
₂} (f (x))) = T^* ([f^-1 (α_{A
₁})] f (x) , [f^-1 (α_{A
₂})] f (x)) = [f^-1 (α_{A
₁}) · f^-1 (α_{A
₂})] _{T
^*} (x) and f^-1 ([β_{A
₁} · β_{A
₂}] _{S
^*}) (x) = [β_{A
₁} · β_{A
₂}] _{S
^*} (f (x)) = S^* (β_{A
₁} (f (x)) , β_{A
₂} (f (x))) = S^* ([f^-1 (β_{A
₁})] f (x) , [f^-1 (β_{A
₂})] f (x)) = [f^-1 (β_{A
₁}) · f^-1 (β_{A
₂})] _{S
^*} (x) . □

Lemma 5.5. [11] Let T and S be a t-norm and a t-conorm respectively. Then we obtain the following equalities: $\begin{matrix} T (T (x, y), T (z, t)) & = & T (T (x, z), T (y, t)), \\ S (S (x, y), S (z, t)) & = & S (S (x, z), S (y, t)) \end{matrix}$ for all x, y, z and t ∈ [0, 1].

Definition 5.6. Let X₁ × X₂ be the cartesian product of KU-algebras X₁ and X₂. If A₁ = (α_{A
₁}, β_{A
₁}) and A₂ = (α_{A
₂}, β_{A
₂}) be two IFSs of X₁ and X₂, respectively, then the Cartesian product of A₁ and A₂ denoted by A = (α_A, β_A), is defined by α_A = α_{A
₁} × α_{A
₂} and β_A = β_{A
₁} × β_{A
₂} such that α_A (x₁, x₂) = (α_{A
₁} × α_{A
₂}) (x₁, x₂) = T (α_{A
₁} (x₁) , α_{A
₂} (x₂)) and β_A (x₁, x₂) = (β_{A
₁} × β_{A
₂}) (x₁, x₂) = S (β_{A
₁} (x₁) , β_{A
₂} (x₂)) for all (x₁, x₂) ∈ X₁ × X₂.

Remark 5.7. Let X and Y be two KU-algebras. Then, we define ∗ on X × Y by (x, y) ∗ (z, t) = (x ∗ z, y ∗ t) for all (x, y), (z, t) ∈X × Y. Clearly, (X × Y, ∗ , (0, 0)) is a KU-algebra.

Definition 5.8. An IFS A₁ × A₂ of X₁ × X₂ is called an intuitionistic fuzzy bi-normed KU-ideal of X₁ × X₂ if for all (x₁, y₁) , (x₂, y₂) and (x₃, y₃) ∈ X₁ × X₂ it satisfies

(α_A
₁ × α_A
₂) (0, 0) ≥ (α_A
₁ × α_A
₂) (x, y) and (β_A
₁ × β_A
₂) (0, 0) ≤ (β_A
₁ × β_A
₂) (x, y),

(α_A
₁ × α_A
₂) ((x₁, y₁) ∗ (x₃, y₃)) ≥ T {(α_A
₁ × α_A
₂) ((x₁, y₁) ∗ ((x₂, y₂) ∗ (x₃, y₃))) , (α_A
₁ × α_A
₂) (x₂, y₂)},

(β_A
₁ × β_A
₂) ((x₁, y₁) ∗ (x₃, y₃)) ≤ S {(β_A
₁ × β_A
₂) ((x₁, y₁) ∗ ((x₂, y₂) ∗ (x₃, y₃))) , (β_A
₁ × β_A
₂) (x₂, y₂)}.

Theorem 5.9. Let A₁ and A₂ be two intuitionistic fuzzy bi-normed KU-ideals of X₁ and X₂, respectively. Then the Cartesian product A₁ × A₂ is an intuitionistic fuzzy bi-normed KU-ideal of X₁ × X₂.

Proof. For any (x, y) ∈ X₁ × X₂, (α_{A
₁} × α_{A
₂}) (0, 0) = T {α_{A
₁} (0) , α_{A
₂} (0)} ≥ T {α_{A
₁} (x) , α_{A
₂} (y)} = (α_{A
₁} × α_{A
₂})(x, y) and (β_{A
₁} × β_{A
₂}) (0, 0) = S {β_{A
₁} (0) , β_{A
₂} (0)}≤S {β_{A
₁} (x) , β_{A
₂} (y)} = (β_{A
₁} × β_{A
₂}) (x, y) .

Now, for any (x₁, y₁) , (x₂, y₂) and (x₃, y₃) ∈ X₁ × X₂, (α_{A
₁} × α_{A
₂}) ((x₁, y₁) ∗ (x₃, y₃)) = (α_{A
₁} × α_{A
₂})(x₁ ∗ x₃, y₁ ∗ y₃) = T {α_{A
₁} (x₁ ∗ x₃) , α_{A
₂} (y₁ ∗ y₃)} ≥ T {T {α_{A
₁} (x₁ ∗ (x₂ ∗ x₃)) , α_{A
₁} (x₂)} , T {α_{A
₂} (y₁ ∗ (y₂ ∗ y₃)) , α_{A
₂} (y₂)}} = T {T {α_{A
₁} (x₁ ∗ (x₂ ∗ x₃)) , α_{A
₂} (y₁ ∗ (y₂ ∗ y₃))} , T {α_{A
₁} (x₂) , α_{A
₂} (y₂)}} = T {(α_{A
₁} × α_{A
₂}) ((x₁ ∗ (x₂ ∗ x₃)) , (y₁ ∗ (y₂ ∗ y₃))) , (α_{A
₁} × α_{A
₂}) (x₂, y₂)} = T {(α_{A
₁} × α_{A
₂}) ((x₁, y₁) ∗ ((x₂, y₂) ∗ (x₃, y₃))) , (α_{A
₁} × α_{A
₂}) (x₂, y₂)} and (β_{A
₁} × β_{A
₂}) ((x₁, y₁) ∗ (x₃, y₃)) = (β_{A
₁} × β_{A
₂}) (x₁ ∗ x₃, y₁ ∗ y₃) = S {β_{A
₁} (x₁ ∗ x₃) , β_{A
₂} (y₁ ∗ y₃)} ≤ S {S {β_{A
₁} (x₁ ∗ (x₂ ∗ x₃)) , β_{A
₁} (x₂)} , S {β_{A
₂} (y₁ ∗ (y₂ ∗ y₃)) , β_{A
₂} (y₂)}} = S {S {β_{A
₁} (x₁ ∗ (x₂ ∗ x₃)) , β_{A
₂} (y₁ ∗ (y₂ ∗ y₃))} , S {β_{A
₁} (x₂) , β_{A
₂} (y₂)}} = S {(β_{A
₁} × β_{A
₂}) ((x₁ ∗ (x₂ ∗ x₃)) , (y₁ ∗ (y₂ ∗ y₃))) , (β_{A
₁} × β_{A
₂}) (x₂, y₂)} = S {(β_{A
₁} × β_{A
₂}) ((x₁, y₁) ∗ ((x₂, y₂) ∗ (x₃, y₃))) , (β_{A
₁} × β_{A
₂}) (x₂, y₂)}. Hence, we have proved that A₁ × A₂ is an intuitionistic fuzzy bi-normed KU-ideal of X₁ × X₂. □

Proposition 5.10. Let A₁ × A₂ be an intuitionistic fuzzy imaginable bi-normed KU-ideal of X₁ × X₂ and (x₁, y₁), (x₂, y₂) ∈X₁ × X₂. If (x₁, y₁) ≤ (x₂, y₂) then (α_{A
₁} × α_{A
₂}) (x₁, y₁) ≥ (α_{A
₁} × α_{A
₂}) (x₂, y₂) and (β_{A
₁} × β_{A
₂}) (x₁, y₁) ≤ (β_{A
₁} × β_{A
₂}) (x₂, y₂).

Proof. Assume that (x₁, y₁), (x₂, y₂) ∈X₁ × X₂ such that (x₁, y₁) ≤ (x₂, y₂). Then (x₂, y₂) ∗ (x₁, y₁) = (0, 0). Now, (α_{A
₁} × α_{A
₂}) (x₂, y₂) ≤ (α_{A
₁} × α_{A
₂}) (0, 0), (β_{A
₁} × β_{A
₂}) (x₂, y₂) ≥ (β_{A
₁} × β_{A
₂}) (0, 0) and (0, 0) ∗ (x₁, y₁) = (x₁, y₁). Therefore, (α_{A
₁} × α_{A
₂}) (x₁, y₁) = (α_{A
₁} × α_{A
₂}) ((0, 0) ∗ (x₁, y₁)) ≥ T {(α_{A
₁} × α_{A
₂}) ((0, 0) ∗ ((x₂, y₂) ∗ (x₁, y₁))) , (α_{A
₁} × α_{A
₂}) (x₂, y₂)} = T {(α_{A
₁} × α_{A
₂}) ((0, 0) ∗ (0, 0)) , (α_{A
₁} ×α_{A
₂}) (x₂, y₂)} = T {(α_{A
₁} × α_{A
₂}) (0, 0) , (α_{A
₁} × α_{A
₂}) (x₂, y₂)} ≥ (α_{A
₁} × α_{A
₂}) (x₂, y₂) and (β_{A
₁} × β_{A
₂}) (x₁, y₁) = (β_{A
₁} × β_{A
₂}) ((0, 0) ∗ (x₁, y₁)) ≤ S {(β_{A
₁} × β_{A
₂}) ((0, 0) ∗ ((x₂, y₂) ∗ (x₁, y₁))) , (β_{A
₁} × β_{A
₂}) (x₂, y₂)} = S {(β_{A
₁} × β_{A
₂}) ((0, 0) ∗ (0, 0)) , (β_{A
₁} × β_{A
₂}) (x₂, y₂)} = S {(β_{A
₁} × β_{A
₂}) (0, 0) , (β_{A
₁} × β_{A
₂}) (x₂, y₂)} ≤ (β_{A
₁} × β_{A
₂}) (x₂, y₂) . This shows that (α_{A
₁} × α_{A
₂}) (x₁, y₁) ≥ (α_{A
₁} × α_{A
₂}) (x₂, y₂) and (β_{A
₁} × β_{A
₂}) (x₁, y₁) ≤ (β_{A
₁} × β_{A
₂}) (x₂, y₂) for all (x₁, y₁), (x₂, y₂) ∈X₁ × X₂.□

Proposition 5.11. Let A₁ × A₂ be an intuitionistic fuzzy imaginable bi-normed KU-ideal of X₁ × X₂ and (x₁, y₁), (x₂, y₂) and (x₃, y₃) ∈ X₁ × X₂. If (x₁, y₁) ∗ (x₂, y₂) ≤ (x₃, y₃) then (α_{A
₁} × α_{A
₂}) (x₂, y₂) ≥ T {(α_{A
₁} × α_{A
₂}) (x₁, y₁) , (α_{A
₁} × α_{A
₃}) (x₂, y₃)} and (β_{A
₁} × β_{A
₂}) (x₂, y₂) ≤ S {(β_{A
₁} × β_{A
₂}) (x₁, y₁) , (β_{A
₁} × β_{A
₂}) (x₃, y₃)}.

Proof. Assume that (x₁, y₁), (x₂, y₂) and (x₃, y₃) ∈ X₁ × X₂ such that (x₁, y₁) ∗ (x₂, y₂) ≤ (x₃, y₃). Then (x₃, y₃) ∗ ((x₁, y₁) ∗ (x₂, y₂)) = (0, 0). Now, (α_{A
₁} × α_{A
₂}) ((x₃, y₃) ∗ (x₂, y₂)) ≥ T {(α_{A
₁} × α_{A
₂}) ((x₃, y₃) ∗ ((x₁, y₁) ∗ (x₂, y₂))) , (α_{A
₁} × α_{A
₂}) (x₁, y₁)} and (β_{A
₁} × β_{A
₂}) ((x₃, y₃) ∗ (x₂, y₂)) ≤ S {(β_{A
₁} × β_{A
₂}) ((x₃, y₃) ∗ ((x₁, y₁) ∗ (x₂, y₂))) , (β_{A
₁} × β_{A
₂}) (x₁, y₁)} .

By putting (x₃, y₃) = (0, 0) in above, we get (α_{A
₁} × α_{A
₂}) ((0, 0) ∗ (x₂, y₂)) = (α_{A
₁} × α_{A
₂}) (x₂, y₂) ≥ T {(α_{A
₁} × α_{A
₂}) ((0, 0) ∗ ((x₁, y₁) ∗ (x₂, y₂))) , (α_{A
₁} × α_{A
₂}) (x₁, y₁)} = T {(α_{A
₁} × α_{A
₂}) ((x₁, y₁) ∗ (x₂, y₂)) , (α_{A
₁} × α_{A
₂}) (x₁, y₁)} ≥ T {T {(α_{A
₁} × α_{A
₂}) ((x₁, y₁) ∗ ((x₃, y₃) ∗ (x₂, y₂))) , (α_{A
₁} × α_{A
₂}) (x₃, y₃)} , (α_{A
₁} × α_{A
₂}) (x₁, y₁)} = T {T {(α_{A
₁} × α_{A
₂}) (0, 0) , (α_{A
₁} × α_{A
₂}) (x₃, y₃)} , (α_{A
₁} × α_{A
₂}) (x₁, y₁)} ≥ T {(α_{A
₁} × α_{A
₂}) (x₃, y₃) , (α_{A
₁} × α_{A
₂}) (x₁, y₁)} = T {(α_{A
₁} × α_{A
₂}) (x₁, y₁) , (α_{A
₁} × α_{A
₂}) (x₃, y₃)} and (β_{A
₁} × β_{A
₂}) ((0, 0) ∗ (x₂, y₂)) = (β_{A
₁} × β_{A
₂}) (x₂, y₂) ≤ S {(β_{A
₁} × β_{A
₂}) ((0, 0) ∗ ((x₁, y₁) ∗ (x₂, y₂))) , (β_{A
₁} × β_{A
₂}) (x₁, y₁)} = S {(β_{A
₁} × β_{A
₂}) ((x₁, y₁) ∗ (x₂, y₂)) , (β_{A
₁} × β_{A
₂}) (x₁, y₁)} ≤ S {S {(β_{A
₁} × β_{A
₂}) ((x₁, y₁) ∗ ((x₃, y₃) ∗ (x₂, y₂))) , (β_{A
₁} × β_{A
₂}) (x₃, y₃)} , (β_{A
₁} × β_{A
₂}) (x₁, y₁) } = S {S {(β_{A
₁} × β_{A
₂}) (0, 0) , (β_{A
₁} × β_{A
₂}) (x₃, y₃)} , (β_{A
₁} × β_{A
₂}) (x₁, y₁)} ≤ S {(β_{A
₁} × β_{A
₂}) (x₃, y₃) , (β_{A
₁} ×β_{A
₂}) (x₁, y₁)} = S {(β_{A
₁} × β_{A
₂}) (x₁, y₁) , (β_{A
₁} × β_{A
₂}) (x₃, y₃)} . Therefore, (α_{A
₁} × α_{A
₂}) (x₂, y₂) ≥ T {(α_{A
₁} × α_{A
₂}) (x₁, y₁) , (α_{A
₁} × α_{A
₂}) (x₃, y₃)} and (β_{A
₁} × β_{A
₂}) (x₂, y₂) ≤ S {(β_{A
₁} × β_{A
₂}) (x₁, y₁) , (β_{A
₁} × β_{A
₂}) (x₃, y₃)}.□

Definition 5.12. Let A₁ × A₂ be an IFS of X₁ × X₂ and $\tilde{s}, \tilde{t} \in [0, 1]$ , then the set $U (α_{A_{1}} \times α_{A_{2}} : \tilde{s}) = {(x_{1}, y_{1}) | (x_{1}, y_{1}) \in X_{1} \times X_{2} & (α_{A_{1}} \times α_{A_{2}}) (x_{1}, y_{1}) \geq \tilde{s}}$ is called an upper $\tilde{s}$ -level of A₁ × A₂ and $L (β_{A_{1}} \times β_{A_{2}} : \tilde{t}) = {(x_{1}, y_{1}) | (x_{1}, y_{1}) \in X_{1} \times X_{2} & (β_{A_{1}} \times β_{A_{2}}) (x_{1}, y_{1}) \leq \tilde{t}}$ is called a lower $\tilde{t}$ -level of A₁ × A₂.

Finally, we prove the following main theorem of this paper. In this theorem, we give a necessary and sufficient condition for the product IFS set A₁ × A₂ of X₁ × X₂ to be an intuitionistic fuzzy bi-normed KU-ideal of X₁ × X₂.

Theorem 5.13. Let A₁ × A₂ be an IFS of X₁ × X₂. Then A₁ × A₂ is an intuitionistic fuzzy bi-normed KU-ideal of X₁ × X₂ if and only if for any $\tilde{s}, \tilde{t} \in [0, 1]$ , the sets $U (α_{A_{1}} \times α_{A_{2}} : \tilde{s})$ and $L (β_{A_{1}} \times β_{A_{2}} : \tilde{t})$ are either empty or KU-ideals of X₁ × X₂.

Proof. Assume that A₁ × A₂ is an intuitionistic fuzzy bi-normed KU-ideal X₁ × X₂. Let $U (α_{A_{1}} \times α_{A_{2}} : \tilde{s})$ , $L (β_{A_{1}} \times β_{A_{2}} : \tilde{t}) \neq φ$ and $(x, y) \in U (α_{A_{1}} \times α_{A_{2}} : \tilde{s})$ , $L (β_{A_{1}} \times β_{A_{2}} : \tilde{t})$ . Then, we have $(α_{A_{1}} \times α_{A_{2}}) (x, y) \geq \tilde{s}$ and $(β_{A_{1}} \times β_{A_{2}}) (x, y) \leq \tilde{t}$ , and so we deduce that $(α_{A_{1}} \times α_{A_{2}}) (0, 0) \geq (α_{A_{1}} \times α_{A_{2}}) (x, y) \geq \tilde{s}$ and $(β_{A_{1}} \times β_{A_{2}}) (0, 0) \leq (β_{A_{1}} \times β_{A_{2}}) (x, y) \leq \tilde{t}$ . This shows that $(0, 0) \in U (α_{A_{1}} \times α_{A_{2}} : \tilde{s})$ , $L (β_{A_{1}} \times β_{A_{2}} : \tilde{t})$ .

Let ((x₁, y₁) ∗ ((x₃, y₃) ∗ (x₂, y₂))) and $(x_{3}, y_{3}) \in U (α_{A_{1}} \times α_{A_{2}} : \tilde{s})$ , $L (β_{A_{1}} \times β_{A_{2}} : \tilde{t})$ . This implies $(α_{A_{1}} \times α_{A_{2}}) ((x_{1}, y_{1}) * ((x_{3}, y_{3}) * (x_{2}, y_{2}))) \geq \tilde{s}$ , $(α_{A_{1}} \times α_{A_{2}}) (x_{3}, y_{3}) \geq \tilde{s}$ , $(β_{A_{1}} \times β_{A_{2}}) ((x_{1}, y_{1}) * ((x_{3}, y_{3}) * (x_{2}, y_{2}))) \leq \tilde{t}$ and $(β_{A_{1}} \times β_{A_{2}}) (x_{3}, y_{3}) \leq \tilde{t}$ . Now, (α_{A
₁} × α_{A
₂}) ((x₁, y₁) ∗ (x₂, y₂)) ≥ T {(α_{A
₁} × α_{A
₂}) ((x₁, y₁) ∗ ((x₃, y₃) ∗ (x₂, y₂))) , (α_{A
₁} × α_{A
₂}) (x₃, $y_{3})} \geq T {\tilde{s}, \tilde{s}} \geq \tilde{s}$ and $(β_{A_{1}} \times β_{A_{2}}) ((x_{1}, y_{1}) * (x_{2}, y_{2})) \leq S {(β_{A_{1}} \times β_{A_{2}}) ((x_{1}, y_{1}) * ((x_{3}, y_{3}) * (x_{2}, y_{2}))), (β_{A_{1}} \times β_{A_{2}}) (x_{3}, y_{3})} \leq S {\tilde{t}, \tilde{t}} \leq \tilde{t}$ . This implies that $(x_{1}, y_{1}) * (x_{2}, y_{2}) \in U (α_{A_{1}} \times α_{A_{2}} : \tilde{s})$ and $L (β_{A_{1}} \times β_{A_{2}} : \tilde{t})$ . Hence, $U (α_{A_{1}} \times α_{A_{2}} : \tilde{s})$ and $L (β_{A_{1}} \times β_{A_{2}} : \tilde{t})$ are KU-ideals of X₁ × X₂.

Conversely, assume that $U (α_{A_{1}} \times α_{A_{2}} : \tilde{s})$ and $L (β_{A_{1}} \times β_{A_{2}} : \tilde{t})$ are KU-ideals of X₁ × X₂. Let (x₁, y₁) ∈ X₁ × X₂ be such that (α_{A
₁} × α_{A
₂}) (0, 0) < (α_{A
₁} × α_{A
₂}) (x₁, y₁) and (β_{A
₁} × β_{A
₂}) (0, 0) > (β_{A
₁} × β_{A
₂}) (x₁, y₁). By putting

$\begin{matrix} \tilde{s_{0}} & = & \frac{1}{2} [(α_{A_{1}} \times α_{A_{2}}) (0, 0) + (α_{A_{1}} \times α_{A_{2}}) (x_{1}, y_{1})] \\ \tilde{t_{0}} & = & \frac{1}{2} [(β_{A_{1}} \times β_{A_{2}}) (0, 0) + (β_{A_{1}} \times β_{A_{2}}) (x_{1}, y_{1})] \end{matrix}$ we get $(α_{A_{1}} \times α_{A_{2}}) (0, 0) < \tilde{s_{0}} < (α_{A_{1}} \times α_{A_{2}}) (x_{1}, y_{1})$ and $(β_{A_{1}} \times β_{A_{2}}) (0, 0) > \tilde{t_{0}} > (β_{A_{1}} \times β_{A_{2}}) (x_{1}, y_{1}) .$ Therefore, $(x_{1}, y_{1}) \in U (α_{A_{1}} \times α_{A_{2}} : \tilde{s})$ , L (β_{A
₁}× $β_{A_{2}} : \tilde{t})$ but $(0, 0) \notin U (α_{A_{1}} \times α_{A_{2}} : \tilde{s})$ , L (β_{A
₁} × β_{A
₂} : $\tilde{t})$ which is a contradiction. Hence(α_{A
₁}× α_{A
₂}) (0, 0) ≥ (α_{A
₁} ×α_{A
₂}) (x, y) and (β_{A
₁} × β_{A
₂}) (0, 0) ≤ (β_{A
₁} × β_{A
₂}) (x, y) for all (x, y) ∈ X₁ × X₂.

Again, we assume that (x₁, y₁), (x₂, y₂) and (x₃, y₃) ∈X₁ × X₂ be such that (α_{A
₁} × α_{A
₂}) ((x₁, y₁) ∗ (x₃, y₃)) < T {(α_{A
₁} × α_{A
₂}) ((x₁, y₁) ∗ ((x₂, y₂) ∗ (x₃, y₃))) , (α_{A
₁} × α_{A
₂}) (x₂, y₂)} and (β_{A
₁} × β_{A
₂}) ((x₁, y₁) ∗ (x₃, y₃)) > S {(β_{A
₁} × β_{A
₂}) ((x₁, y₁) ∗ ((x₂, y₂) ∗ (x₃, y₃))) , (β_{A
₁} × β_{A
₂}) (x₂, y₂)}. By putting

$\begin{matrix} \tilde{s_{1}} = \frac{1}{2} [(α_{A_{1}} \times α_{A_{2}}) ((x_{1}, y_{1}) * (x_{3}, y_{3})) \\ + T {(α_{A_{1}} \times α_{A_{2}}) ((x_{1}, y_{1}) * ((x_{2}, y_{2}) * (x_{3}, y_{3}))), \\ (α_{A_{1}} \times α_{A_{2}}) (x_{2}, y_{2})}], \\ \tilde{t_{1}} = \frac{1}{2} [(β_{A_{1}} \times β_{A_{2}}) ((x_{1}, y_{1}) * (x_{3}, y_{3})) \\ + S {(β_{A_{1}} \times β_{A_{2}}) ((x_{1}, y_{1}) * ((x_{2}, y_{2}) * (x_{3}, y_{3}))), \\ (β_{A_{1}} \times β_{A_{2}}) (x_{2}, y_{2})}] \end{matrix}$

Then, we get $(α_{A_{1}} \times α_{A_{2}}) ((x_{1}, y_{1}) * (x_{3}, y_{3})) < \tilde{s_{1}} < T {(α_{A_{1}} \times α_{A_{2}}) ((x_{1}, y_{1}) * ((x_{2}, y_{2}) * (x_{3}, y_{3}))),$ (α_{A
₁} × α_{A
₂}) (x₂, y₂)} and $(β_{A_{1}} \times β_{A_{2}}) ((x_{1}, y_{1}) * (x_{3}, y_{3})) > \tilde{t_{1}} > S {(β_{A_{1}} \times β_{A_{2}}) ((x_{1}, y_{1}) * ((x_{2}, y_{2}) * (x_{3}, y_{3}))), (β_{A_{1}} \times β_{A_{2}}) (x_{2}, y_{2})}$ .

This shows that $(x_{1}, y_{1}) * ((x_{2}, y_{2}) * (x_{3}, y_{3})) \in U (α_{A_{1}} \times α_{A_{2}} : \tilde{s})$ , $L (β_{A_{1}} \times β_{A_{2}} : \tilde{t})$ and ((x₂, y₂) $\in U (α_{A_{1}} \times α_{A_{2}} : \tilde{s})$ , $L (β_{A_{1}} \times β_{A_{2}} : \tilde{t})$ but $(x_{1}, y_{1}) * (x_{3}, y_{3}) \notin U (α_{A_{1}} \times α_{A_{2}} : \tilde{s})$ , $L (β_{A_{1}} \times β_{A_{2}} : \tilde{t})$ which is a contradiction. Therefore, (α_{A
₁} × α_{A
₂}) ((x₁, y₁) ∗ (x₃, y₃)) ≥ T {(α_{A
₁} × α_{A
₂}) ((x₁, y₁) ∗ ((x₂, y₂) ∗ (x₃, y₃))) , (α_{A
₁} × α_{A
₂}) (x₂, y₂)} and (β_{A
₁} × β_{A
₂}) ((x₁, y₁) ∗ (x₃, y₃)) ≤ S {(β_{A
₁} × β_{A
₂}) ((x₁, y₁) ∗ ((x₂, y₂) ∗ (x₃, y₃))) , (β_{A
₁} × β_{A
₂}) (x₂, y₂)}. Hence, A₁ × A₂ is an intuitionistic fuzzy bi-normed KU-ideal of X₁ × X₂. □

The relationship between the intuitionistic fuzzy bi-normed KU-ideals [A₁ · A₂] _(T,S) and A₁ × A₂ can be described in the following diagram

where I = [0, 1] and g : X → X × X is defined by g (x) = (x, x). It is not difficult to see that [A₁ · A₂] _(T,S) is the preimage of A₁ × A₂ under g.

6 Conclusions and future work

Recently, in [31], the authors have already studied the intuitionistic fuzzy KU-subalgebras of KU-algebras with respect to t-norm and t-conorm. In this paper, we characterize the intuitionistic fuzzy bi-normed KU-ideals of a KU-algebra. By using the imaginable property we are able to introduce the intuitionistic fuzzy imaginable bi-normed KU-ideals of a KU-algebra X. Finally, we established the direct products and (T, S)-products of intuitionistic fuzzy bi-normedKU-ideals.

We believe that our results presented in this paper will give a foundation for further study the algebraic theory of KU-algebras. In our future study of fuzzy structure of KU-algebras, the following topics will be considered and discussed.

to find the interval-valued intuitionistic fuzzybi-normed KU-subalgebras of KU-algebras,

to find the interval-valued intuitionistic fuzzybi-normed KU-ideals of KU-algebras.

Footnotes

Acknowledgments

The authors are highly grateful to referees and Professor S. Solovjovs, Associate Editor, for their valuable comments and suggestions for improving the paper.

References

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