Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 531 https://internationalpubls.com On Neutrosophic Ideal of KK-algebras A. Ibrahim 1 and B. Kavitha 2 1 Assistant Professor, P.G. and Research Department of Mathematics, H.H. The Rajah’s College, Pudukkottai, Affiliated to Bharathidasan University, Trichirappalli, Tamilnadu, India. Email: dribra@hhrc.ac.in; dribrahimaadhil@gmail.com 2 Research Scholar, P.G. and Research Department of Mathematics, H.H. The Rajah’s College, Pudukkottai, Affiliated to Bharathidasan University, Trichirappalli, Tamilnadu, India. Email:bkavitha835@gmail.com Article History: Received: 04-06-2024 Revised: 03-07-2024 Accepted: 30-07-2024 Abstract: In this paper, we introduce the notion of a neutrosophic ideal KK-algebra. Also, we investigate some properties of the neutrosophic ideal with suitable illustrations. Further, we describe how to deal with the homomorphism of the image and the inverse image of the neutrosophic ideal of the KK-algebra. AMS Mathematical Subject Classification (2020): 06F35, 03B47, 03B52, 03E70. Keywords: KK-algebra; Ideal; Neutrosophic Set; Neutrosophic ideal; Neutrosophic subalgebra. 1. Introduction: The concept of KK-akgebra was first developed by Asawasamrit and A. Sudprasert [3]. Asawasamrit and A. Sudprasert [4-6] established the concepts of ideals, subalgebras of KK-algebras, then they examined the relationships between them by using the idea of KK-algebra homomorphism and looked into a few associated properties. In 1965, Zadeh [11] introduced the concept of a fuzzy set. U. Leerawat and C. Prabpayak [8] introduced the idea of KU-algebras and examined certain associated properties and provided the homomorphism of KU-algebras. Fuzzy ideals of KK-algebras have been introduced by Huda Ali Faleh, Dr. Ahmed Hamzah Abed, and Dr. Areej Tawfeeq Hameed [2]. Samarandache [10] in 1998 first developed the concept of neutrosophic sets. The authors [7] looked into some properties and presented the notation of a neutrosophic ideal of BN- algebra. In this work, we first introduce the notion of the neutrosophic ideal of KK-algebra's and then look into a number of fundamental properties that are connected to it. For the neutrosophic ideal, We explain how to handle the image and inverse image homomorphism. 2. Preliminaries In order to better understand the primary findings, we go over the basic definitions of KK-algebra, ideals, and ideal characteristics in this section. We also go over the concepts of neutrosophic sets and neutrosophic ideals of KK-algebra. Definition 2.1[2]: An algebra ( , , 0) of type (2, 0) is called a KK-algebra, if it satisfied the following axioms for all Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 532 https://internationalpubls.com (i) (ii) (iii) . Definition 2.2[2]: A binary relation on KK-algebra ( , ,0), such that if and only if for all Proposition 2.3[2]: A any KK-algebra ( , , 0), the following axioms are hold for all (i) (ii) (iii) (iv) ( ) (v) (vi) (vii) implies (viii) implies . Definition 2.4[2]: Let ( , , 0) be a KK-algebra, and let be a nonempty subset of is called subalgebra of if for all Definition 2.5[2]: An ideal of is a non-empty subset of a KK-algebra ( , , 0) that satisfies the following for all (i) (ii) and imply . Proposition 2.6[2]: Each ideal in the KK-algebra ( , , 0) is a subalgebra of . Proposition 2.7[2]: Let a family of ideals of KK-algebra be denoted by .The intersection of any set of ideals of KK-algebra is also is an ideal. Definition 2.8[2]: Let ( and ( , , ) be an any two KK-algebras. Then, the mapping is called homomorphism, if it satisfied for all Definition 2.9 [10]: Let be the discourse universe. A neutrosophic set of is characterized by a truth membership function , an indeterminacy membership function , and a falsity membership function , where , , and are real standard elements of [0, 1]. It can be written as { ) Where ] [ . There is no restriction on the sum of and so + Definition 2.10 [7]: A neutrosophic set of BN-algebra ( , , 0), if is called an neutrosophic ideal of BN-algebra then it satisfies the following for all Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 533 https://internationalpubls.com (i) (ii) ; Definition 2.11[7]: Let be any neutrosophic set. If for any [0, 3], then an neutrosophic level set of is defined by = 3. Main results This part presents the key findings of the study, starting with an idea of neutrosophic ideal of KK algebra, and an explanation of neutrosophic ideal. Furthermore, various properties of the neutrosophic ideal in KK-algebra are investigated. Definition 3.1: A non empty subset of a KK-algebra is called neutrosophic ideal of KK-algebra, if it satisfies the following for all (i) , (ii) ; ; . Example 3.2: Consider a set Define a binary operation on given by the following Table 3.1, and neutrosophic set by the Table 3.2 as shown below: 0.7 0.5 0.5 0.5 0.5 0.6 0.4 0.4 0.4 0.4 0.2 0.1 0.1 0.1 0.1 Table 3.1: Operation Table 3.2: Neutrosophic set It is easily verified that is a neutrosophic ideal of , and that it satisfies the conditions of definition 3.1. Definition 3.3: Let ( , 0) be a KK-algebra, and let be a nonempty neutrosophic subset of is called neutrosophic sub algebra of if it satisfies the following axioms for all (i) (ii) (iii) . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 534 https://internationalpubls.com Example 3.4: Consider a set be a neutrosophic subset of Define a binary operation on given by the following Table 3.3, and neutrosophic set by the Table 3.4 as shown below: Table 3.3: Operation 0.7 0.5 0.5 0.6 0.4 0.4 0.2 0.1 0.1 Table 3.4: Neutrosophic set It is easily verified that is neutrosophic sub algebra of , and that it satisfies the conditions of definition 3.3. Proposition 3.5: Let be a neutrosophic ideal in KK-algebra and if then for all Proof: Let be a neutrosophic ideal of KK-algebra and . Then from the definition 2.2, we have for all From (ii) of definition 3.1we have, Then, we get Similarly, we can prove for Next, from (ii) of the definition 3.1, we have Hence, we get . ∎ Definition 3.6: Let be a nonempty set and be a neutrosophic subset of X, for [ ] the set is called a level subset of N. Theorem 3.7: Let be a neutrosophic subset of KK-algebra . If N is a neutrosophic sub algebra of if and only if the level set is a subalgebra of for every [ ]. Proof: Let be neutrosophic subset of KK-algebra and also neutrosophic sub algebra of Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 535 https://internationalpubls.com KK-algebra . Let be such that and then and . Since N is neutrosophic sub algebra. It follows that and that Hence is a sub algebra of Similarly, we can prove for and . Conversely, If is not true. Then there exists such that Putting then and . Hence, and which imply that and since is a subalgebra it follows that , and that This is contradiction. Therefore, is neutrosophic sub algebra of Similarly, we can prove for and .∎ Theorem 3.8: Let be a neutrosophic ideal of KK-algebra be a neutrosophic ideal of if and only if for every [ ] is an ideal of Proof: Let be a neutrosophic ideal of KK-algebra . Then, from the definition of 3.1 we have for all Therefore, for and so , Let be such that and then and Since is a neutrosophic ideal, then from (ii) of the definition 3.1, it follows that and, we have that . Hence, is an ideal of . Similarly, we can prove for and . Conversely, we need to show that (i) and (ii) of definition 3.1 are true. If (i) of definition 3.1 is not satisfied, then there exists such that If we take = then and 0 then and as is an ideal of X we have and so This is a contradiction. Therefore we have If (ii) is not true then there exists such that Putting = then and } , Hence, which imply that , and . Since is an ideal. It follows that and . This is a contradiction. Thus, is neutrosophic ideal of . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 536 https://internationalpubls.com Similarly, we can prove for and . ∎ Proposition 3.9: Every neutrosophic ideal of KK-algebra is neutrosophic sub algebra of Proof: Assume that be an neutrosophic ideal of a KK-algebra Next based on theorem 3.8 every [ ] is an ideal of . According to proposition 2.6 for every t [ ] is a subalgebra of . Thus, from the theorem 3.7, is neutrosophic subalgebra of KK-algebra . ∎ Theorem 3.10: Let I be an ideal of KK-algebra of Then for any fixed number t in an open interval (0, 1), there exists neutrosophic ideal of such that Proof: Define [ ] by ) { Where is a fixed number in (0, 1) clearly for all . Let , if , then and so If Then clearly If and then since is an ideal, then . Hence, is neutrosophic ideal of .It is clear that . Similarly, we can prove for and . ∎ Theorem 3.11: Let be neutrosophic ideal of a KK-algebra and assume that , be level ideals of ,as well as < then the following are equivalent (i) = (ii) There is no such that . Proof: Let be neutrosophic ideal of a KK-algebra . Assume that for and that there exists such that then is proper subset of This is a contradiction. Therefore, there is no such that Conversely, suppose that there is no such that . It follows that < then . Let then and because does not lie between and hence .This implies that therefore = . ∎ Proposition 3.12: The intersection of any set of neutrosophic ideals of KK-algebra X is also neutrosophic ideal. Proof: For any , let { be a family of neutrosophic ideals of KK-algebra then, , ⋂ ) (0) ⋂ ⋂ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 537 https://internationalpubls.com = = {⋂ ⋂ } Similarly, we can prove for and .∎ Definition 3.13: Let ( , , ) and ( , , ) be an any two KK-algebras and ( , , ) be a mapping between the corresponding nonempty sets and . For every neutrosophic subset of , the neutrosophic subset of defined" by = { is called as the image of under f . Similarly, is the subset of then the neutrosophic subset = ( That is, [ ] for all is called pre image of under Theorem 3.14: The neutrosophic ideal is also its homomorphic pre-image. Proof: A homomorphism of KK-algebras, denoted by where is the neutrosophic ideal of and is the pre image of under , then [ ] for every . Since and is neutrosophic ideal of it follows that [ ]= for every where is the zero element of But, ( ) and so for all Let’s now then we obtain )} } That is . Similarly, we can prove for and .∎ Definition 3.15 : A neutrosophic subset of a set has supremum property if for any subset of , there exist such that ( ) = { ( )/ }. Theorem 3.16: Let ( , , ) be a homomorphism between KK-algebras and respectively. For every neutrosophic ideal N in X with supremum property, then is a neutrosophic ideal of . Proof: Let ( , , 0) and ( , , ) be an any two KK-algebras and ( , , ) satisfies a homomorphism property, from the definition of 3.15, = { / ( )} for all ( ) We have to prove , Assume that the onto homomorphism of the KK-algebra is , Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 7s (2024) 538 https://internationalpubls.com is neutrosophic ideal of with supremum property and is the image of under . Since is neutrosophic ideal of we have (0) ( ) for all Note that 0 ), where and are the zero elements of and respectively. Thus we have (0 ) . Which implies that for any . For any , let , be such that ( ) [ ] [ [ ] [ ] [ ] ( ). Then, ( ) ( ) ( ), ( )} { (t) , ( )} { ) , ) } Thus, Therefore, is neutrosophic ideal of . Similarly, we can prove for and . ∎ 4. Conclusions This paper starts by discussing the concepts of KK-algebra and the neutrosophic ideal with appropriate examples. Then, we look into various fundamental aspects of the neutrosophic ideal in KK-algebra. We also need to deal with the homomorphism of image and inverse image of neutrosophic ideals in KK-algebra. References [1] Andrzej Walendziak and Grzegorz Dymek(2015), Fuzzy Ideals of BN-Algebras, Hindawi Publishing Corporation the Scientific World Journal, Volume 2015, (9 pages). [2] Dr. Areej Tawfeeq Hameed1, Huda Ali Faleh and Dr. Ahmed Hamzah Abed, Fuzzy Ideals of KK-algebra, Journal of Physics: Conference Series, 1804 (2021) 012066. [3] S. Asawasamrit and A. Sudprasert, A Structure of KK-algebras and its properties, International Journal of Mathematical Analysis, Volume 6, Number 21 (2012), 1035-1044. [4] S. 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[10] Smarandache Florentin (2005), Neutrosophic set-a generalization of the intuitionistic fuzzy set, International Journal of Pure and Applied Mathematics, Volume 24, 287-297. [11] L.A. Zadeh, Fuzzy sets, information and control, Volume 8 (1965), 338- 353.