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Journal of Intelligent & Fuzzy Systems 30 (2016) 1169–1180 DOI:10.3233/IFS-151841 IOS Press

Atanassov’s intuitionistic fuzzy bi-normed KU-ideals of a KU-algebra Tapan Senapatia,∗ and K.P. Shumb a Department

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of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, India b Institute of Mathematics, Yunnan University, Kunming, People’s Republic of China

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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 KUalgebra. 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.

1. Introduction

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Keywords: KU-algebra, T -norm, T -conorm, intuitionistic fuzzy bi-normed KU-ideal, homomorphism, upper(lower)-level cut

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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 related t-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∗ Corresponding author. Tapan Senapati, Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, India. Tel.: +91 963 543 0583; E-mail: [email protected].

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 KUalgebra. 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 KUideals in KU-algebras and discussed some special properties. M. Akram et al. [1] introduced the notion of ˜ δ)-fuzzy ˜ interval-valued (θ, 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]

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T. Senapati and K.P. Shum / Atanassov’s intuitionistic fuzzy bi-normed KU-ideals of a KU-algebra

tial 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, (i) (ii) (iii) (iv) (v)

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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, 1. 2. 3. 4.

(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 KUalgebra unless otherwise specified. For the sake of brevity, we call X a KU-algebra. We now define a par-

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.

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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, αA (x) >: x ∈ X} where αA (x) is called the membership value of x in A and 0 ≤ αA (x) ≤ 1. For any fuzzy sets A and B of a set X, we define A ∩ B = min{αA (x), αB (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 TM (x, y) = min(x, y), the product TP (x, y) = x.y and the Lukasiewicz t-norm TL (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 SM (x, y) = max(x, y), the probabilistic sum SP (x, y) = x + y − x.y and the Lukasiewicz t-conorm or (bounded sum) SL (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.

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introduced the notion of fuzzy KU-subalgebras and KU-ideals of KU-algebras with respect to a given t-norm and obtained some of their properties. T. Senapati [31] also introduced the intuitionistic fuzzy binormed, 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)-normed KU-ideals.

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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) αA (x ∗ y) ≥ T {αA (x), αA (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) αA (0) ≥ αA (x) and (T3) αA (x ∗ z) ≥ T {αA (x ∗ (y ∗ z)), αA (y)} for all x, y, z ∈ X.

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

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(TS1) αA (x ∗ y) ≥ T {αA (x), αA (y)}, (TS2) βA (x ∗ y) ≤ S{βA (x), βA (y)}.

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Definition 2.6. [31] An intuitionistic fuzzy set A = (αA , βA ) 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:

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 otherwise specified. Definition 3.1. [31] An intuitionistic fuzzy set A = (αA , βA ) 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:

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c 0 d 0

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0 0

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Let Tm , Sm : [0, 1] × [0, 1] → [0, 1] be functions defined by Tm (x, y) = max(x + y − 1, 0) and Sm (x, y) = min(x + y, 1) for all x, y ∈ [0, 1]. Then Tm is a t-norm and Sm is a s-norm. Define an IFS A = (αA , βA ) in X by ⎧ ⎪ ⎨ 0.9, if x = 0 αA (x) = 0.7, if x = a, c ⎪ ⎩ 0.5, if x = b, d

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Definition 2.5. [3] The intuitionistic fuzzy sets defined on a non-empty set X as objects having the form A = {x, αA (x), βA (x) : x ∈ X}, where the functions αA (x) : X → [0, 1] and βA (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 ≤ αA (x) + βA (x) ≤ 1 for all x ∈ X. Obviously, when βA (x) = 1 − αA (x) for every x ∈ X, the set A becomes a fuzzy set. For the sake of simplicity, we shall use the symbol A = (αA , βA ) for the intuitionistic fuzzy subset A = {x, αA (x), βA (x) : x ∈ X}.

We now illustrate the above definitions by using some examples.

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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(αA ) ⊆ P .

(TS3) αA (0) ≥ αA (x) and βA (0) ≤ βA (x), (TS4) αA (x ∗ z) ≥ T {αA (x ∗ (y ∗ z)), αA (y)}, (TS5) βA (x ∗ z) ≤ S{βA (x ∗ (y ∗ z)), βA (y)}.

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We now give the following definitions:

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and

⎧ ⎪ ⎨ 0.1, if x = 0 βA (x) = 0.3, if x = a, c ⎪ ⎩ 0.4, if x = b, d.

Then A is clearly a bi-normed KU-ideal. Hence, A is an an intuitionistic fuzzy (Tm , Sm )-normed KU-ideal of X. Definition 3.3. An intuitionistic fuzzy bi-normed KUideal A = (αA , βA ) is called an intuitionistic fuzzy imaginable bi-normed KU-ideal of X if αA and βA satisfy the imaginable property with respect to T and S respectively. Example 3.4. Let Tm be a t-norm, Sm a s-norm and X={0, a, b, c} be a KU-algebra in Example 3.2. Define an IFS A = (αA , βA ) in X by αA (0) = 1, βA (0) = 0, αA (x) = 0 and βA (x) = 1 if x ∈ {a, b, c, d}. Then, it is easy to verify that αA (x ∗ y) ≥ Tm {αA (x), αA (y)} and βA (x ∗ y) ≤ Sm {βA (x), βA (y)} for all x, y ∈ X. Also, we have Im(αA ) ⊆ Tm and Im(βA ) ⊆ Sm . Hence, A is an intuitionistic fuzzy imaginable (Tm , Sm )-normed KU-ideal of X.


Theorem 3.5. Let A = (αA , βA ) be an intuitionistic fuzzy bi-normed KU-ideal of X. Then the sets IαA and IβA are KU-ideals of X. Proof. Let A be an intuitionistic fuzzy bi-normed KU-ideal of X. Then it is obvious that 0 ∈ IαA , IβA . Let x, y, z ∈ X be such that x ∗ (y ∗ z) ∈ IαA , IβA and y ∈ IαA , IβA . If αA (x ∗ (y ∗ z)) = αA (0) = αA (y) and βA (x ∗ (y ∗ z)) = βA (0) = βA (y), then we have αA (x ∗ z) ≥ T {αA (x ∗ (y ∗ z)), αA (y)} ≥ αA (0) and βA (x ∗ z) ≤ S{βA (x ∗ (y ∗ z)), βA (y)} ≤ βA (0). Since A is an intuitionistic fuzzy bi-normed KU-ideal of X, αA (x ∗ z) = αA (0) and βA (x ∗ z) = βA (0) i.e., x ∗ z ∈ IαA , IβA . Hence, IαA and IβA are indeed KU-ideals of X.

Proposition 3.9. Let A be an intuitionistic fuzzy binormed KU-ideal of X and x, y, z ∈ X. Then the following statments hold: (i) αA (x ∗ (x ∗ y)) ≥ αA (y) and βA (x ∗ (x ∗ y)) ≤ βA (y). (ii) If x ≤ y then αA (x) ≥ αA (y) and βA (x) ≤ βA (y). (iii) If x ∗ y ≤ z then αA (y) ≥ T {αA (x), αA (z)} and βA (y) ≤ S{βA (x), βA (z)}.

Proof. (i) Taking z = x ∗ y in (TS4), (TS5) and using (TS3) and x ∗ 0 = x, we get αA (x ∗ (x ∗ y)) ≥ T {αA (x ∗ (y ∗ (x ∗ y))), αA (y)} = T {αA (x ∗ (x ∗ (y ∗ y))), αA (y)} = T {αA (x ∗ (x ∗ 0)), αA (y)} = T {αA (0), αA (y)} = αA (y) and βA (x ∗ (x ∗ y)) ≤ S{βA (x ∗ (y ∗ (x ∗ y))), βA (y)} = S{βA (x ∗ (x ∗ (y ∗ y))), βA (y)} = S{βA (x ∗ (x ∗ 0)), βA (y)} = S{βA (0), βA (y)} = βA (y). (ii) Let x, y ∈ X be such that x ≤ y. Then y ∗ x = 0. This implies αA (x) = αA (0 ∗ x) ≥ T {αA (0 ∗ (y ∗ x)), αA (y)} = T {αA (0 ∗ 0), αA (y)} = T {αA (0), αA (y)} ≥ αA (y) and βA (x) = βA (0 ∗ x) ≤ S{βA (0 ∗ (y ∗ x)), βA (y)} = S{βA (0 ∗ 0), βA (y)} = S{βA (0), βA (y)} ≤ βA (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: αA (z ∗ y) ≥ T {αA (z ∗ (x ∗ y)), αA (x)}, if we put z = 0 then αA (0 ∗ y) = αA (y) ≥ T {αA (0 ∗ (x ∗ y)), αA (x)} = T {αA (x ∗ y), αA (x)} ≥ T {T {αA (x ∗ (z ∗ y)), αA (z)}, αA (x)} = T {T {αA (z ∗ (x ∗ y)), αA (z)}, αA (x)} = T {T {αA (0), αA (z)}, αA (x)} = T {αA (z), αA (x)}. Again, from (TS5), we have the following inequalities: βA (z ∗ y) ≤ S{βA (z ∗ (x ∗ y)), βA (x)}, if we put z = 0 then βA (0 ∗ y) = βA (y) ≤ S{βA (0 ∗ (x ∗ y)), βA (x)} = S{βA (x ∗ y), βA (x)} ≤ S{S{βA (x ∗ (z ∗ y)), βA (z)}, βA (x)} = S{S{βA (z ∗ (x ∗ y)), βA (z)}, βA (x)} = S{S{βA (0), βA (z)}, βA (x)} = S{βA (z), βA (x)}. Therefore, αA (y) ≥ T {αA (x), αA (z)} and βA (y) ≤ {βA (x), βA (z)}, for all x, y, z ∈ X.

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

Theorem 3.8. Let {Ai : i = 1, 2, 3, 4, . . .} be a family of intuitionistic fuzzy bi-normed KU-ideals of a (KU-algebra X. Then Ai is an intuitionistic fuzzzy bi-normed KU-ideal of X, where Ai = (min αAi (x), max βAi (x)).

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The sets {x | x ∈ X & αA (x) = αA (0)} and {x | x ∈ X & βA (x) = βA (0)} are denoted by IαA and IβA , respectively. The above two sets are clearly the KUideal of the KU-algebra X.

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

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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, αA1 ∩A2 (x ∗ z) = min{αA1 (x ∗ z), αA2 (x ∗ z)} ≥ min{T {αA1 (x ∗ (y ∗ z)), αA1 (y)}, T {αA2 (x ∗ (y ∗ z)), αA2 (y)}} ≥ T {min{αA1 (x ∗ (y ∗ z)), αA2 (x ∗ (y ∗ z))}, min{αA1 (y), αA2 (y)}} = T {αA1 ∩A2 (x ∗ (y ∗ z)), αA1 ∩A2 (y)} and βA1 ∩A2 (x ∗ z) = max{βA1 (x ∗ z), βA2 (x ∗ z)} ≤ max{T {βA1 (x ∗ (y ∗ z)), βA1 (y)}, T {βA2 (x ∗ (y ∗ z)), βA2 (y)}} ≤ T {max{βA1 (x ∗ (y ∗ z)), βA2 (x ∗ (y ∗ z))}, max{βA1 (y), βA2 (y)}} = T {βA1 ∩A2 (x ∗ (y ∗ z)), βA1 ∩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:


s˜ 0 =

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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 βA (0) = 1 − βA (0) ≥ 1 − βA (x) = βA (x) and βA (x ∗ z) = 1 − βA (x ∗ z) ≥ 1 − S{βA (x ∗ (y ∗ z)), βA (y)} = T {1 − βA (x ∗ (y ∗ z)), 1 − βA (y)} = T {βA (x ∗ (y ∗ z)), βA (y)}. Hence, A2 is a T -fuzzy KU-ideal of X. Conversely, assume that A1 and A2 are T -fuzzy KU-ideals of X. Then αA (0) ≥ αA (x) and βA (0) ≤ βA (x) for all x ∈ X. For every x, y, z ∈ X,αA (x ∗ z) ≥ T {αA (x ∗ (y ∗ z)), αA (y)} and 1 − βA (x ∗ z) = βA (x ∗ z) ≥ T {βA (x ∗ (y ∗ z)), βA (y)} = T {1 − βA (x ∗ (y ∗ z)), 1 − βA (y)} = 1 − S{βA (x ∗ (y ∗ z))βA (y)}. That is, βA (x ∗ z) ≤ S{βA (x ∗ (y ∗ z)), βA (y)}. Hence, we deduce that A = (αA , βA ) is an intuitionistic fuzzy bi-normed KU-ideal of X. The two operators A and A on IFS as follows:

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Theorem 3.10. An IFS A = (αA , βA ) is an intuitionistic fuzzy bi-normed KU-ideal of X if and only if the fuzzy sets A1 = {x | x ∈ A & αA (x)} and A2 = {x | x ∈ A & βA (x)} are either empty or T -fuzzy KU-ideals of X.

αA (x) ≥ s˜ . By (TS4) we get αA (0) = αA (y ∗ 0) ≥ T {αA (y ∗ (x ∗ 0)), αA (x)} = T {αA (y ∗ 0), αA (x)} = T {αA (0), αA (x)} ≥ s˜ . Thus, 0 ∈ U(αA : s˜ ). Now letting x ∗ (y ∗ z), y ∈ U(αA : s˜ ), This implies that αA (x ∗ z) ≥ T {αA (x ∗ (y ∗ z)), αA (y)} ≥ s˜ . Therefore, x ∗ z ∈ U(αA : s˜ ). Hence, U(αA : s˜ ) is a KU-ideal of X. Again, let ˜t ∈ [0, 1] be such that L(βA : ˜t ) = / φ and x ∈ L(βA : ˜t ), then βA (x) ≤ ˜t . Then, by (TS5) we get the following equalities: βA (0) = βA (y ∗ 0) ≤ S{βA (y ∗ (x ∗ 0)), βA (x)} = S{βA (y ∗ 0), βA (x)} = S{βA (0), βA (x)} ≤ ˜t . Thus, 0 ∈ L(βA : ˜t ). Now letting x ∗ (y ∗ z), y ∈ L(βA : ˜t ), This implies that βA (x ∗ z) ≤ S{βA (x ∗ (y ∗ z)), βA (y)} ≤ ˜t . Therefore, x ∗ z ∈ L(βA : ˜t ). Hence, L(βA : ˜t ) is a KU-ideal of X. Conversely, suppose that for every s˜ , ˜t ∈ [0, 1], U(αA : s˜ ) and L(βA : ˜t ) are KU-ideals of X. Then, in the contrary, we consider the followings: Let x0 , y0 , z0 ∈ X be such that αA (x0 ∗ z0 ) < T {αA (x0 ∗ (y0 ∗ z0 )), αA (y0 )} and βA (x0 ∗ z0 ) > S{βA (x0 ∗ (y0 ∗ z0 )), βA (y0 )}. Assume that

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We now state the following theorem.

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Definition 3.11. [3] Let A = (αA , β A ) be an IFS defined on X. The operators A and A are defined as A = (αA , αA ) and A = (βA , βA ) respectively.

Proof. Straightforward.

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Theorem 3.12. If A = (αA , βA ) is an intuitionistic fuzzy bi-normed fuzzy KU-ideal of X, then (i) A, and (ii) A, both are intuitionistic fuzzy bi-normed KU-ideals.

Let A = (αA , βA ) be an intuitionistic fuzzy KU-ideal of X. For s˜ , ˜t ∈ [0, 1], the set U(αA : s˜ ) = {x | x ∈ X & αA (x) ≥ s˜ } is called the upper s˜ -level of A and L(βA : ˜t ) = {x | x ∈ X & βA (x) ≤ ˜t } is called the lower ˜t -level of A. Theorem 3.13. Let A = (αA , βA ) be an IFS in X. Then A is an intuitionistic fuzzy bi-normed KU-ideal of X if and oly if for all s˜ , ˜t ∈ [0, 1], the upper level set U(αA : s˜ ) and the lower level set L(βA : ˜t ) are either empty or KU-ideals of X. Proof. Assume that A is an intuitionistic fuzzy binormed KU-ideal of X and x, y, z ∈ X. Let s˜ ∈ [0, 1] be such that U(αA : s˜ ) = / φ and x ∈ U(αA : s˜ ). Then

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1 αA (x0 ∗ z0 ) + T {αA (x0 ∗ (y0 ∗ z0 )), αA (y0 )} 2

and ˜t0 =

1 βA (x0 ∗ z0 ) + S{βA (x0 ∗ (y0 ∗ z0 )), βA (y0 )} . 2

Then, we have αA (x0 ∗ z0 ) < s˜ 0 ≤ T {αA (x0 ∗ (y0 ∗ z0 )), αA (y0 )} ≤ min{αA (x0 ∗ (y0 ∗ z0 )), αA (y0 )} and so x0 ∗ z0 ∈ / U(αA : s˜ ) but x0 ∗ (y0 ∗ z0 ), y0 ∈ U(αA : s˜ ). Also, we have βA (x0 ∗ z0 ) > ˜t0 ≥ S{βA (x0 ∗ (y0 ∗ z0 )), βA (y0 )} ≥ max{βA (x0 ∗ (y0 ∗ z0 )), βA (y0 )} and / L(βA : ˜t ) but x0 ∗ (y0 ∗ z0 ), y0 ∈ L(βA : so x0 ∗ z0 ∈ ˜t ). This is a contradiction and hence we see that αA and βA 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


αA (x) =

n n+1

if x ∈ Sn \ Sn+1

1

if x ∈

1 n+1

if x ∈ Sn \ Sn+1

αA (x ∗ z) s ≥ = min{αA (x ∗ (y ∗ z)), αA (y)} s+1 ≥ T {αA (x ∗ (y ∗ z)), αA (y)}

∩∞ n=0 Sn

and βA (x) =

0

βA (x ∗ z)

where n = 0, 1, 2, . . . and S0 stands for X. Since 0 ∈ ∩∞ n=0 Sn , αA (0) = 1 ≥ αA (x) and βA (0) = 0 ≤ βA (x) for all x ∈ X. Let x, y, z ∈ X. Assume that x ∗ (y ∗ z) ∈ Sn \ Sn+1 and y ∈ Sk \ Sk+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 ∈ Sn , so x ∗ z ∈ Sn because Sn is a KU-ideal of X. Hence,

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

n = min{αA (x ∗ (y ∗ z)), αA (y)} n+1 ≥ T {αA (x ∗ (y ∗ z)), αA (y)}

1 βA (x ∗ z) ≤ = max{βA (x ∗ (y ∗ z)), βA (y)} n+1 ≤ S{βA (x ∗ (y ∗ z)), βA (y)}.

TH

∞ If x ∗ (y ∗ z), y ∈ ∩∞ n=0 Sn , then x ∗ z ∈ ∩n=0 Sn . Thus we deduce that

αA (x ∗ z) = 1 = min{αA (x ∗ (y ∗ z)), αA (y)} ≥ T {αA (x ∗ (y ∗ z)), αA (y)}

AU

βA (x ∗ z) = 0 = max{βA (x ∗ (y ∗ z)), βA (y)} ≤ S{βA (x ∗ (y ∗ z)), βA (y)}.

∞ If x ∗ (y ∗ z) ∈ / ∩∞ n=0 Sn and y ∈ ∩n=0 Sn , then there exists a positive integer r such that x ∗ (y ∗ z) ∈ Sr \ Sr+1 . It follows that x ∗ z ∈ Sr so that

αA (x ∗ z) ≥

r = min{αA (x ∗ (y ∗ z)), αA (y)} r+1 ≥ T {αA (x ∗ (y ∗ z)), αA (y)}

βA (x ∗ z) ≤

1 = max{βA (x ∗ (y ∗ z)), βA (y)} r+1 ≤ S{βA (x ∗ (y ∗ z)), βA (y)}.

/ Finally suppose that x ∗ (y ∗ z) ∈ ∩∞ n=0 Sn and y ∈ Then y ∈ Ss \ Ss+1 for some positive integer s. It follows that x ∗ z ∈ Ss , and hence ∩∞ n=0 Sn .

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.

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 (αB ), f −1 (βB )) in X with the membership function and non-membership function respectively are given by f −1 (αB )(x) = αB (f (x)) and f −1 (βB )(x) = βB (f (x)). It can be shown that f −1 (B) is an IFS We prove in below the following Proposition.

OR

αA (x ∗ z) ≥

1 = max{βA (x ∗ (y ∗ z)), βA (y)} s+1 ≤ S{βA (x ∗ (y ∗ z)), βA (y)}.

≤

if x ∈ ∩∞ n=0 Sn

PY

CO

1174

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 (αB )(0) = αB (f (0)) ≥ αB (f (x)) = f −1 (αB )(x) and f −1 (βB )(0) = βB (f (0)) ≤ βB (f (x)) = f −1 (βB )(x). Now let x, y, z ∈ X, then f −1 (αB )(x ∗ z) = αB (f (x ∗ z)) = αB (f (x) ∗ f (z)) ≥ T {αB (f (x) ∗ (f (y) ∗ f (z))), αB (f (y))} = T {αB (f (x ∗ (y ∗ z))), αB (f (y))} = T {f −1 (αB )(x ∗ (y ∗ z)), f −1 (αB )(y)} and f −1 (βB )(x ∗ z) = βB (f (x ∗ z)) = βB (f (x) ∗ f (z)) ≤ S{βB (f (x) ∗ (f (y) ∗ f (z))), βB (f (y))} = S{βB (f (x ∗ (y ∗ z))), βB (f (y))} = S{f −1 (βB )(x ∗ (y ∗ z)), f −1 (βB )(y)}. This completes the proof. Definition 4.2. Let f be a mapping from the set X to the set Y . If A = (αA , βA ) is an IFS in X and B is the image of A, then B is given by


αB (y) =

⎧ sup αA (x), ⎪ ⎪ ⎨ x∈f −1 (y)

αA (x0 ∗ (y0 ∗ z0 )) = αB {f (x0 ∗ (y0 ∗ z0 ))} = αB (x ∗ (y ∗ z ))

if f −1 (y) = {x ∈ X, f (x) = y} = / φ ⎪ ⎪ ⎩ 0, otherwise

= =

and ⎧ inf βA (x), ⎪ ⎪ ⎨ x∈f −1 (y)

t0 ∈T

= βB (x ∗ (y ∗ z )) =

=

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

αA (x ∗ z ) =

αB (x ) and βB (0 ) ≤

AU

TH

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 αA (0) ≥ αA (x) and βA (0) ≤ βA (x), for all x ∈ X. Note that 0 ∈ f −1 (0 ) where 0, 0 are the zero of X and Y respectively. Thus, αB (0 ) = sup αA (t) = t∈f −1 (0 )

βA (x) which implies that inf

t∈f −1 (x )

βA (t)

βA (t) = βA (0) ≤ sup

t∈f −1 (x ) = βB (x ).

αA (t) =

For any x , y , z ∈ Y , let x0 ∈ f −1 (x ), y0 ∈ f −1 (y ) and z0 ∈ f −1 (z ) be such that αA (x0 ∗ y0 ) = βA (x0 ∗ y0 ) = αA (y0 ) = βA (y0 ) =

sup t∈f −1 (x ∗z )

inf

t∈f −1 (x ∗z )

sup t∈f −1 (y )

sup t∈f −1 (y )

αA (t), βA (t),

αA (t), βA (t),

βA (t).

sup t∈f −1 (x ∗z )

αA (t) = αA (x0 ∗ z0 )

≥ T {αA (x0 ∗ (y0 ∗ z0 )), αA (y0 )} =T sup αA (t), sup

OR

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 .

inf

inf

t∈f −1 (x ∗(y ∗z ))

βA (x0 ∗ (y0 ∗ z0 ))

Then

t0 ∈T

t∈f −1 (0 ) αB (0 ) ≥

inf

(x0 ∗(y0 ∗z0 ))∈f −1 (x ∗(y ∗z ))

CO

and βA (t0 ) = inf βA (t) respectively.

=

αA (t)

βA (x0 ∗ (y0 ∗ z0 )) = βB {f (x0 ∗ (y0 ∗ z0 ))}

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 αA (t0 ) = sup αA (t)

αA (0) ≥ αA (x) and βB

sup t∈f −1 (x ∗(y ∗z ))

αA (x0 ∗ (y0 ∗ z0 ))

and

if f −1 (y) = {x ∈ X, f (x) = y} = / φ ⎪ ⎪ ⎩ 0, otherwise.

(0 )

sup (x0 ∗(y0 ∗z0 ))∈f −1 (x ∗(y ∗z ))

PY

βB (y) =

1175

t∈f −1 (x ∗(y ∗z ))

t∈f −1 (y )

αA (t)

= T {αA (x ∗ (y ∗ z )), αA (y )}

and βA (x ∗ z ) =

inf

t∈f −1 (x ∗z )

βA (t) = βA (x0 ∗ z0 )

≤ S{βA (x0 ∗ (y0 ∗ z0 )), βA (y0 )} βA (t), inf =S inf t∈f −1 (x ∗(y ∗z ))

t∈f −1 (y )

βA (t)

= S{βA (x ∗ (y ∗ z )), βA (y )}. Thus, B is is an intuitionistic fuzzy bi-normed KUideal 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 KUideals 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.

1176


Definition 5.1. Let A1 = (αA1 , βA1 ) and A2 = (αA2 , βA2 ) be two intuitionistic fuzzy subsets of a KUalgebra X. Then the (T, S)-product of A1 and A2 , denoted by

[βA1 .βA2 ]S ∗ (x ∗ z) = S ∗ {βA1 (x ∗ z), βA2 (x ∗ z)} ≤ S ∗ {S{βA1 (x ∗ (y ∗ z)), βA1 (y)}, S{βA2 (x ∗ (y ∗ z)), βA2 (y)}} ∗

≤ S(S (βA1 (x ∗ (y ∗ z)), βA2 (x ∗ (y ∗ z))),

[A1 · A2 ](T,S) = {[αA1 · αA2 ]T , [βA1 · βA2 ]S }

S ∗ (βA1 (y), βA2 (y)))

and is defined by

= S{[βA1 .βA2 ]S ∗ (x ∗ (y ∗ z)), [βA1 .βA2 ]S ∗ (y)}.

[αA1 · αA2 ]T (x) = T (αA1 (x), αA2 (x)),

(i) [αA1 .αA2 ]T (0) ≥ [αA1 .αA2 ]T (x) and [βA1 .βA2 ]S (0) ≤ [βA1 .βA2 ]S (x), (ii) [αA1 .αA2 ]T (x ∗ z) ≥ T {[αA1 .αA2 ]T (x ∗ (y ∗ z)), [αA1 .αA2 ]T (y)}, (iii) [βA1 .βA2 ]S (x ∗ z) ≤ S{[βA1 .βA2 ]S (x ∗ (y ∗ z)), [βA1 .βA2 ]S (y)}.

OR

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

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 ).

CO

for all x ∈ X.

PY

Hence, [A1 · A2 ](T ∗ ,S ∗ ) is an intuitionistic fuzzy binormed KU-ideal of X.

[βA1 · βA2 ]S (x) = S(βA1 (x), βA2 (x)),

TH

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., T ∗ (T (a, b), T (c, d)) ≥ T (T ∗ (a, c), T ∗ (b, d)) and S ∗ (S(a, b), S(c, d)) ≤ S(S ∗ (a, c), S ∗ (b, d))

AU

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 [αA1 .αA2 ]T (0) = T {αA1 (0), αA2 (0)} ≥ T {αA1 (x), αA2 (x)} = [αA1 .αA2 ]T (x), [βA1 .βA2 ]S (0) = S{βA1 (0), βA2 (0)} ≤ S{βA1 (x), βA2 (x)} = [βA1 .βA2 ]S (x) and [αA1 .αA2 ]T ∗ (x ∗ z) = T ∗ {αA1 (x ∗ z), αA2 (x ∗ z)} ≥ T ∗ {T {αA1 (x ∗ (y ∗ z)), αA1 (y)}, T {αA2 (x ∗ (y ∗ z)), αA2 (y)}} ≥ T (T ∗ (αA1 (x ∗ (y ∗ z)), αA2 (x ∗ (y ∗ z))), T ∗ (αA1 (y), αA2 (y))) = T {[αA1 .αA2 ]T ∗ (x ∗ (y ∗ z)), [αA1 .αA2 ]T ∗ (y)},

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 ([αA1 · αA2 ]T ∗ ), f −1 ([βA1 · βA2 ]S ∗ )} is the (T ∗ , S ∗ )-product of f −1 (A1 ) and f −1 (A2 ). Then we have f −1 ([αA1 · αA2 ]T ∗ ) = [f −1 (αA1 ) · f −1 (αA2 )]T ∗ and f −1 ([βA1 · βA2 ]S ∗ ) = [f −1 (βA1 ) · f −1 (βA2 )]S ∗ . Proof. For any x ∈ X, we derive the following equalities: f −1 ([αA1 · αA2 ]T ∗ )(x) = ∗ [αA1 · αA2 ]T ∗ (f (x)) = T (αA1 (f (x)), αA2 (f (x))) = T ∗ ([f −1 (αA1 )]f (x), [f −1 (αA2 )]f (x)) = [f −1 (αA1 ) · f −1 ([βA1 · βA2 ]S ∗ )(x) = f −1 (αA2 )]T ∗ (x) and ∗ [βA1 · βA2 ]S ∗ (f (x)) = S (βA1 (f (x)), βA2 (f (x))) = S ∗ ([f −1 (βA1 )]f (x), [f −1 (βA2 )] f (x)) = [f −1 (βA1 ) · f −1 (βA2 )]S ∗ (x).


S(S(x, y), S(z, t)) = S(S(x, z), S(y, t)) 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 = (αA1 , βA1 ) and A2 = (αA2 , βA2 ) be two IFSs of X1 and X2 , respectively, then the Cartesian product of A1 and A2 denoted by A = (αA , βA ), is defined by αA = αA1 × αA2 and βA = βA1 × βA2 such that αA (x1 , x2 ) = (αA1 × αA2 )(x1 , x2 ) = T (αA1 (x1 ), αA2 (x2 )) and βA (x1 , x2 ) = (βA1 × βA2 )(x1 , x2 ) = S(βA1 (x1 ), βA2 (x2 )) for all (x1 , x2 ) ∈ 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 (αA1 × αA2 )(x1 , y1 ) ≥ (αA1 × αA2 )(x2 , y2 ) and (βA1 × βA2 )(x1 , y1 ) ≤ (βA1 × βA2 )(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, (αA1 × αA2 )(x2 , y2 ) ≤ (αA1 × αA2 )(0, 0), and (βA1 × βA2 )(x2 , y2 ) ≥ (βA1 × βA2 )(0, 0) (0, 0) ∗ (x1 , y1 ) = (x1 , y1 ). Therefore, (αA1 × αA2 )(x1 , y1 ) = (αA1 × αA2 )((0, 0) ∗ (x1 , y1 )) ≥ T {(αA1 × αA2 )((0, 0) ∗ ((x2 , y2 ) ∗ (x1 , y1 ))), (αA1 × αA2 )(x2 , y2 )} = T {(αA1 × αA2 )((0, 0) ∗ (0, 0)), (αA1 ×αA2 )(x2 , y2 )} = T {(αA1 × αA2 )(0, 0), (αA1 × and αA2 )(x2 , y2 )} ≥ (αA1 × αA2 )(x2 , y2 ) (βA1 × βA2 )(x1 , y1 ) = (βA1 × βA2 )((0, 0) ∗ (x1 , y1 )) ≤ S{(βA1 × βA2 )((0, 0) ∗ ((x2 , y2 ) ∗ (x1 , y1 ))), (βA1 × βA2 )(x2 , y2 )} = S{(βA1 × βA2 )((0, 0) ∗ (0, 0)), (βA1 × βA2 )(x2 , y2 )} = S{(βA1 × βA2 )(0, 0), (βA1 × βA2 )(x2 , y2 )} ≤ (βA1 × βA2 )(x2 , y2 ). This shows that (αA1 × αA2 )(x1 , y1 ) ≥ and (βA1 × βA2 )(x1 , y1 ) ≤ (αA1 × αA2 )(x2 , y2 ) (βA1 × βA2 )(x2 , y2 ) for all (x1 , y1 ), (x2 , y2 ) ∈ X1 × X2 .

OR

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.

PY

T (T (x, y), T (z, t)) = T (T (x, z), T (y, t)),

(y2 ∗ y3 ))}, T {αA1 (x2 ), αA2 (y2 )}} = T {(αA1 × αA2 )((x1 ∗ (x2 ∗ x3 )), (y1 ∗ (y2 ∗ y3 ))), (αA1 × αA2 )(x2 , y2 )} = T {(αA1 × αA2 )((x1 , y1 ) ∗ ((x2 , y2 ) ∗ (x3 , y3 ))), (αA1 × αA2 )(x2 , y2 )} and (βA1 × βA2 )((x1 , y1 ) ∗ (x3 , y3 )) = (βA1 × βA2 )(x1 ∗ x3 , y1 ∗ y3 ) = S{βA1 (x1 ∗ x3 ), βA2 (y1 ∗ y3 )} ≤ S{S{βA1 (x1 ∗ (x2 ∗ x3 )), βA1 (x2 )}, S{βA2 (y1 ∗ (y2 ∗ y3 )), βA2 (y2 )}} = S{S{βA1 (x1 ∗ (x2 ∗ x3 )), βA2 (y1 ∗ (y2 ∗ y3 ))}, S{βA1 (x2 ), βA2 (y2 )}} = S{(βA1 × βA2 )((x1 ∗ (x2 ∗ x3 )), (y1 ∗ (y2 ∗ y3 ))), (βA1 × βA2 )(x2 , y2 )} = S{(βA1 × βA2 )((x1 , y1 ) ∗ ((x2 , y2 ) ∗ (x3 , y3 ))), (βA1 × βA2 )(x2 , y2 )}. Hence, we have proved that A1 × A2 is an intuitionistic fuzzy bi-normed KU-ideal of X1 × X 2 .

CO

Lemma 5.5. [11] Let T and S be a t-norm and a t-conorm respectively. Then we obtain the following equalities:

1177

TH

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

AU

(i) (αA1 × αA2 )(0, 0) ≥ (αA1 × αA2 )(x, y) and (βA1 × βA2 )(0, 0) ≤ (βA1 × βA2 )(x, y), (ii) (αA1 × αA2 )((x1 , y1 ) ∗ (x3 , y3 )) ≥ T {(αA1 × αA2 )((x1 , y1 ) ∗ ((x2 , y2 ) ∗ (x3 , y3 ))), (αA1 × αA2 )(x2 , y2 )}, (iii) (βA1 × βA2 )((x1 , y1 ) ∗ (x3 , y3 )) ≤ S{(βA1 × βA2 )((x1 , y1 ) ∗ ((x2 , y2 ) ∗ (x3 , y3 ))), (βA1 × βA2 )(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 , (αA1 × αA2 )(0, 0) = T {αA1 (0), αA2 (0)} ≥ T {αA1 (x), αA2 (y)} = (αA1 ×αA2 ) (x, y) and (βA1 × βA2 )(0, 0) = S{βA1 (0), βA2 (0)} ≤ S{βA1 (x), βA2 (y)} = (βA1 × βA2 )(x, y). Now, for any (x1 , y1 ), (x2 , y2 ) and (x3 , y3 ) ∈ X1 × X2 , (αA1 × αA2 )((x1 , y1 ) ∗ (x3 , y3 )) = (αA1 × αA2 ) (x1 ∗ x3 , y1 ∗ y3 ) = T {αA1 (x1 ∗ x3 ), αA2 (y1 ∗ y3 )} ≥ T {T {αA1 (x1 ∗ (x2 ∗ x3 )), αA1 (x2 )}, T {αA2 (y1 ∗ (y2 ∗ y3 )), αA2 (y2 )}} = T {T {αA1 (x1 ∗ (x2 ∗ x3 )), αA2 (y1 ∗

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 (αA1 × αA2 )(x2 , y2 ) ≥ T {(αA1 × αA2 )(x1 , y1 ), (αA1 × αA3 )(x2 , y3 )} and (βA1 × βA2 )(x2 , y2 ) ≤ S{(βA1 × βA2 )(x1 , y1 ), (βA1 × βA2 )(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, (αA1 ×

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deduce that (αA1 × αA2 )(0, 0) ≥ (αA1 × αA2 )(x, y) ≥ s˜ and (βA1 × βA2 )(0, 0) ≤ (βA1 × βA2 )(x, y) ≤ ˜t . This shows that (0, 0) ∈ U(αA1 × αA2 : s˜ ), L(βA1 × βA2 : ˜t ). Let ((x1 , y1 ) ∗ ((x3 , y3 ) ∗ (x2 , y2 ))) and (x3 , y3 ) ∈ U(αA1 × αA2 : s˜ ), L(βA1 × βA2 : ˜t ). This implies (αA1 × αA2 )((x1 , y1 ) ∗ ((x3 , y3 ) ∗ (x2 , y2 ))) ≥ s˜ , (αA1 × αA2 )(x3 , y3 ) ≥ s˜ , (βA1 × βA2 )((x1 , y1 ) ∗ ((x3 , y3 ) ∗ (x2 , y2 ))) ≤ ˜t and (βA1 × βA2 )(x3 , y3 ) ≤ ˜t . Now, (αA1 × αA2 )((x1 , y1 ) ∗ (x2 , y2 )) ≥ T {(αA1 × αA2 )((x1 , y1 ) ∗ ((x3 , y3 ) ∗ (x2 , y2 ))), (αA1 × αA2 )(x3 , and (βA1 × βA2 )((x1 , y1 ) ∗ y3 )} ≥ T {˜s, s˜ } ≥ s˜ (x2 , y2 )) ≤ S{(βA1 × βA2 )((x1 , y1 ) ∗ ((x3 , y3 ) ∗ This (x2 , y2 ))), (βA1 × βA2 )(x3 , y3 )} ≤ S{˜t , ˜t } ≤ ˜t . implies that (x1 , y1 ) ∗ (x2 , y2 ) ∈ U(αA1 × αA2 : s˜ ) and L(βA1 × βA2 : ˜t ). Hence, U(αA1 × αA2 : s˜ ) and L(βA1 × βA2 : ˜t ) are KU-ideals of X1 × X2 . Conversely, assume that U(αA1 × αA2 : s˜ ) and L(βA1 × βA2 : ˜t ) are KU-ideals of X1 × X2 . Let (x1 , y1 ) ∈ X1 × X2 be such that (αA1 × αA2 )(0, 0) < (αA1 × αA2 )(x1 , y1 ) and (βA1 × βA2 )(0, 0) > (βA1 × βA2 )(x1 , y1 ). By putting

Definition 5.12. Let A1 × A2 be an IFS of X1 × X2 and s˜ , ˜t ∈ [0, 1], then the set U(αA1 × αA2 : s˜ ) = {(x1 , y1 ) | (x1 , y1 ) ∈ X1 × X2 & (αA1 × αA2 )(x1 , y1 ) ≥ s˜ } is called an upper s˜ -level of A1 × A2 and L(βA1 × βA2 : ˜t ) = {(x1 , y1 ) | (x1 , y1 ) ∈ X1 × X2 & (βA1 × βA2 )(x1 , y1 ) ≤ ˜t } is called a lower ˜t -level of A1 × A2 .

we get (αA1 × αA2 )(0, 0) < s˜0 < (αA1 × αA2 )(x1 , y1 ) and (βA1 × βA2 )(0, 0) > t˜0 > (βA1 × βA2 )(x1 , y1 ). Therefore, (x1 , y1 ) ∈ U(αA1 × αA2 : s˜ ), L(βA1 × / U(αA1 × αA2 : s˜ ), L(βA1 × βA2 : βA2 : ˜t ) but (0, 0) ∈ ˜t ) which is a contradiction. Hence (αA1 × αA2 )(0, 0) ≥ (αA1 × αA2 )(x, y) and (βA1 × βA2 )(0, 0) ≤ (βA1 × βA2 )(x, y) for all (x, y) ∈ X1 × X2 . Again, we assume that (x1 , y1 ), (x2 , y2 ) and (x3 , y3 ) ∈ X1 × X2 be such that (αA1 × αA2 )((x1 , y1 ) ∗ (x3 , y3 )) < T {(αA1 × αA2 )((x1 , y1 ) ∗ ((x2 , y2 ) ∗ (x3 , y3 ))), (αA1 × αA2 )(x2 , y2 )} and (βA1 × βA2 )((x1 , y1 ) ∗ (x3 , y3 )) > S{(βA1 × βA2 )((x1 , y1 ) ∗ ((x2 , y2 ) ∗ (x3 , y3 ))), (βA1 × βA2 )(x2 , y2 )}. By putting

CO

PY

αA2 )((x3 , y3 ) ∗ (x2 , y2 )) ≥ T {(αA1 × αA2 )((x3 , y3 ) ∗ ((x1 , y1 ) ∗ (x2 , y2 ))), (αA1 × αA2 )(x1 , y1 )} and (βA1 × βA2 )((x3 , y3 ) ∗ (x2 , y2 )) ≤ S{(βA1 × βA2 )((x3 , y3 ) ∗ ((x1 , y1 ) ∗ (x2 , y2 ))), (βA1 × βA2 )(x1 , y1 )}. By putting (x3 , y3 ) = (0, 0) in above, we get (αA1 × αA2 )((0, 0) ∗ (x2 , y2 )) = (αA1 × αA2 )(x2 , y2 ) ≥ T {(αA1 × αA2 )((0, 0) ∗ ((x1 , y1 ) ∗ (x2 , y2 ))), (αA1 × αA2 )(x1 , y1 )} = T {(αA1 × αA2 )((x1 , y1 ) ∗ (x2 , y2 )), (αA1 × αA2 )(x1 , y1 )} ≥ T {T {(αA1 × αA2 )((x1 , y1 ) ∗ ((x3 , y3 ) ∗ (x2 , y2 ))), (αA1 × αA2 )(x3 , y3 )}, (αA1 × αA2 )(x1 , y1 )} = T {T {(αA1 × αA2 )(0, 0), (αA1 × αA2 )(x3 , y3 )}, (αA1 × αA2 )(x1 , y1 )} ≥ T {(αA1 × αA2 )(x3 , y3 ), (αA1 × αA2 )(x1 , y1 )} = T {(αA1 × αA2 )(x1 , y1 ), (αA1 × αA2 )(x3 , y3 )} and (βA1 × βA2 ) ((0, 0) ∗ (x2 , y2 )) = (βA1 × βA2 )(x2 , y2 ) ≤ S{(βA1 × βA2 )((0, 0) ∗ ((x1 , y1 ) ∗ (x2 , y2 ))), (βA1 × βA2 )(x1 , y1 )} = S{(βA1 × βA2 )((x1 , y1 ) ∗ (x2 , y2 )), (βA1 × βA2 )(x1 , y1 )} ≤ S{S{(βA1 × βA2 )((x1 , y1 ) ∗ ((x3 , y3 ) ∗(x2 , y2 ))), (βA1 × βA2 )(x3 , y3 )}, (βA1 × βA2 )(x1 , y1 ) } = S{S{(βA1 × βA2 )(0, 0), (βA1 × βA2 )(x3 , y3 )}, (βA1 × βA2 )(x1 , y1 )} ≤ S{(βA1 × βA2 )(x3 , y3 ), (βA1 ×βA2 )(x1 , y1 )} = S{(βA1 × βA2 )(x1 , y1 ), (βA1 × Therefore, (αA1 × αA2 )(x2 , y2 ) ≥ βA2 )(x3 , y3 )}. T {(αA1 × αA2 )(x1 , y1 ), (αA1 × αA2 )(x3 , y3 )} and (βA1 × βA2 )(x2 , y2 ) ≤ S{(βA1 × βA2 )(x1 , y1 ), (βA1 × βA2 )(x3 , y3 )}.

1 [(αA1 × αA2 )(0, 0) + (αA1 × αA2 )(x1 , y1 )] 2 1 t˜0 = [(βA1 × βA2 )(0, 0) + (βA1 × βA2 )(x1 , y1 )] 2

AU

TH

OR

s˜0 =

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 KUideal of X1 × X2 . Theorem 5.13. Let A1 × A2 be an IFS of X1 × X2 . Then A1 × A2 is an intuitionistic fuzzy bi-normed KUideal of X1 × X2 if and only if for any s˜ , ˜t ∈ [0, 1], the sets U(αA1 × αA2 : s˜ ) and L(βA1 × βA2 : ˜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(αA1 × / φ and (x, y) ∈ U(αA1 × αA2 : s˜ ), L(βA1 × βA2 : ˜t ) = αA2 : s˜ ), L(βA1 × βA2 : ˜t ). Then, we have (αA1 × αA2 )(x, y) ≥ s˜ and (βA1 × βA2 )(x, y) ≤ ˜t , and so we

1 [(αA1 × αA2 )((x1 , y1 ) ∗ (x3 , y3 )) 2 +T {(αA1 × αA2 )((x1 , y1 ) ∗ ((x2 , y2 ) ∗ (x3 , y3 ))),

s˜1 =

(αA1 × αA2 )(x2 , y2 )}], 1 [(βA1 × βA2 )((x1 , y1 ) ∗ (x3 , y3 )) 2 +S{(βA1 × βA2 )((x1 , y1 ) ∗ ((x2 , y2 ) ∗ (x3 , y3 ))),

t˜1 =

(βA1 × βA2 )(x2 , y2 )}]


Acknowledgments

PY

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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OR

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

(i) to find the interval-valued intuitionistic fuzzy bi-normed KU-subalgebras of KU-algebras, (ii) to find the interval-valued intuitionistic fuzzy bi-normed KU-ideals of KU-algebras.

CO

Then, we get (αA1 × αA2 )((x1 , y1 ) ∗ (x3 , y3 )) < s˜1 < T {(αA1 × αA2 )((x1 , y1 ) ∗ ((x2 , y2 ) ∗ (x3 , y3 ))), (αA1 × αA2 )(x2 , y2 )} and (βA1 × βA2 )((x1 , y1 ) ∗ (x3 , y3 )) > t˜1 > S{(βA1 × βA2 )((x1 , y1 ) ∗ ((x2 , y2 ) ∗ (x3 , y3 ))), (βA1 × βA2 )(x2 , y2 )}. This shows that (x1 , y1 ) ∗ ((x2 , y2 ) ∗ (x3 , y3 )) ∈ U(αA1 × αA2 : s˜ ), L(βA1 × βA2 : ˜t ) and ((x2 , y2 ) ∈ U(αA1 × αA2 : s˜ ), L(βA1 × βA2 : ˜t ) but (x1 , y1 ) ∗ / U(αA1 × αA2 : s˜ ), L(βA1 × βA2 : ˜t ) which (x3 , y3 ) ∈ is a contradiction. Therefore, (αA1 × αA2 )((x1 , y1 ) ∗ (x3 , y3 )) ≥ T {(αA1 × αA2 )((x1 , y1 ) ∗ ((x2 , y2 ) ∗ (x3 , y3 ))), (αA1 × αA2 )(x2 , y2 )} and (βA1 × βA2 )((x1 , y1 ) ∗ (x3 , y3 )) ≤ S{(βA1 × βA2 )((x1 , y1 ) ∗ ((x2 , y2 ) ∗ (x3 , y3 ))), (βA1 × βA2 )(x2 , y2 )}. Hence, A1 × A2 is an intuitionistic fuzzy bi-normed KU-ideal of X1 × X2 .

6. Conclusions and future work

AU

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.

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 KUideals 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-normed KU-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.

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T. Senapati and K.P. Shum / Atanassov’s intuitionistic fuzzy bi-normed KU-ideals of a KU-algebra [25] T. Senapati, M. Bhowmik and M. Pal, Fuzzy dot subalgebras and fuzzy dot ideals of B-algebras, Journal of Uncertain Systems 8(1) (2014), 22–30. [26] T. Senapati, Bipolar fuzzy structure of BG-subalgebras, The Journal of Fuzzy Mathematics 23(1) (2015), 209–220. [27] T. Senapati, M. Bhowmik and M. Pal, Interval-valued intuitionistic fuzzy BG-subalgebras, The Journal of Fuzzy Mathematics 20(3) (2012), 707–720. [28] T. Senapati, M. Bhowmik and M. Pal, Fuzzy dot structure of BG-algebras, Fuzzy Inf Eng 6(3) (2014), 315–329. [29] T. Senapati, T-fuzzy KU-subalgebras of KU-algebras, Ann Fuzzy Math Inform 10(2) (2015), 261–270. [30] T. Senapati, T-fuzzy KU-ideals of KU-algebras, Submitted. [31] T. Senapat and K.P. Shum, Atanassov’s intuitionistic fuzzy binormed KU-subalgebras of KU-algebras, Submitted. [32] N. Yaqoob, S.M. Mostafa and M.A. Ansari, On cubic KU-ideals of KU-algebras, ISRN Algebra Article ID 935905 (2013), 10. [33] L.A. Zadeh, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

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[18] C. Prabpyak and U. Leerawat, On ideals and congruences in KU-algebras, Scientia Magna 5(1) (2009), 54–57. [19] C. Prabpayak and U. Leerawat, On isimorphism of KUalgebras, Scientia Magna 5(3) (2009), 25–31. [20] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific Journal of Mathematics 10 (1960), 313–334. [21] T. Senapati, M. Bhowmik, M. Pal and B. Davvaz, Fuzzy translations of fuzzy H-ideals in BCK=BCI-algebras, Journal of the Indonesian Mathematical Society 21(1) (2015), 45–58. [22] T. Senapati, M. Bhowmik and M. Pal, Atanassov’s intuitionistic fuzzy translations of intuitionistic fuzzy H-ideals in BCK/BCIalgebras, Notes on Intuitionistic Fuzzy Sets 19(1) (2013), 32–47. [23] T. Senapati, C. Jana, M. Bhowmik and M. Pal, L-fuzzy G-subalgebras of G-algebras, Journal of the Egyptian Mathematical Society 23(2) (2015), 219–223. [24] T. Senapati, C.S. Kim, M. Bhowmik and M. Pal, Cubic subalgebras and cubic closed ideals of B-algebras, Fuzzy Information and Engineering 7(2) (2015), 129–149.

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