In this paper, we discuss about some properties such as fuzzy translation in fuzzy d-ideals and fuzzy d-subalgebr

K.R., Sobha

Keerthana, R G

English

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Ratio Mathematica                                                                            Volume 44, 2022     Fuzzy Translation in Fuzzy d-Ideals and  Fuzzy d-Subalgebra   R. G. Keerthana1    K. R. Sobha2   Abstract In this paper, we discuss about some properties such as fuzzy translation in fuzzy d-ideals and fuzzy d-subalgebra.  Keywords: d - algebra, d - subalgebra, d - ideal, fuzzy d - ideal, fuzzy – 𝛼 – translation, fuzzy d - subalgebra.  2010 AMS Subject Classification: 94D05, 08A723.          1Research Scholar, Reg.No.20213182092001, Sree Ayyappa College for Women, Chunkankadai, Nagercoil-62900. [Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli -627012, Tamilnadu, India.]. keerthanarajagopal.rg@gmail.com  2Assistant Professor, Department of Mathematics, Sree Ayyappa College for Women, Chunkankadai, Nagercoil. [Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli - 627012, Tamilnadu, India.]  vijayakumar.sobha9@gmail.com. 3Received on June 18th, 2022. Accepted on Aug 10th, 2022. Published on Nov 30th, 2022. doi: 10.23755/rm.v44i0.885. ISSN: 1592 -7415. eISSN: 2282 – 8214. ©The Authors. This paper is published under the CC - BY licence agreement. 24R. G. Keerthana, K. R. Sobha   1. Introduction Fuzzy set theory was introduced by Zadeh in 1965 [6]. The study of fuzzy subsets and its applications to various mathematical contexts has given rise to what is now commonly called fuzzy mathematics.  It forms a branch of mathematics including fuzzy set theory and fuzzy logic.  Fuzzy set theory was guided by the assumption that the classical sets were not natural appropriate or useful notions in describing the real-life problems because every object encountered in the real physical world carries some degree of fuzziness.  Hence fuzzy set has become strong area of research in engineering, medical science, graph theory etc. Algebraic structures play an important role in mathematics with wide range of application in many disciplines such as computer sciences, control engineering, theoretical physics, information systems and topological spaces. Since these ideas have been applied to other algebraic structures such as group, semigroup, ring, modules, vector spaces and topologies. It gives enthusiasm to the researchers to view various concepts and results from the area of abstract algebra in the broader frame work of fuzzy setting.  Fuzzy algebra is an important branch of fuzzy mathematics. In 1996, J. Negger and H.S. Kim [3] introduced the class of d-algebra which is a generalization of BCK-algebras and investigated relation between d-algebra and BCK-algebra. M. Akram and K.H. Dar [1] introduced the concepts fuzzy d-algebra, fuzzy subalgebra and fuzzy d-ideals of d-algebra.  2. Preliminaries Definition:2.1 A d-algebra is a non-empty set 𝑋 with a constant 0 and a binary operation ∗ satisfies the following axioms: i.𝑥 ∗ 𝑥 = 0 ii.0 ∗ 𝑥 = 0 iii.𝑥 ∗ 𝑦 = 0 and 𝑦 ∗ 𝑥 = 0 ⇒ 𝑥 = 𝑦, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦 ∈ 𝑋.  Definition:2.2 A non-empty subset of a d-algebra 𝑋 is called d-subalgebra of 𝑋 if 𝑥 ∗ 𝑦 ∈ 𝑋, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦 ∈ 𝑋.  Definition: 2.3 Let (𝑋,∗ ,0) be a d-algebra and 𝑎 ∈ 𝑋. Define 𝑎 ∗ 𝑋 = {𝑎 ∗ 𝑥/𝑥 ∈ 𝑋}.   Then 𝑋 is said to be edge if 𝑎 ∗ 𝑋 = {0, 𝑎 } for all 𝑎 ∈ 𝑋.  25Fuzzy Translation in Fuzzy d-Ideals and Fuzzy d-Subalgebra    Definition: 2.4 Let 𝑋 be a d-algebra and 𝐼 be a subset of 𝑋, then 𝐼 is called d-ideal of 𝑋 if it satisfies the following conditions: i.0 ∈ 𝐼 ii.𝑥 ∗ 𝑦 ∈ 𝐼 𝑎𝑛𝑑 𝑦 ∈ 𝐼 ⇒ 𝑥 ∈ 𝐼 iii.x ∈ I and y ∈ X ⇒ x ∗ y ∈ I.  Definition: 2.5 A fuzzy subset 𝜇𝐴 of 𝑋 is called a fuzzy d-ideal of 𝑋 if it satisfies the following condition: i.𝜇𝐴(0) ≥ 𝜇𝐴(𝑥) ii.𝜇𝐴(𝑥) ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥 ∗ 𝑦), 𝜇𝐴(𝑦)} iii.𝜇𝐴(𝑥 ∗ 𝑦) ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)}    𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦 ∈ 𝑋.  3. Fuzzy Translation in Fuzzy d - Ideals  Definition: 3.1 Let 𝜇𝐴 be a fuzzy subset of 𝑋 and 𝛼 ∈ [0, 𝑇]. A mapping (𝜇𝐴)𝛼𝑇 : 𝑋 →[0,1] is said to be a fuzzy-𝛼-translation of 𝜇𝐴 if it satisfies (𝜇𝐴)𝛼𝑇 (𝑥) = 𝜇𝐴(𝑥) +𝛼, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑋.  Theorem: 3.2 If 𝜇𝐴 is a fuzzy d-ideal of 𝑋, then the fuzzy-𝛼-translation (𝜇𝐴)𝛼𝑇  of 𝜇𝐴 is a fuzzy d-ideal of X, for all 𝛼 ∈ [0,1]. Proof: Let 𝜇𝐴 be a fuzzy d-ideal of 𝑋 and let 𝛼 ∈ [0,1] Then  (𝜇𝐴)𝛼𝑇 (0) = 𝜇𝐴(0) + 𝛼 ≥ 𝜇𝐴(𝑥) + 𝛼 = (𝜇𝐴)𝛼𝑇 (𝑥) (𝜇𝐴)𝛼𝑇 (0) ≥ (𝜇𝐴)𝛼𝑇 (𝑥) (𝜇𝐴)𝛼𝑇 (𝑥) = 𝜇𝐴(𝑥) + 𝛼 ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥 ∗ 𝑦), 𝜇𝐴(𝑦)} + 𝛼 = 𝑚𝑖𝑛{𝜇𝐴(𝑥 ∗ 𝑦) + 𝛼, 𝜇𝐴(𝑦) + 𝛼} = 𝑚𝑖𝑛{(𝜇𝐴)𝛼𝑇 (𝑥 ∗ 𝑦), (𝜇𝐴)𝛼𝑇 (𝑦)} (𝜇𝐴)𝛼𝑇 (𝑥) ≥ 𝑚𝑖𝑛{(𝜇𝐴)𝛼𝑇 (𝑥 ∗ 𝑦), (𝜇𝐴)𝛼𝑇 (𝑦)} (𝜇𝐴)𝛼𝑇 (𝑥 ∗ 𝑦) = 𝜇𝐴(𝑥 ∗ 𝑦) + 𝛼 ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)} + 𝛼 = 𝑚𝑖𝑛{𝜇𝐴(𝑥) + 𝛼, 𝜇𝐴(𝑦) + 𝛼} = 𝑚𝑖𝑛{(𝜇𝐴)𝛼𝑇 (𝑥), (𝜇𝐴)𝛼𝑇 (𝑦)} (𝜇𝐴)𝛼𝑇 (𝑥 ∗ 𝑦) ≥ 𝑚𝑖𝑛{(𝜇𝐴)𝛼𝑇 (𝑥), (𝜇𝐴)𝛼𝑇 (𝑦)}   Hence(𝜇𝐴)𝛼𝑇  of 𝜇𝐴 is a fuzzy d-ideal of X for all 𝛼 ∈ [0,1].  26R. G. Keerthana, K. R. Sobha  Theorem: 3.3 Let 𝜇𝐴 be a fuzzy subset of X such that the fuzzy-𝛼-translation (𝜇𝐴)𝛼𝑇  of 𝜇𝐴 is a fuzzy d-ideal of 𝑋for some 𝛼 ∈ [0,1], then 𝜇𝐴 is a fuzzy d-ideal of 𝑋. Proof: Let (𝜇𝐴)𝛼𝑇   is a fuzzy d-ideal of 𝑋 for some 𝛼 ∈ [0,1] Let 𝑥, 𝑦 ∈ 𝑋 𝜇𝐴(0) + 𝛼 = (𝜇𝐴)𝛼𝑇 (0) ≥ (𝜇𝐴)𝛼𝑇 (𝑥) = 𝜇𝐴(𝑥) + 𝛼 𝜇𝐴(0) ≥ 𝜇𝐴(𝑥) 𝜇𝐴(𝑥) + 𝛼 = (𝜇𝐴)𝛼𝑇 (𝑥) ≥ 𝑚𝑖𝑛{(𝜇𝐴)𝛼𝑇 (𝑥 ∗ 𝑦), (𝜇𝐴)𝛼𝑇 (𝑦)} = 𝑚𝑖𝑛{𝜇𝐴(𝑥 ∗ 𝑦) + 𝛼, 𝜇𝐴(𝑦) + 𝛼} = 𝑚𝑖𝑛{𝜇𝐴(𝑥 ∗ 𝑦), 𝜇𝐴(𝑦)} + 𝛼 𝜇𝐴(𝑥) ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥 ∗ 𝑦), 𝜇𝐴(𝑦)} 𝜇𝐴(𝑥 ∗ 𝑦) + 𝛼 = (𝜇𝐴)𝛼𝑇 (𝑥 ∗ 𝑦) ≥ 𝑚𝑖𝑛{(𝜇𝐴)𝛼𝑇 (𝑥), (𝜇𝐴)𝛼𝑇 (𝑦)} = 𝑚𝑖𝑛{𝜇𝐴(𝑥) + 𝛼, 𝜇𝐴(𝑦) + 𝛼} = 𝑚𝑖𝑛{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)} + 𝛼 𝜇𝐴(𝑥 ∗ 𝑦) ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)} Hence 𝜇𝐴 is a fuzzy d-ideal of 𝑋.  4. Fuzzy Translation in Fuzzy d - Subalgebra Definition: 4.1 A fuzzy set 𝜇𝐴in d-algebra 𝑋 is called fuzzy d-subalgebra of 𝑋 if it satisfies, 𝜇𝐴(𝑥𝑦) ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)}     𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥, 𝑦 ∈ 𝑋.  Theorem: 4.2 If 𝜇𝐴 is a fuzzy d-subalgebra of 𝑋, then the fuzzy-𝛼-translation 𝜇𝛼𝑇 of 𝜇𝐴 is also a fuzzy d-subalgebra of X for all 𝛼 ∈ [0,1]. Proof: Let 𝑥, 𝑦 ∈ 𝑋 and let 𝛼 ∈ [0,1] Let 𝜇𝐴 be a fuzzy d-subalgebra of 𝑋 . Then    𝜇𝐴(𝑥𝑦) ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)} (𝜇𝐴)𝛼𝑇 (𝑥𝑦) = 𝜇𝐴(𝑥𝑦) + 𝛼 ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)} + 𝛼 = 𝑚𝑖𝑛{𝜇𝐴(𝑥) + 𝛼, 𝜇𝐴(𝑦) + 𝛼} = 𝑚𝑖𝑛{(𝜇𝐴)𝛼𝑇 (𝑥), (𝜇𝐴)𝛼𝑇 (𝑦)} (𝜇𝐴)𝛼𝑇 (𝑥𝑦) ≥ 𝑚𝑖𝑛{(𝜇𝐴)𝛼𝑇 (𝑥), (𝜇𝐴)𝛼𝑇 (𝑦)}  Hence (𝜇𝐴)𝛼𝑇  of 𝜇𝐴 is a fuzzy d-subalgebra of X for all 𝛼 ∈ [0,1].  27Fuzzy Translation in Fuzzy d-Ideals and Fuzzy d-Subalgebra    Theorem: 4.3 Let 𝜇𝐴 be a fuzzy subset of X such that the fuzzy-𝛼-translation (𝜇𝐴)𝛼𝑇  of 𝜇 is a fuzzy d-subalgebra of 𝑋 for some 𝛼 ∈ [0,1], then 𝜇𝐴 is a fuzzy d-subalgebra of 𝑋. Proof: Let (𝜇𝐴)𝛼𝑇  is a fuzzy d-subalgebra of 𝑋 for some 𝛼 ∈ [0,1]. Let 𝑥, 𝑦 ∈ 𝑋. 𝜇𝐴(𝑥𝑦) + 𝛼 = (𝜇𝐴)𝛼𝑇 (𝑥𝑦) ≥ 𝑚𝑖𝑛{(𝜇𝐴)𝛼𝑇 (𝑥), (𝜇𝐴)𝛼𝑇 (𝑦)} = 𝑚𝑖𝑛{𝜇𝐴(𝑥) + 𝛼, 𝜇𝐴(𝑦) + 𝛼} = 𝑚𝑖𝑛{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)} + 𝛼 𝜇𝐴(𝑥𝑦) ≥ 𝑚𝑖𝑛{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)} Hence 𝜇𝐴 is a fuzzy d-subalgebra of 𝑋.  5. Conclusions In this paper, we have given some ideas on fuzzy translation in fuzzy d-ideals and fuzzy d-subalgebras. Our further research will be focus on dot product and level set.  References [1] M. Akram and K.H. Dar, On fuzzy d-algebra, Punjab University Journal of Math, 37 (2005), 61-76. [2] J. Neggers; Y.B. Jun; H.S. Kim, “On d – Ideals in d – algebras”, Mathematica Slovaca.49 (1999), No.3, 243-251 [3] J. Neggers and H.S. Kim, “On d – algebra”, Math. Slovaca 49 (1999) no.1, 19 – 26. [4] T. Priya and T. Ramachandran, Homomorphism and Cartesian Product of fuzzy Ps-algebra, Applied Mathematical Sciences, 8, Vol (67) (2014) 3321 – 3330. [5] T. Priya and T. Ramachandran, Homomorphism and Cartesian Product on Fuzzy translation and fuzzy multiplication of Ps-algebras, Annals of Pure and mathematics, Vol.8, No.1, 2014, 93 – 104. [6] L.A. Zadeh,” Fuzzy set”, Inform and Control.8 (1965), 338 – 353.      28

Fuzzy Translationin Fuzzy d-Ideals and  Fuzzy d-Subalgebra

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Fuzzy Translationin Fuzzy d-Ideals and Fuzzy d-Subalgebra

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