An intuitionistic fuzzy magnified translation of KM ideals on K-Algebra

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Fuzziness is found in engineering, medicine, manufacturing and other areas in our daily life. Recently, Fuzziness is applied to find the solution of human decision, reasoning and learning. The concept of intuitionistic fuzzification of subalgebra and KM ideals of the K-algebra are considered for testing the satisfaction of the properties.

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International Journal of Mechanical Engineering and Technology (IJMET) Volume 10, Issue 03, March 2019, pp. 1339-1350. Article ID: IJMET_10_03_135 Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=3 ISSN Print: 0976-6340 and ISSN Online: 0976-6359 © IAEME Publication Scopus Indexed AN INTUITIONISTIC FUZZY MAGNIFIED TRANSLATION OF KM IDEALS ON K- ALGEBRA S. Kailasavalli Assistant Professors, PSNA College of Engineering and Technology, Dindigul-624622, Tamilnadu, India. C.Yamini Assistant Professors, PSNA College of Engineering and Technology, Dindigul-624622, Tamilnadu, India. M.Meenakshi Assistant Professors, PSNA College of Engineering and Technology, Dindigul-624622, Tamilnadu, India. VE.Jayanthi Professor and Head, Department of Bio medical Engineering, PSNA College of Engineering and Technology, Dindigul-624622, Tamilnadu, India. ABSTRACT Fuzziness is found in engineering, medicine, manufacturing and other areas in our daily life. Recently, Fuzziness is applied to find the solution of human decision, reasoning and learning. The concept of intuitionistic fuzzification of subalgebra and KM ideals of the K-algebra are considered for testing the satisfaction of the properties. In addition to Intuitionistic fuzzy magnified translations to intuitionistic fuzzy KM ideals in K-algebra also introduced and their several related properties are tested and proofs are presented in this work. Keywords: K-algebra, fuzzy KM ideals, intuitionistic fuzzy KM ideal, intuitionistic fuzzy magnified translation of KM ideals. Cite this Article S. Kailasavalli, C.Yamini, M.Meenakshi and VE.Jayanthi, an Intuitionistic Fuzzy Magnified Translation of Km Ideals on K-Algebra, International Journal of Mechanical Engineering and Technology, 10(3), 2019, pp. 1339-1350. http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&IType=3 http://www.iaeme.com/IJMET/index.asp 1339 editor@iaeme.com
S. Kailasavalli, C.Yamini, M.Meenakshi and VE.Jayanthi 1. INTRODUCTION The fuzzy set theory domain is a wide range and its information is incomplete or inaccurate such as bioinformatics. Initially, the notion of the fuzzy sets and its functions are introduced by Zadeh[1]. Extension properties of BCK-Algebras are described and it is implicative [2].Intuitionistic fuzzy set (IFS) is a powerful tool to deal with imprecision. Assigning each element as a membership degree in IFS is the extension of Zadeh’s fuzzy set and it is successfully applied and tested by Atanassov[3]. IFS are applied in various fields in artificial intelligence system of intuitionistic fuzzy for expert systems; neural networks decision making, machine leaming and semantic representations. The concepts of fuzzy magnified translation on groups are introduced [4]. Intuitionistic fuzzy magnified translation notion and its various properties are discussed [5],[6], [7] and [8].The K-Algebras and BCI Algebras are introduced and the concepts are clearly discussed [9]. Fuzzy ideals of K Algebras are introduced and their properties are verified [10] and [11] and the extension of Fuzzy KM an ideal on K-algebras is also introduced and checked [12]. In this paper, Intuitionistic fuzzy and Intuitionistic fuzzy magnified translations basic concepts are presented in section 2. Intuitionistic fuzzy is defined in KM ideal and some theorems are verified in section 3 and its morphism properties proof are in section 4. The following section 5 consists of Intuitionistic fuzzy magnified translations in KM ideal are defined and their morphism properties are discussed in section 6. 2. PRELIMINARIES Basic concepts of fuzzy K-algebra and its KM ideal are defined and discussed in this section for obtaining the results of intuitionistic fuzzy and intuitionistic fuzzy magnified translations under homomorphism and epimorphism properties. Definition 2.1: Let (G, ·, e) be a group with the identity e such that x2 ≠ e for some x(≠ e) ∈ G. A K-algebra built on G (briefly, K-algebra) is a structure K = (G, ., ⨀, e) where “ ” is a binary operation on G which is induced from the operation “·”, that satisfies the following: (k1) (∀a, x, y ∈ G) ((a x) (a y) = (a (y-1 x-1))⨀a), (k2) (∀a, x ∈ G) (a (a x) = (a x-1) a), (k3) (∀a ∈ G) (a a = e), (k4) (∀a ∈ G) (a e = a), (k5) (∀a ∈ G) (e a = a-1). If G is abelian, then conditions (k1) and (k2) are replaced by: (k1’) (∀a, x, y ∈ G) ((a x) (a y) = y x), (k2’) (∀a, x ∈ G) (a (a x) = x), respectively. A nonempty subset H of a K-algebra K is called a subalgebra of K if it satisfies: • (∀a, b ∈ H) (a b ∈ H). Note that every subalgebra of a K-algebra K contains the identity e of the group (G, ·). Definition 2.2: A fuzzy set η in a k-algebra k is called a fuzzy ideal of k if it satisfies: (i) (∀x∈ 𝐺)(η(e)≥ η(x)) (ii) (∀x, 𝑦 ∈ 𝐺) (η(y)≥min{η(y ⨀x), η((x⨀(x ⨀ y))}) http://www.iaeme.com/IJMET/index.asp 1340 editor@iaeme.com
An Intuitionistic Fuzzy Magnified Translation of Km Ideals on K-Algebra Definition 2.3: An intuitionistic fuzzy set A = {: x∈ 𝑅} is called an intuitionistic fuzzy subalgebra X if it satisfies i. η𝐴 (x*y)≥ min{η𝐴 (x),η𝐴 (y)} and ii. ϑ𝐴 (x*y) ≤ max{ϑ𝐴 (x),ϑ𝐴 (y)} for all x,y∈ X. Definition 2.4: An intuitionistic fuzzy set A = {/x∈ 𝑋} in X is called an intuitionistic fuzzy ideal of X if it satisfies I. η𝐴 (0) ≥ η𝐴 (x), ϑ𝐴 (0) ≤ ϑ𝐴 (x) II. η𝐴 (x) ≥ min{η𝐴 (x*y), η𝐴 (y)} III. ϑ𝐴 (x) ≤ max{ϑ𝐴 (x*y),ϑ𝐴 (y)} for all x,y∈ X. 3. INTUITIONISTIC FUZZY KM-IDEAL: Intuitionistic Fuzzy is the Extensions of fuzzy is defined and its subdivion of KM ideals are described below. Definition 3.1: Let (X,⊙,0) be a K-algebra, if the following conditions are satisfied, for all x,y,z𝜖X I. η(0) ≥ η(x). II. 𝜇(𝑦 ⊙ 𝑧) ≥min{ 𝜇(𝑥 ⊙ 𝑦), 𝜇(𝑧 ⊙ 𝑥) } III. then, a fuzzy subset ηin x is called a fuzzy KM-Ideal of X Definition 3.2: An intuitionistic fuzzy set A = {/x∈ 𝑋} in K-algebra X is called an intuitionistic fuzzy KM-ideal of A if I. η𝐴 (0) ≥ η𝐴 (x) and ϑ𝐴 (0) ≥ ϑ𝐴 (x) II. η𝐴 (y⨀ z) ≥ min{η𝐴 (x⨀ y), η𝐴 (z⨀ x)} III. ϑ𝐴 (y⨀ z) ≤ max {ϑ𝐴 (x⨀ y), ϑ𝐴 (z⨀ x)} Theorem 3.3: Let A be a nonempty subset and intuitionistic fuzzy KM ideals of a k-algebra. Then so is A = η𝐴 and A = (x, ϑ𝐴 , ̅̅̅ (x, η𝐴 , ̅̅̅) 𝜗𝐴 ) Proof: We know that,η𝐴 (0) ≥ η𝐴 (x). This implies that 1- ̅̅̅ η𝐴 (0) ≥ 1- ̅̅̅ η𝐴 (x) η𝐴 (0) ≤ ̅̅̅ ̅̅̅ η𝐴 (x) for every a∈ 𝐴. Let us consider ∀ a, b, c ∈ 𝑋. Then η𝐴 (y⨀ z) ≥ min{η𝐴 (x⨀ y), η𝐴 (z⨀ x)} η𝐴 (y⨀ z) ≥ min {1- ̅̅̅ ⟹ 1- ̅̅̅ η𝐴 (x⨀ y), 1- ̅̅̅ η𝐴 (z⨀ x)} ⟹η ̅̅̅ 𝐴 (y⨀ z) ≤ 1- {η ̅̅̅ 𝐴 η𝐴 (z⨀ x)} (x⨀ y), ̅̅̅ ⟹η ̅̅̅ 𝐴 (y⨀ z) ≤ max {η ̅̅̅ 𝐴 η𝐴 (z⨀ x)} (x⨀ y), ̅̅̅ Hence A = (x, η𝐴 , ̅̅̅) η𝐴 is an intuitionistic fuzzy KM ideal of X. http://www.iaeme.com/IJMET/index.asp 1341 editor@iaeme.com
S. Kailasavalli, C.Yamini, M.Meenakshi and VE.Jayanthi To prove the second part, Let A be an intuitionistic fuzzy KM ideal of a K-algebra X. Nowϑ𝐴 (0) ≤ ϑ𝐴 (x) ⟹ 1- ̅̅̅ ϑ𝐴 (0) ≤ 1- ̅̅̅ ϑ𝐴 (x) ⟹ϑ̅̅̅ ̅̅̅ 𝐴 (0) ≥ ϑ𝐴 (x) ∀x∈ X. Now consider ∀ a, b, c ∈ A. Then we have ϑ𝐴 (y⨀ z) ≤ max {ϑ𝐴 (x⨀ y), ϑ𝐴 (z⨀ x)} ⟹ 1- ̅̅̅ ϑ𝐴 (y⨀ z) ≤ max { 1- ̅̅̅ ϑ𝐴 (x⨀ y), 1- ̅̅̅ ϑ𝐴 (z⨀ x)} ⟹ϑ̅̅̅ ̅̅̅ ̅̅̅ 𝐴 (y⨀ z) ≥ 1- max { ϑ𝐴 (x⨀ y), ϑ𝐴 (z⨀ x)} ⟹ϑ̅̅̅ ̅̅̅ ̅̅̅ 𝐴 (y⨀ z) ≥ min {ϑ𝐴 (x⨀ y), ϑ𝐴 (z⨀ x)} Hence A = (x, ϑ𝐴 , ̅̅̅ ϑ𝐴 ) is an intuitionistic fuzzy KM ideal of X. Definition 3.4 An ideal A of a KM ideal is said to be closed if 0*x ∈ X, ∀ x ∈ X. Theorem3.5: Let A be a nonempty set and an intuitionistic fuzzy closed KM-ideal of a K-algebra A. Then A η𝐴 and A = (x, ϑ𝐴 , ̅̅̅ = (x, η𝐴 , ̅̅̅) ϑ𝐴 ) is also a closed intuitionisticfuzzy closed KM-ideal. Proof: First we prove A = (x, η𝐴 , ̅̅̅) η𝐴 is a intuitionistic closed fuzzy KM-ideal . For all x ∈ X we have η𝐴 (0*x) ≤ η𝐴 (x)⟹ 1- ̅̅̅(0*x) η𝐴 ≤ 1-η ̅̅̅(x)⟹η 𝐴 ̅̅̅(0*x) 𝐴 η𝐴 ≥ ̅̅̅(x) Therefore A = (x, η𝐴 , ̅̅̅) η𝐴 is a intuitionistic closed KM-ideal of A. To prove the second part, let x ∈ X, then we have ϑ𝐴 (0*x) ≥ ϑ𝐴 (x) ⟹ 1- ̅̅̅ϑ𝐴 (0*x) ≥ 1- ̅̅̅ ϑ𝐴 (x) ⟹ϑ̅̅̅ ̅̅̅ 𝐴 (0*x) ≤ ϑ𝐴 (x) ∀ a, b, c ∈A. ̅̅̅ Hence A = (x, ϑ𝐴 , ϑ𝐴 ) is an intuitionistic fuzzy closed KM-ideal of A. Lemma 3.6: Let A= (η𝐴 , ϑ𝐴 ) be an intuitionistic fuzzy KM ideal of X. If y ≤ x in X, then η𝐴 (y) ≥ η𝐴 (x), ϑ𝐴 (y) ≤ ϑ𝐴 (x) that is η𝐴 is order-reserving and ϑ𝐴 is order-preserving. PROOF Let x, y ∈ X be such that y ≤ x. Then y⨀ x = 0 and so η𝐴 (y) = η𝐴 (0⨀ y) ≥ min (η𝐴 (x⨀ 0), η𝐴 (y⨀ x)) ≥ min (η𝐴 (x), η𝐴 (0)) = η𝐴 (x) Hence η𝐴 (y) ≥ η𝐴 (x) Similarly ϑ𝐴 (y) = ϑ𝐴 (0⨀ y) ≤ max (ϑ𝐴 (x⨀ 0), ϑ𝐴 (y⨀ x)) http://www.iaeme.com/IJMET/index.asp 1342 editor@iaeme.com
An Intuitionistic Fuzzy Magnified Translation of Km Ideals on K-Algebra ≤ max (ϑ𝐴 (x), ϑ𝐴 (0))= ϑ𝐴 (x) Hence ϑ𝐴 (y) ≤ ϑ𝐴 (x) Lemma 3.7: Let an intuitionistic fuzzy set A= (η𝐴 , ϑ𝐴 ) in X be an intuitionistic fuzzy ideal of X. If the inequality (x⨀ y) ≤ z holds in X, then η𝐴 (y⨀ z) ≥ min{η𝐴 (x⨀ y), η𝐴 (z⨀ x)} ϑ𝐴 (y⨀ z) ≤ max {ϑ𝐴 (x⨀ y), ϑ𝐴 (z⨀ x)} Proof: Let x, y, z ∈ X be such that x⨀ y ≤ z. Then (x⨀ y) ⨀ z = 0 and thus η𝐴 (y⨀ z) ≥ min{η𝐴 (x⨀ y), η𝐴 (z⨀ x)} ≥ min {min { η𝐴 ((x⨀ y) ⨀ z), η𝐴 (z⨀ x)}, η𝐴 (x⨀ y)} = min {min {η𝐴 (0), η𝐴 (z⨀ x), η𝐴 (x⨀ y)} = min {η𝐴 (x⨀ y), η𝐴 (z⨀ x)} ϑ𝐴 (y⨀ z) ≤ max {ϑ𝐴 (x⨀ y), ϑ𝐴 (z⨀ x)} ≤ max {max{ϑ𝐴 ((x⨀ y) ⨀ z), ϑ𝐴 (z⨀ x), ϑ𝐴 (x⨀ y)} = max {max{ϑ𝐴 (0), ϑ𝐴 (z⨀ x)}, ϑ𝐴 (x⨀ y)} = max {ϑ𝐴 (z⨀ x), ϑ𝐴 (x⨀ y)} = max {ϑ𝐴 (x⨀ y), ϑ𝐴 (z⨀ x)} Hence the proof. Theorem 3.8: An intuitionistic fuzzy set A= (η𝐴 , ϑ𝐴 ) is an intuitionistic fuzzy KM ideal of X if and only if the fuzzy set η𝐴 and ̅̅̅ ϑ𝐴 are fuzzy KM ideal of X. Proof: Let A= (η𝐴 , ϑ𝐴 ) be an intuitionistic fuzzy ideal of X. Clearly, η𝐴 is a fuzzy KM ideal of X. For every x, y, z ∈X we have ̅̅̅ ϑ𝐴 (0) = 1- ϑ𝐴 (0) ≥ 1- ϑ𝐴 (x) = ̅̅̅ ϑ𝐴 (x). ̅̅̅ ϑ𝐴 (y⨀ z) = 1- ϑ𝐴 (x) ≥ 1- max{ϑ𝐴 (x⨀ y),ϑ𝐴 (z⨀ x) } = min {1- ϑ𝐴 (x⨀ y), 1-ϑ𝐴 (z⨀ x)} = min { ̅̅̅ ϑ𝐴 (x⨀ y), ̅̅̅ ϑ𝐴 (z⨀ x)} Hence ̅̅̅ ϑ𝐴 is a fuzzy KM ideal of X. Conversely, assume that η𝐴 and ̅̅̅ ϑ𝐴 are fuzzy KM ideal of X. For every x, y, z ∈X we get η (0) ≥ η (x), 1- ϑ𝐴 (0) = ̅̅̅ 𝐴 𝐴 ϑ𝐴 (0) ≥ ̅̅̅ ϑ𝐴 (x) = 1- ϑ𝐴 (x) That is, ϑ𝐴 (0) ≤ ϑ𝐴 (x), η𝐴 (y⨀ z) ≥ min{η𝐴 (x⨀ y), η𝐴 (z⨀ x)} http://www.iaeme.com/IJMET/index.asp 1343 editor@iaeme.com
S. Kailasavalli, C.Yamini, M.Meenakshi and VE.Jayanthi 1 − ϑ𝐴 (y⨀ z) = ϑ̅̅̅ ̅̅̅ ̅̅̅ 𝐴 (y⨀ z) ≥ min { ϑ𝐴 (x⨀ y), ϑ𝐴 (z⨀ x)} ≥ min {1- ϑ𝐴 (x⨀ y), 1-ϑ𝐴 (z⨀ x)} = 1- max{ϑ𝐴 (x⨀ y),ϑ𝐴 (z⨀ x) } That is, ϑ𝐴 (y⨀ z) ≤max{ϑ𝐴 (x⨀ y),ϑ𝐴 (z⨀ x) } Hence A= (η𝐴 , ϑ𝐴 ) is an intuitionistic fuzzy ideal of X. 4. INTUITIONISTIC FUZZY KM IDEAL ON HOMOMORPHISM Definition 4.1: A mapping f: X → Y be a mapping of K-algebras is called a homomorphism if f(x⨀ y) = f(x) ⨀ f(y) ∀ x, y ∈ X. Definition 4.2: A mapping f : X → Y be a mapping of K-algebras and α𝐴 be a fuzzy set of Y. The map α𝐴 f is the pre-image of α𝐴 under f if α𝐴 f (x) = α𝐴 (f(x)). Definition 4.3: If f: X → Y is a homomorphism of K-algebras, then f(0) = 0. Let f: X → Y be a homomorphism of K- algebra for any intuitionistic fuzzy set A= (η𝐴 , ϑ𝐴 ) in Y, we define new intuitionistic fuzzy setAf = (η𝐴 f, ϑ𝐴 f) in X by η𝐴 f (x) = η𝐴 (f(x)) ϑ𝐴 f (x) = ϑ𝐴 (f(x)) ∀ x ∈ X. Theorem 4.4: Let f: X → Y be a homomorphism of K-algebras. If an intuitionistic fuzzy set A= (η𝐴 , ϑ𝐴 ) in Y is an intuitionistic fuzzy KM ideal of Y then an intuitionistic fuzzy set Af = (η𝐴 f, ϑ𝐴 f) in X is an intuitionistic fuzzy KM ideal of X. Proof: We first prove that η𝐴 f (x) = η𝐴 (f(x)) ≤ η𝐴 (0) = η𝐴 𝑓(0)= η𝐴 f (0) Similarly ϑ𝐴 f (x) = ϑ𝐴 (f(x)) ≥ ϑ𝐴 (0)= ϑ𝐴 𝑓(0)= ϑ𝐴 f (0) ∀ x ∈ X. Let x, y,z∈ X. Then min{η𝐴 f (x⨀ y), η𝐴 f (z⨀ x)} = min{ η𝐴 (𝑓(x⨀ y)), η𝐴 (𝑓(z⨀ x))} = min{ η𝐴 (f(x) ⨀ f(y)), η𝐴 (f(z) ⨀ f(x)} ≤ η𝐴 (f(y) ⨀ f(z)) ≤ η𝐴 f(y ⨀ z) = η𝐴 f (y⨀ z) max{ ϑ𝐴 f (x⨀ y), ϑ𝐴 f (z⨀ x)} = max{ ϑ𝐴 (𝑓(x⨀ y)), ϑ𝐴 (𝑓(z⨀ x))} = max{ ϑ𝐴 (f(x) ⨀ f(y)), ϑ𝐴 (f(z) ⨀ f(x)} ≥ ϑ𝐴 (f(y) ⨀ f(z)) ≥ ϑ𝐴 f(y ⨀ z) = ϑ𝐴 f (y⨀ z) Hence Af = (η𝐴 f, ϑ𝐴 f) is an intuitionistic fuzzy KM ideal of X. http://www.iaeme.com/IJMET/index.asp 1344 editor@iaeme.com
An Intuitionistic Fuzzy Magnified Translation of Km Ideals on K-Algebra Theorem 4.5: Let f: X → Y be an epimorphism of K-algebras and let A= (η𝐴 , ϑ𝐴 ) be an intuitionistic fuzzy set in Y. If Af = (η𝐴 f, ϑ𝐴 f) is an intuitionistic fuzzy KM ideal of X, then A= (η𝐴 , ϑ𝐴 ) is an intuitionistic fuzzy KM ideal of Y. Proof: For any x ∈ Y, there exists a ∈ X such that f(a) = x. Then η𝐴 (x) = η𝐴 (f(a)) = η𝐴 f (a) ≤ η𝐴 f (0)= η𝐴 (f(0)) = η𝐴 (0) ϑ𝐴 (x) = ϑ𝐴 (f(a)) = ϑ𝐴 f (a) ≥ ϑ𝐴 f (0)= ϑ𝐴 (f(0)) = ϑ𝐴 (0) Let x, y, z ∈ Y. Then f(a) = x and f(b) = y and f(c) = z for some a, b, c ∈ X. It follows that η𝐴 (y⨀z) = η𝐴 (f(b) ⨀ f(c)) = η𝐴 (f(b⨀ c)) = η𝐴 f (b⨀ c) ≥ min{ η𝐴 f (a⨀ b), η𝐴 f (c⨀ a)} ≥ min{ η𝐴 (f(a⨀ b), η𝐴 (f(c⨀ a))} = min{η𝐴 (f(a) ⨀ f(b)), η𝐴 (f(c) ⨀ f(a)} = min{η𝐴 (x⨀ y), η𝐴 (z⨀ x)} ϑ𝐴 (y ⨀z) =ϑ𝐴 ( f(b) ⨀ f(c)) = ϑ𝐴 f(b ⨀ c) = ϑ𝐴 f (b ⨀ c) ≤ max{ ϑ𝐴 f (a⨀ b), ϑ𝐴 f (c⨀ a)} ≤ max{ ϑ𝐴 (f(a⨀ b), ϑ𝐴 (f(c⨀ a))} = max{ϑ𝐴 (f(a) ⨀ f(b)), ϑ𝐴 (f(c) ⨀ f(a)} = max{ϑ𝐴 (x⨀ y), ϑ𝐴 (z⨀ x)} This completes the proof. 5. INTUITIONISTIC FUZZY MAGNIFIED TRANSLATION ON KM IDEAL Definition 5.1: Let A= (η𝐴 , ϑ𝐴 ) be an intuitionistic fuzzy subset of X and β∈ [0,1] and α ∈[0,1- sup{ η𝐴 (x) + ϑ𝐴 (x): x∈ 𝑋, 0
S. Kailasavalli, C.Yamini, M.Meenakshi and VE.Jayanthi Let T be an intuitionistic fuzzy KM ideal of X, then the intuitionistic fuzzy magnified translation of T is an intuitionistic fuzzy KM ideal of X and α ∈[0, T]. Proof: Let x, y, z ∈ X. Then assume that η ∈ X and α ∈[0, T]. Then η𝑇 (0) = βη𝐴 (0) + α≥ βη𝐴 (x) + α= η𝑇 (x) and η𝑇 (y⨀ z) = βη𝐴 (y⨀z) + α ≥ β min { η𝐴 (x⨀ y), η𝐴 (z⨀ x)} + α = min { βη𝐴 (x⨀ y) + α, βη𝐴 (z⨀ x) + α} = min (η𝑇 (x⨀ y), η𝑇 (z⨀ x)} for all x, y, z ∈ X. Also, ϑ 𝑇 (0) = βϑ𝐴 (0) + α ≤ βϑ𝐴 (x) + α=ϑ 𝑇 (x) Andϑ 𝑇 (y⨀ z) = βϑ𝐴 (y⨀ z) + α ≤ β max { ϑ𝐴 (x⨀ y), ϑ𝐴 (z⨀ x)} + α = max{ βϑ𝐴 (x⨀ y) + α, βϑ𝐴 (z⨀ x) + α} = max (ϑ 𝑇 (x⨀ y), ϑ 𝑇 (z⨀ x)} for all x, y, z ∈ X. Therefore, T is an intuitionistic fuzzy magnified translation of KM ideal of X. Theorem 5.3: Let Abe an intuitionistic fuzzy subset of K – algebra such that the intuitionistic fuzzy magnified 𝛼 translation T of A is an intuitionistic fuzzy KM ideal of X for some α ∈[0, T]. Then A is an intuitionistic fuzzy KM ideal of K- algebra on X. Proof: Assume that η𝑇(β,α) is a intuitionistic fuzzy KM ideal of X for some α ∈[0, T]. Let x, y, z ∈ X. Then βη𝐴 (0) + α = η𝑇(β,α) (0) ≥ η𝑇(β,α) (𝑥) = βη𝐴 (x) + α and so η(0)≥ η(x). Now, we have βη𝐴 (y⨀ x) + α = η𝑇(β,α) (y⨀ x) ≥ min {η𝑇(β,α) (x⨀ y), η𝑇(β,α) (z⨀ x)} + α = min { βη𝐴 (x⨀ y) + α, βη𝐴 (z⨀ x) + α} = min { βη𝐴 (x⨀ y) , βη𝐴 (z⨀ x)} + α = β min {η𝐴 (x⨀ y) ,η𝐴 (z⨀ x)} + α which implies that η𝐴 (y⨀ x) ≥ min {η𝐴 (x⨀ y) ,η𝐴 (z⨀ x)} and βϑ𝐴 (0) + α = ϑ𝑇(β,α) (0) ≤ ϑ𝑇(β,α) (𝑥) = βϑ𝐴 (x) + α and so ϑ (0)≤ ϑ(x). βϑ𝐴 (y⨀ z) + α = ϑ𝑇(β,α) (y⨀ z) ≤ max {ϑ𝑇(β,α) (x⨀ y), ϑ𝑇(β,α) (z⨀ x)} + α = max { βϑ𝐴 (x⨀ y) + α, βϑ𝐴 (z⨀ x) + α} = max { βϑ𝐴 (x⨀ y) , βϑ𝐴 (z⨀ x)} + α = β max {ϑ𝐴 (x⨀ y) ,ϑ𝐴 (z⨀ x)} + α which implies that ϑ𝐴 (y⨀ z) ≤ max {ϑ𝐴 (x⨀ y) ,ϑ𝐴 (z⨀ x)}. http://www.iaeme.com/IJMET/index.asp 1346 editor@iaeme.com
An Intuitionistic Fuzzy Magnified Translation of Km Ideals on K-Algebra Hence A is an intuitionistic fuzzy KM ideal of K- algebra. Lemma 5.4: If 𝐴𝑇(𝛽,𝛼) = (𝜇(𝛽,𝛼) 𝐴 𝐴 , 𝜗(𝛽,𝛼) ) is intuitionistic fuzzy magnified translation KM ideal and y≤ 𝑥 in 𝐴 𝐴 𝐴 𝐴 𝐴 X, then 𝜇(𝛽,𝛼) (𝑦) ≥ 𝜇(𝛽,𝛼) (𝑥), 𝜗(𝛽,𝛼) (𝑦) ≤ 𝜗(𝛽,𝛼) (𝑥) . That is 𝜇(𝛽,𝛼) is order reserving and 𝐴 𝜗(𝛽,𝛼) is order preserving. Proof: Let x,y∈ 𝑋 be such that y≤ 𝑥, by lemma (3.6) 𝜇𝐴 (𝑦) ≥ 𝜗𝐴 (𝑥) and 𝐴 𝐴 𝜇(𝛽,𝛼) (𝑦) = 𝛽𝜇𝐴 (𝑦) +𝛼 ≥ 𝛽𝜇𝐴 (𝑥) +𝛼 = 𝜇(𝛽,𝛼) (𝑥) Also 𝐴 𝐴 𝜗(𝛽,𝛼) (𝑦) = 𝛽𝜗𝐴 (𝑦) +𝛼 ≤ 𝛽𝜗𝐴 (𝑥) +𝛼= 𝜗(𝛽,𝛼) (𝑥) Hence the proof. Lemma 5.5: If 𝐴𝑇(𝛽,𝛼) = (𝜇(𝛽,𝛼) 𝐴 𝐴 , 𝜗(𝛽,𝛼) ) is intuitionistic fuzzy magnified translation KM ideal and the 𝐴 𝐴 𝐴 𝐴 inequality 𝑥⨀𝑦 ≤ 𝑧 hold in X, then 𝜇(𝛽,𝛼) (𝑥) ≥ min(𝜇(𝛽,𝛼) (𝑦), 𝜇(𝛽,𝛼) (𝑧)) and 𝜗(𝛽,𝛼) (𝑥) ≤ 𝐴 𝐴 max(𝜗(𝛽,𝛼) (𝑦), 𝜗(𝛽,𝛼) (𝑧)). Proof: Let x,y,z∈ 𝑋 be such that 𝑥⨀𝑦 ≤ 𝑧. Thus 𝐴 𝜇(𝛽,𝛼) (𝑥) = 𝛽𝜇𝐴 (𝑦) +𝛼 ≥ 𝛽(min(𝜇𝐴 (𝑦) , 𝜇𝐴 (𝑧))+𝛼 = 𝑚𝑖𝑛(𝛽𝜇𝐴 (𝑦) + 𝛼 , 𝛽𝜇𝐴 (𝑧)+𝛼) 𝐴 𝐴 = min(𝜇(𝛽,𝛼) (𝑦), 𝜇(𝛽,𝛼) (𝑧)) Also 𝐴 𝜗(𝛽,𝛼) (𝑥) = 𝛽𝜗𝐴 (𝑦) +𝛼 ≤ 𝛽(max(𝜗𝐴 (𝑦) , 𝜗𝐴 (𝑧))+𝛼 = 𝑚𝑎𝑥(𝛽𝜗𝐴 (𝑦) + 𝛼 , 𝛽𝜗𝐴 (𝑧)+𝛼) 𝐴 𝐴 = max(𝜗(𝛽,𝛼) (𝑦), 𝜗(𝛽,𝛼) (𝑧)). Hence the proof. 6. INTUITIONISTIC FUZZY MAGNIFIED TRANSLATION ON KM IDEAL ON HOMOMORPHISM http://www.iaeme.com/IJMET/index.asp 1347 editor@iaeme.com
S. Kailasavalli, C.Yamini, M.Meenakshi and VE.Jayanthi Definition 6.1: Let f: X→ 𝑌 be a homomorphism of K-Algebra, for any intuitionistic fuzzy magnified 𝑓 translation T of A in Y. We define intuitionistic fuzzy magnified Translation 𝐴(𝛽,𝛼) = 𝑓 𝑓 (𝜇(𝛽,𝛼) , 𝜗(𝛽,𝛼) ) in X by 𝑓 𝑓 𝜇(𝛽,𝛼) (𝑥) = 𝜇(𝛽,𝛼) 𝑓(𝑥) and 𝜗(𝛽,𝛼) (𝑥) = 𝜗(𝛽,𝛼) 𝑓(𝑥). Theorem 6.2: Let f: X→ 𝑌be a homomorphism of K-Algebra. If an intuitionistic fuzzy set 𝐴(𝛽,𝛼) = 𝑓 (𝜇(𝛽,𝛼) , 𝜗(𝛽,𝛼) ) in Y is an intuitionistic fuzzy magnified translation KM-Ideal of Y then 𝐴(𝛽,𝛼) = 𝑓 𝑓 (𝜇(𝛽,𝛼) , 𝜗(𝛽,𝛼) ) is an intuitionistic fuzzy magnified translation KM-Ideal of X. Proof: For all x,y,z∈ 𝑋, we have 𝑓 𝑓 𝜇(𝛽,𝛼) (𝑥) = 𝜇(𝛽,𝛼) 𝑓(𝑥) ≤ 𝜇(𝛽,𝛼) (0) ≤ 𝜇(𝛽,𝛼) 𝑓(0) = 𝜇(𝛽,𝛼) (0) And 𝑓 𝑓 𝜗(𝛽,𝛼) (𝑥) = 𝜗(𝛽,𝛼) 𝑓(𝑥) ≥ 𝜗(𝛽,𝛼) (0) ≥ 𝜗(𝛽,𝛼) 𝑓(0) = 𝜗(𝛽,𝛼) (0) Now, 𝑓 𝜇(𝛽,𝛼) (𝑦 ⊙ 𝑧) = 𝜇(𝛽,𝛼) 𝑓(𝑦 ⊙ 𝑧) = 𝜇(𝛽,𝛼) (𝑓(𝑦) ⊙ 𝑓(𝑧)) ≥ min{𝜇(𝛽,𝛼) (𝑓(𝑥) ⊙ 𝑓(𝑦)), 𝜇(𝛽,𝛼) (𝑓(𝑧) ⊙ 𝑓(𝑥))} = min{𝜇(𝛽,𝛼) (𝑓(𝑥 ⊙ 𝑦)), 𝜇(𝛽,𝛼) (𝑓(𝑧 ⊙ 𝑥))} 𝑓 𝑓 = 𝑚𝑖𝑛 {𝜇(𝛽,𝛼) (𝑥 ⊙ 𝑦), 𝜇(𝛽,𝛼) (𝑧 ⊙ 𝑥)} 𝑓 𝜗(𝛽,𝛼) (𝑦 ⊙ 𝑧) = 𝜗(𝛽,𝛼) 𝑓(𝑦 ⊙ 𝑧) = 𝜗(𝛽,𝛼) (𝑓(𝑦) ⊙ 𝑓(𝑧)) ≤ max{𝜗(𝛽,𝛼) (𝑓(𝑥) ⊙ 𝑓(𝑦)), 𝜗(𝛽,𝛼) (𝑓(𝑧) ⊙ 𝑓(𝑥))} = max{𝜗(𝛽,𝛼) (𝑓(𝑥 ⊙ 𝑦)), 𝜗(𝛽,𝛼) (𝑓(𝑧 ⊙ 𝑥))} 𝑓 𝑓 = 𝑚𝑎𝑥 {𝜗(𝛽,𝛼) (𝑥 ⊙ 𝑦), 𝜗(𝛽,𝛼) (𝑧 ⊙ 𝑥)} 𝑓 𝑓 𝑓 Hence 𝐴(𝛽,𝛼) = (𝜇(𝛽,𝛼) , 𝜗(𝛽,𝛼) ) is an intuitionistic fuzzy magnified translation KM-Ideal of X. Theorem 6.3: Let f: X→ 𝑌be an epimorphism of K-Algebra. If intuitionistic fuzzy set 𝐴(𝛽,𝛼) = 𝑓 𝑓 𝑓 (𝜇(𝛽,𝛼) , 𝜗(𝛽,𝛼) )a intuitionistic fuzzy magnified translation in Y. If 𝐴(𝛽,𝛼) = (𝜇(𝛽,𝛼) , 𝜗(𝛽,𝛼) ) intuitionistic fuzzy magnified translation KM-Ideal of X, then 𝐴(𝛽,𝛼) = (𝜇(𝛽,𝛼) , 𝜗(𝛽,𝛼) )is intuitionistic fuzzy magnified translation of KM-Ideal in Y. Proof: For any x∈ 𝑦, there exists a∈ 𝑥such that f(a) =x 𝑓 𝑓 𝜇(𝛽,𝛼) (𝑥) = 𝜇(𝛽,𝛼) (𝑓(𝑎)) = 𝜇(𝛽,𝛼) (𝑎) ≤ 𝜇(𝛽,𝛼) (0) ≤ 𝜇(𝛽,𝛼) 𝑓(0) = 𝜇(𝛽,𝛼) (0) http://www.iaeme.com/IJMET/index.asp 1348 editor@iaeme.com
An Intuitionistic Fuzzy Magnified Translation of Km Ideals on K-Algebra And 𝑓 𝑓 𝜗(𝛽,𝛼) (𝑥) = 𝜗(𝛽,𝛼) (𝑓(𝑎)) = 𝜗(𝛽,𝛼) (𝑎) ≥ 𝜗(𝛽,𝛼) (0) ≥ 𝜗(𝛽,𝛼) 𝑓(0) = 𝜗(𝛽,𝛼) (0) Now, let x,y,z Y. Then f(a)=x, f(b)=y, f(c)=z for some a, 𝑏, 𝑐 ∈ 𝑧. It follows that 𝜇(𝛽,𝛼) (𝑦 ⊙ 𝑧) = 𝜇(𝛽,𝛼) (𝑓(𝑏) ⊙ 𝑓(𝑐)) = 𝜇(𝛽,𝛼) (𝑓(𝑏 ⊙ 𝑐)) 𝑓 = 𝜇(𝛽,𝛼) (𝑏 ⊙ 𝑐) 𝑓 𝑓 ≥ min {𝜇(𝛽,𝛼) (𝑎 ⊙ 𝑏), 𝜇(𝛽,𝛼) (𝑐 ⊙ 𝑎)} = min {𝜇(𝛽,𝛼) (𝑓(𝑎 ⊙ 𝑏)), 𝜇(𝛽,𝛼) (𝑓(𝑐 ⊙ 𝑎))} = min{𝜇(𝛽,𝛼) (𝑓(𝑎) ⊙ 𝑓(𝑏)), 𝜇(𝛽,𝛼) (𝑓(𝑐) ⊙ 𝑓(𝑎))} = min{𝜇(𝛽,𝛼) (𝑥 ⊙ 𝑦), 𝜇(𝛽,𝛼) (𝑧 ⊙ 𝑥)} and 𝜗(𝛽,𝛼) (𝑦 ⊙ 𝑧) = 𝜗(𝛽,𝛼) (𝑓(𝑏) ⊙ 𝑓(𝑐)) = 𝜗(𝛽,𝛼) (𝑓(𝑏 ⊙ 𝑐)) 𝑓 = 𝜗(𝛽,𝛼) (𝑏 ⊙ 𝑐) 𝑓 𝑓 ≤ max {𝜗(𝛽,𝛼) (𝑎 ⊙ 𝑏), 𝜗(𝛽,𝛼) (𝑐 ⊙ 𝑎)} =max{𝜗(𝛽,𝛼) (𝑓(𝑎 ⊙ 𝑏)), 𝜗(𝛽,𝛼) (𝑓(𝑐 ⊙ 𝑎))} =max{𝜗(𝛽,𝛼) (𝑓(𝑎) ⊙ 𝑓(𝑏)), 𝜗(𝛽,𝛼) (𝑓(𝑐) ⊙ 𝑓(𝑎))} = max{𝜗(𝛽,𝛼) (𝑥 ⊙ 𝑦), 𝜗(𝛽,𝛼) (𝑧 ⊙ 𝑥)} Hence the proof. 6. CONCLUSION The concept of intuitionistic fuzzification of subalgebra and KM ideals of the K-algebra are tested with some of the properties and the results are presented in this paper. Intuitionistic fuzzy magnified translations to intuitionistic fuzzy KM ideals in K-algebra properties are tested and some of the properties are proved. This work is useful for the real time application of decision making for identifying the diseases. REFERENCES: [1] L.A. Zadeh, “ Fuzzy set”, Inform control, 8 (1965),338-353. [2] K.Iseki and S.Tanaka, “An introduction to the theory of BCK-Algebras, Math, Japonica 23 (1978), no.1, 1-26. [3] K.Atanassov “ Intutionistic Fuzzy set”, Fuzzy sets and systems, 20(1), (1986), 87-86 [4] S.K.Majunder, S.K.Sardar, “ Fuzzy Magnified translation on groups” Journal of Mathematics, North Bengal University,1(2),(2008),(17-24). http://www.iaeme.com/IJMET/index.asp 1349 editor@iaeme.com
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