EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6392 ISSN 1307-5543 – ejpam.com Published by New York Business Global Neutrosophic Bi-Ideals in INK-Algebras Mounikalakshmi Remala1, Eswarlal Tamma1,∗, Yella Bhargavi1 1 Department of Engineering Mathematics, College of Engineering, Koneru Lakshmaiah Education Foundation, Vaddeswaram, Andhra Pradesh, India Abstract. The main aim of this research article is to identify the bi-ideals in INK-algebra. This paper introduces the notion of neutrosophic bi-ideals in INK-algebra and discusses basic operations such as order-reversing and order-preserving properties. It is shown that the intersection and union (with containment) of two neutrosophic bi-ideals result in another neutrosophic bi-ideal. Further, the paper explores homomorphisms and epimorphisms between neutrosophic bi-ideals in INK- algebras. It is demonstrated that the direct product of two neutrosophic sets in an INK-subalgebra remains within the same structure. Observations regarding the direct product of neutrosophic bi- ideals in INK-algebras are provided. Finally, an application related to neutrosophic bi-ideals is discussed. 2020 Mathematics Subject Classifications: 03E72, 08A72, 06F25 Key Words and Phrases: INK-algebra, neutrosophic set, neutrosophic INK-algebra, neutro- sophic bi-ideal, homomorphism of neutrosophic bi-ideal, direct product of neutrosophic bi-ideal 1. Introduction Iseki and Tanaka [1, 2] have worked on the concept of BCK and BCI algebras to pick-up their characteristics and applications. There exists an immense area of empirical applica- tions in fuzzy sets, Intuitionistic fuzzy sets and Neutrosophic sets. The literature works of Fuzzy sub algebras and fuzzy K-ideals in INK-algebras, fuzzy p-ideal in INK-algebra, Fuzzy translation of INK-ideal of INK-algebras also they proposed On intuitionistic fuzzy INK-ideals of INK-algebras, Direct product of intuitionistic fuzzy K-ideals of INK-algebras, Intuitionistic fuzzy translation on INK-algebra and also they have discussed neutosophic set in INK-algebra, neutrosophic h-ideal in INK-algebra have discussed by Kaviyarasu and Indhira, [3–7] and homomorphism and anti-homomorphism of neutrosophic INK-algebras have done by Mounikalakshmi, Eswarlal, Venkata Kalyani and Aiyred Iampan [8]. Re- cently, Kaviyarasu and Rajeshwari [9] discussed Translation of neutrosophic INK-algebras and Mounikalakshmi, Eswarlal [10, 11] worked on bipolar fuzzy INK subalgebras of INK ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6392 Email addresses: mouninaidu0521@gmail.com (M. Remala), eswarlal@kluniversity.in (E. Tamma) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Remala, E. Tamma, Y. Bhargavi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6392 2 of 20 algebras and Implicative ideals and Positive Implicative INK-Ideals in Neutrosophic INK- algebra. Moreover, the work of INK-algebra has been explored in different aspects of fuzzy sets, bipolar fuzzy sets, intuitionistic fuzzy sets and also in neutrosophic sets. Zadeh, traversed the notion of fuzzy sets in 1965, which is an extension of classical set theory that deals with vagueness and uncertainty in the given data. Classical set theory asserts that an element is either a member of a set or not. Fuzzy set theory discourses this by introducing membership values scaling from 0 to 1, which have been used to represent the degree to which an element is affiliated with a set. Fuzzy set theory has innumerable implementations in diverse fields, comprises of artificial intelligence, decision- making and control systems that allow the both modeling and handling of vague and imprecise information. Moreover, Kuroki [12, 13] discussed his work on fuzzy bi-ideals in semigroups and Fuzzy generalized bi-ideals in semigroups. Yiarayong [14] have done results on fuzzy bi-ideal theory applied on semi-groups. Eventually, Atanassov proposed the generalization of fuzzy set, which is Intuitionistic Fuzzy Set(IFS) in the year 1980, which gives information about a fresh parameter known as “non-membership degree,” where fuzzy tells us about the membership degree but in IFS gives information about uncertainty and vagueness regarding membership degrees and non-membership degrees. Intuitionistic fuzzy sets have applications in decision-making where uncertainty plays a significant role. Various approaches of IFS include expert systems, risk assessment and medical diagnosis, where precise decisions are difficult to maintain on vague or uncertain data or information. Young Bae Jun and Kyung Ho Kim [15] have done work on Intuitionistic fuzzy ideals of BCK-algebra. Few authors, Kim and Lee [16, 17] have discussed results of On intuitionistic fuzzy bi-ideals of semi-groups, Interval valued intuitionistic fuzzy bi-ideals of semigroups. Also, Bhargavi [18, 19] and Raagamayi [20] have discussed the results on Vague bi-ideals. Sindhu and Himaya Jaleela Begun [21] have worked on Intuitionistic fuzzy bi-ideals of BCK-algebras. Sequentially, Vinnela and Raagamayi [22] have worked on Bipolar fuzzy bi-ideals of gamma near ring. Sub-sequentially few more authors have completed work on bi-ideals [7, 23–25] like bi-ideals in semi-groups, fuzzy bi-ideals and generalized fuzzy bi-ideals in semigroups also [14, 26, 27]. Later, the neutrosophic sets were introduced by smarandache in 1990, which is the generalization of classical sets, fuzzy sets and Intuitionistic fuzzy sets by involving a new parameter called “indeterminancy.” Neutrosophic sets are particularly useful in situations where uncertainty exist also holds the basic components of truth, falsehood and indetermi- nacy are simultaneously considered. Neutrosophic sets have plenty of approaches in many fields like decision making, expert systems, image processing and fuzzy logic thus enables us to do effective modeling and analysis in situations where classical set theory falls short. Some authors De Gruyter [28] have discussed few results on MBJ-neutrosophic ideals of BCK/BCI-algebras. Muhuuddin and Young Bae Jun [29] have done further results of neu- trosophic subalgebras in BCK/BCI-algebras based on neutrosophic points. Now in this article we look over the concept of bi-ideals in INK-algebra in terms of neutrosophic sets. M. Remala, E. Tamma, Y. Bhargavi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6392 3 of 20 2. Preliminaries In this segment, we have used the some of the definitions which are being utilized for this work. Definition 2.1. [30] An INK-algebra ( ¨̈U, •, 0) is said to be an INK-algebra if it satisfies the following conditions for any E, 3, Ä ∈ ¨̈U : • INK-1: ((E • 3) • (E • Ä)) • (Ä • 3) = 0. • INK-2: ((E • Ä) • (3 • Ä)) • (E • 3) = 0. • INK-3: E • 0 = E. • INK-4: E • 3 = 0 and 3 • E = 0 imply E = 3. Note. In ¨̈U we can interpret ≤ by E ≤ 3 if and only if E • 3 = 0. Definition 2.2. [30] Let Y be a non-empty subset of a INK-algebra ¨̈U , then Y is said to be an INK-sub-algebra of ¨̈U , if E• 3∈ Y where E, 3 ∈ ¨̈U . Definition 2.3. [3] A fuzzy set g in a INK-algebra ¨̈U is known as FINK-subalgebra of ¨̈U if g(E• 3) ≥ min{g(E), g(3)} ∀ E, 3∈ ¨̈U . Definition 2.4. [3] Let fuzzy set g in INK-algebra ¨̈U is known as fuzzy-ideal, if it satisfies: FID-1: g(0) ≥ g(E) FID-2: g(E) ≥ min{g(E• 3), g(3)} ∈ ¨̈U . Definition 2.5. [3] Let F be a non-empty subset of a INK-algebra ¨̈U . Then F is defined as INK-ideal of ¨̈U if (i) 0 ∈ F , (ii) ((Ä • E) • (Ä • 3)) ∈ F and 3 ∈ F imply E ∈ F for all E, 3,Ä ∈ ¨̈U . Definition 2.6. [6] A neutrosophic set g=(gT , gI , gF ) in X is called a neutrosophic INK sub-algebra of ¨̈U if it satisfies the following condition, for all E, 3, Ä∈ ¨̈U . (i) gT (E• 3) ≥ min {gT (E), gT (3)} (ii) gF (E• 3) ≤ max {gF (E), gF (3)} (iii) gF (E• 3) ≤ max {gF (E), gF (3)} Example 2.1. Consider the INK-algebra ¨̈U = {0, a, b} with the following Cayley table. • 0 a b 0 0 b a a a 0 b b b a 0 M. Remala, E. Tamma, Y. Bhargavi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6392 4 of 20 A neutrosophic set g = (gT , gI , gF ) on ¨̈U is defined by 0 a b gT 0.2 0.5 0.6 gI 0.9 0.8 0.8 gF 0.8 0.5 0.4 Then g=(gT , gI , gF ) be a neutrosophic sub-algebra. Let us take (randomly) for truth- membership degree E= a & 3= b. So, gT (a • b) ≥ min {gT (a), gT (b)} gT (b) ≥ min {gT (a), gT (b)} = 0.6 > 0.5 Also, gI(a • b) ≤ max {gI (a), gI (b)} gI(b) ≤ max {gI (a), gI (b)} = 0.8 = 0.8 Similarly, gF (a • b) ≤ max {gF (a), gF (b)} gF (b) ≤ max {gF (a), gF (b)} = 0.4 < 0.5 Likely, for all outcomes the above condition satisfied. Definition 2.7. [6] A neutrosophic set g = (gT , gI , gF ) in ¨̈U is termed as neutrosophic ideal of ¨̈U if it satisfies the following condition, for all E, 3 ∈ ¨̈U : (i) gT (0) ≥ gT (E), gI(0) ≤ gI(E), gF (0) ≤ gF (E). (ii) gT (E) ≥ min{ gT (E • 3), gT (3) }. (iii) gI(E) ≤ max{ gI(E • 3), gI(3) }. (iv) gF (E) ≤ max{ gF (E • 3), gF (3) }. Definition 2.8. [6] A neutrosophic set g = (gT , gI , gF ) in X is termed as neutrosophic INK-ideal of ¨̈U if it satisfies the following condition, for all E, 3,Ä ∈ ¨̈U : (i) gT (0) ≥ gT (E), gI(0) ≤ gI(E), gF (0) ≤ gF (E). (ii) gT (E) ≥ min{ gT ((Ä • E) • (Ä • 3)), gT (3) }. (iii) gI(E) ≤ max{ gI((Ä • E) • (Ä • 3)), gI(3) }. (iv) gF (E) ≤ max{ gF ((Ä • E) • (Ä • 3)), gF (3) }. Definition 2.9. [6] Let g=(gT , gI , gF ) and h=(hT , hI , hF ) be two neutrosophic sets in ¨̈U , then the union and intersection are defined by (i) g ∪ h (E) = {< E, max{Tg(E), Th(E)}, min {I g(E), Ih(E)}, min {F g(E), Fh(E)}} (ii) g ∩ h (E) = {< E, min{Tg(E), Th(E)}, max {I g(E), Ih(E)}, max {F g(E), Fh(E)}}. Definition 2.10. [8] A Mapping ¢ : ¨̈U → ˘̈̈ U of INK-algebras is known as homomorphism if ¢(E• 3) = ¢(E) • ¢(3) for all E, 3∈ ¨̈U . If ¢ : ¨̈U → ˘̈̈ U is a homomorphism then ¢(0)=0. M. Remala, E. Tamma, Y. Bhargavi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6392 5 of 20 3. Intersection and Union on Neutrosophic Bi-ideals Definition 3.1. An INK-algebra ¨̈U is said to be associative INK algebra if it satisfies (E• 3)•Ä= E• (3•Ä) Definition 3.2. Let F be a non-empty subset of a INK algebra ¨̈U . Then F is defined as Bi-ideal of INK-algebra ¨̈U if (i) 0 ∈ F, (ii) E• 3• Ä∈ F and Ä∈ F imply that E• 3∈ F for all E, 3, Ä∈ ¨̈U . Definition 3.3. A Neutrosophic set g=(gT , gI , gF ) is called a neutrosophic Bi-ideal of INK-algebra ¨̈U if it satisfies (i) gT (0) ≥ gT (E), gI(0) ≤ gI(E), gF (0) ≤ gF (E) (ii) gT (E• 3) ≥ min {gT (E• 3• Ä), gT (Ä)}. (iii) gI(E• 3) ≤ max {gI(E• 3• Ä), gI(Ä)}. (iv) gF (E• 3) ≤ max {gF (E• 3• Ä), gF (Ä)} for all E, 3, Ä∈ ¨̈U . Example 3.1. Consider INK-algebra ¨̈U={0, 2, 4, 6} with Cayley table • 0 2 4 6 0 0 0 0 0 2 2 0 0 2 4 4 2 0 4 6 6 6 6 0 • 0 2 4 6 gT 0.8 0.7 0.6 0.5 gI 0.7 0.8 0.8 0.9 gF 0.4 0.5 0.5 0.8 Let us take (randomly) E= 0, 3= 4 , Ä= 6. So, gT (0 • 4) ≥ min {gT (0 • 4 • 6), gT (6)} gT (0) ≥ min {gT (0), gT (6)} = 0.8 > 0.5 Also, gI(0 • 4) ≤ max {gI(0 • 4 • 6), gI(6)} gI(0) ≤ max {gI(0), gI(6)} = 0.7 < 0.9 Similarly, gF (0 • 4) ≤ max {gF (0 • 4 • 6), gF (6)} gF (0) ≤ max {gF (0), gF (6)} = 0.4 < 0.8 Likely, for all outcomes neytrosophic bi-ideal condition satisfied. Then g=(gT , gI , gF ) be neutrosophic bi-ideal of INK-algebra of ¨̈U . Lemma 3.1. Let neutrosophic set g=(gT , gI , gF ) in INK-algebra ¨̈U is an neutrosophic bi-ideal of ¨̈U . If the inequality E• 3≤ Äholds in ¨̈U , then M. Remala, E. Tamma, Y. Bhargavi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6392 6 of 20 (i) gT (E• 3) ≥ min {gT (E), gT (Ä)}. (ii) gI(E• 3) ≤ max {gI (E), gI (Ä)}. (iii) gF (E• 3) ≤ max {gF (E), gF (Ä)}. Proof . Let E, 3, Ä∈ ¨̈U be such that E• 3≤ Ä. By the definition of partial order in INK algebra. We have E• 3≤ Ä=⇒ (E• 3) • Ä=0 =⇒ E• 3• Ä=0. Since g=(gT , gI , gF ) is an neutrosophic Bi-ideal of INK algebra ¨̈U , by def 3.3.1 it satisfies the following conditions for all E, 3, Ä∈ ¨̈U Now gT (E • 3) ≥ min{gT (E • 3 • Ä), gT (Ä)} (3.1) Since gT (E • 3) ≥ min{gT (0), gT (Ä)} (3.2) Now, Using the property of Neutrosophic bi-ideal that gT (0) ≥ gT (E), we obtain min{gT (0), gT (Ä)} ≥ min{gT (E), gT (Ä)} (3.3) substitute (3.3) into (3.2) then we get, gT (E• 3) ≥ min{ gT (E), gT (Ä)}. Therefore gI(E • 3) ≤ max{gI(E • 3 • Ä), gI(Ä)} (3.4) Since E• 3• Ä=0, it follows (3.1) gI(E • 3) ≤ max{gI(0), gI(Ä)} (3.5) Now, Using the property of Neutrosophic bi-ideal that gI (0) ≤ gI(E), we obtain max{gI(0), gI(Ä)} ≤ max{gI(E), gI(Ä)} (3.6) substitute (3.6) into (3.5) then we get, gI(E• 3) ≤ max{ gI(E), gT (Ä)}. Also (iii) gF (E • 3) ≤ max{gF (E • 3 • Ä), gF (Ä)} (3.7) Since E• 3• Ä=0, it follows (3.1) gF (E • 3) ≤ max{gF (0), gF (Ä)} (3.8) Now, Using the property of Neutrosophic bi-ideal that gF (0) ≤ gF (E), we obtain max{gF (0), gF (Ä)} ≤ max{gF (E), gF (Ä)} (3.9) substitute (3.9) into (3.8) then we get, gF (E• 3) ≤ max{ gF (E), gF (Ä)} Hence Proved. Lemma 3.2. Let NS g=(gT , gI , gF ) be a Neutrosophic Bi-ideal of ¨̈U . If the inequality E• 3≤ Ä holds in ¨̈U then gT (E• 3) ≥ gT (Ä), gI(E• 3) ≤ gI(Ä) and gF (E• 3) ≤ gF (Ä) that gT is order reversing and gI , gF are order preserving. Proof . By using INK ordering, E ≤ 3 ⇔ E • 3 = 0. So the condition E • 3 ≤ Ä implies (E•3)•Ä = 0 ⇒ E•3•Ä = 0. Let g = (gT , gI , gF ) be a Neutrosophic Bi-ideal of INK-algebra ¨̈U for all E, 3,Ä ∈ ¨̈U . M. Remala, E. Tamma, Y. Bhargavi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6392 7 of 20 (i) gT (E • 3) ≥ min{gT (E • 3 •Ä), gT (Ä)}. Since gT (E • 3 •Ä) = gT (0) and gT (0) ≥ gT (Ä) (from the definition), we get gT (E • 3) ≥ min{gT (0), gT (Ä)} = gT (Ä). (ii) gI(E • 3) ≤ max{gI(E • 3 • Ä), gI(Ä)}. Since gI(E • 3 • Ä) = gI(0) and gI(0) ≤ gI(Ä) (from the definition), we obtain gI(E • 3) ≤ max{gI(0), gI(Ä)} = gI(Ä). (iii) gF (E•3) ≤ max{gF (E•3•Ä), gF (Ä)}. Since gF (E•3•Ä) = gF (0) and gF (0) ≤ gF (Ä) (from the definition), we get gF (E • 3) ≤ max{gF (0), gF (Ä)} = gF (Ä). Hence proved. Theorem 3.1. Every neutrosophic bi-ideal of ¨̈U is an neutrosophic sub-algebra of ¨̈U . Proof . Let neutrosophic set g=(gT , gI , gF ) be a neutrosophic bi-ideal of ¨̈U . Since E • 3 ≤ E for all E, 3,Ä ∈ ¨̈U . Consider E • 3 • 3 ≤ E for all E, 3,Ä ∈ ¨̈U , (E • (3 • 3) • E) = 0⇒(E • 0 • E) = ((E • 0) • E) = 0. It follows that, gT (E • 3 • 3) ≥ gT (E), gI(E • 3 • 3) ≤ gI(E) and gF (E • 3 • 3) ≤ gF (E). By Considering the Bi-ideal conditions, and substitute Ä = 3 for all E, 3,Ä ∈ ¨̈U . (i) gT (E • 3) ≥ min { gT (E • 3 • Ä), gT (Ä)} = min{gT (E • 3 • 3), gT (3)} (ii) gI(E • 3) ≤ max { gI(E • 3 • Ä), gI(Ä)} = max{gI(E • 3 • 3), gI(3)} (iii) Therefore, gF (E • 3) ≤ max {gF (E • 3 • Ä), gF (Ä)} = max{gF (E • 3 • 3), gF (3)}. Theorem 3.2. Let g=(gT , gI , gF ) and h=(hT , hI , hF ) be two neutrosophic bi-ideals of INK-algebra ¨̈U . Then g ∩ h = (Tg∩h(E), Ig∩h(E), Fg∩h(E)) is a neutrosophic bi-ideal of INK sub-algebra of ¨̈U . Proof . Let g and h be two neutrosophic bi ideals of INK-algebra ¨̈U . (i) Truth membership Tg∩h(E• 3) = min{gT (E• 3), hT (E• 3)} ≥ min {min{gT (E• 3• Ä), gT (Ä)}, min{hT (E• 3• Ä), hT (Ä)} = min{min{ gT (E• 3• Ä), hT (E• 3• Ä)}, min{gT (Ä), hT (Ä)} ≥ min{Tg∩h (E• 3• Ä) , Tg∩h (Ä)} (ii) Indeterminacy membership Ig∩h(E• 3) = max{gI(E• 3), hI(E• 3)} ≤ max {max{gI(E• 3• Ä), gI(Ä)}, max{hI(E• 3• Ä), hI(Ä)} = max{max{ gI(E• 3• Ä), hI(E• 3• Ä)}, max{gI(Ä), hI(Ä)} ≤ max{Ig∩h (E• 3• Ä) , Ig∩h (Ä)} M. Remala, E. Tamma, Y. Bhargavi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6392 8 of 20 (iii) (Falsehood membership) Fg∩h(E• 3) = max{gF (E• 3), hF (E• 3)} ≤ max {max{gF (E• 3• Ä), gF (Ä)}, max{hF (E• 3• Ä), hF (Ä)} = max{max{ gF (E• 3• Ä), hF (E• 3• Ä)}, max{gF (Ä), hF (Ä)} ≤ max{Fg∩h (E• 3• Ä) , Fg∩h (Ä)} Remark 3.1. The union of neutrosophic bi-ideal of INK-sub algebra of INK-algebra ¨̈U need not be a union of neutrosophic bi ideal. Example 3.2. Consider INK sub algebra ¨̈U = {0, 2, 4, 6} with the following Cayley table. • 0 2 4 6 0 0 0 0 0 2 2 0 0 2 4 4 2 0 4 6 6 6 6 0 Define Neutrosophic bi-ideal g = (gT , gI , gF ) by the component memberships below. • 0 2 4 6 gT 0.4 0.6 0.3 0.5 gI 0.3 0.8 0.5 0.6 gF 0.4 0.6 0.4 0.3 Define Neutrosophic bi-ideal h = (hT , hI , hF ) by the component memberships below. • 0 2 4 6 hT 0.6 0.4 0.4 0.4 hI 0.7 0.8 0.5 0.4 hF 0.8 0.2 0.4 0.5 Clearly, g and h are two Neutrosophic bi-ideals of INK sub-algebras. Here Tg∪h(2 • 4) = 0.4 but it is not greater than or equal to i.e., 0.5= min {Tg∪h(2 • 4 • 6), Tg∪h(6)}. Similarly, for Ig∪h(2 • 4) = 0.7 but it is not less than or equal to i.e., 0.4= max {Ig∪h(2 • 4 • 6), Tg∪h(6)}. Also for, Ig∪h(2 • 4) = 0.8 but it is not less than or equal to i.e., 0.4= max {Ig∪h(2 • 4 • 6), Tg∪h(6)}. Therefore, g ∪ h = (Tg∪h, Ig∪h, Fg∪h) is not a neutrosophic bi-ideal of INK sub-algebra. Thus, union of neutrosophic bi-ideal of INK-sub algebras is not a neutrosophic bi-ideal. In particular, that follows Theorem 3.3. Let g = (gT , gI , gF ) and h = (hT , hI , hF ) be two neutrosophic bi-ideals of INK sub algebras of INK algebra ¨̈U , then g∪h is a neutrosophic bi-ideal of INK sub-algebra only if g ⊆ h or h ⊆ g. Proof . Suppose g ⊆ h. Let E, 3 ∈ ¨̈U . M. Remala, E. Tamma, Y. Bhargavi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6392 9 of 20 (i) Truth membership Tg∪h(E • 3) = max{ gT (E • 3), hT (E • 3) } = hT (E • 3) ≥ min{hT (E • 3 • Ä), hT (Ä) } ≥ min { max{gT (E • 3 • Ä), hT (E • 3 • Ä)} • max{gT (Ä), hT (Ä)} } = max{Tg∪h(E • 3 • Ä), Tg∪h(Ä) }. (ii) Indeterminacy membership Ig∪h(E • 3) = min{ gI(E • 3), hI(E • 3) } = hI(E • 3) ≤ max{hI(E • 3 • Ä), hI(Ä) } ≤ max { min{gI(E • 3 • Ä), hI(E • 3 • Ä)} • min{gI(Ä), hI(Ä)} } = min{ Ig∪h(E • 3 • Ä), Ig∪h(Ä) }. (iii) Falsehood membership Fg∪h(E • 3) = min{ gF (E • 3), hF (E • 3) } = hF (E • 3) ≤ max{hF (E • 3 • Ä), hF (Ä) } ≤ max { min{gF (E • 3 • Ä), hF (E • 3 • Ä)} • min{gF (Ä), hF (Ä)} } = min{Fg∪h(E • 3 • Ä), Fg∪h(Ä) }. This completes the verification under the assumption g ⊆ h. 4. Homomorphism of Neutrosophic Bi-ideal of INK sub-algebra Definition 4.1. Let ¢ : ¨̈U → ˘̈̈ U be a homomorphism of INK-algebra and g = (gT , gI , gF ) be a neutrosophic set in X̆, then the neutrosophic set g[¢] = (gT [¢], gI [¢], gF [¢]) in ¨̈U is defined by neutrosophic set such that for every E ∈ ¨̈U is called pre-image of g under ¢. gT [¢] : ¨̈U → [0, 1], gT [¢](E) = gT (¢(E)). gI [¢] : ¨̈U → [0, 1], gI [¢](E) = gI(¢(E)). gF [¢] : ¨̈U → [0, 1], gF [¢](E) = gF (¢(E)). Proof . Let ¢ : ¨̈U → ˘̈̈ U be a homomorphism of INK-algebra. If g = (gT , gI , gF ) be a neutrosophic bi-ideal in INK-algebra Y , and g[¢] = (gT [¢], gI [¢], gF [¢]) be the pre-image of g under ¢ is defined by We first have that gT [¢](E • 3) = gT (¢(E • 3)) ≥ gT (0) = gT (¢(0)), gI [¢](E • 3) = gI(¢(E • 3)) ≤ gI(0) = gI(¢(0)), gF [¢](E • 3) = gF (¢(E • 3)) ≤ gF (0) = gF (¢(0)) for all E ∈ ¨̈U. M. Remala, E. Tamma, Y. Bhargavi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6392 10 of 20 Consider gT [¢](E • 3) = gT (¢(E • 3)) ≥ min{ gT [¢](E • 3 • Ä), gT [¢](Ä) } ≥ min{ gT (¢(E • 3 • Ä)), gT (¢(Ä)) } ≥ min{ gT (¢((E • 3) • Ä)), gT (¢(Ä)) } ≥ min{ gT ((¢(E) • ¢(3)) • (¢(Ä))), gT (¢(Ä)) } ≥ min{ gT (¢(E) • (¢(3))), gT (¢(Ä)), gT (¢(Ä)) } ≥ min{ gT (¢(E • 3)) }. Also for, gI [¢](E • 3) = gI(¢(E • 3)) ≤ max{ gI [¢](E • 3 • Ä), gI [¢](Ä) } ≤ max{ gI(¢(E • 3 • Ä)), gI(¢(Ä)) } ≤ max{ gI(¢((E • 3) • Ä))), gI(¢(Ä)) } ≤ max{ gI((¢(E) • ¢(3)) • (¢(Ä))), gI(¢(Ä)) } ≤ max{ gI(¢(E) • ¢(3)), gI(¢(Ä)), gI(¢(Ä)) } ≤ max{ gI(¢(E • 3)) }. Similarly, gF [¢](E • 3) = gF (¢(E • 3)) ≤ max{ gF [¢](E • 3 • Ä), gF [¢](Ä) } ≤ max{ gF (¢(E • 3 • Ä)), gF (¢(Ä)) } ≤ max{ gF (¢((E • 3) • Ä))), gF (¢(Ä)) } ≤ max{ gF ((¢(E) • ¢(3)) • (¢(Ä))), gF (¢(Ä)) } ≤ max{ gF (¢(E) • ¢(3)), gF (¢(Ä)), gF (¢(Ä)) } ≤ max{ gF (¢(E • 3)) }. Theorem 4.1. Let ¢ : ¨̈U → ˘̈̈ U be an epimorphism of INK-algebra. If g[¢] = (gT [¢], gI [¢], gF [¢]) is an neutrosophic bi-ideal of INK sub-algebra of INK-algebra ¨̈U , then g = (gT , gI , gF ) is an neutrosophic bi-ideal of INK sub-algebra of INK-algebra ¨̈U . Proof . Let E, 3 ∈ ¨̈U , there exist E, 3 ∈ X such that ¢(E • 3) = E • 3. Then gT (E • 3) = gT (¢)(E • 3), gT [¢](E • 3) ≥ gT = gT (¢(0)) = gT (0), gI(E • 3) = gI(¢)(E • 3), gI [¢](E • 3) ≤ gI = gI(¢(0)) = gI(0), gF (E • 3) = gF (¢)(E • 3), gF [¢](E • 3) ≤ gF = gF (¢(0)) = gF (0). Consider gT (E • 3) = gT (¢)(E • 3) = gT [¢](E • 3) ≥ min{ gT ([¢](E • 3) • [¢](Ä)), gT [¢](Ä) } ≥ min{ gT ([¢](E • 3 • Ä)), gT [¢](Ä) } ≥ min{ gT ((¢(E • 3 • Ä)), gT (¢(Ä)) } ≥ min{ gT (E • 3 • Ä), gT (Ä) }. M. Remala, E. Tamma, Y. Bhargavi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6392 11 of 20 Also for gI(E • 3) = gI(¢)(E • 3) = gI [¢](E • 3) ≤ max{ gI([¢](E • 3) • [¢](Ä)), gI [¢](Ä) } ≤ max{ gI([¢](E • 3 • Ä)), gI [¢](Ä) } ≤ max{ gI((¢(E • 3 • Ä)), gI(¢(Ä)) } ≤ max{ gI(E • 3 • Ä), gI(Ä) }. Similarly, gF (E • 3) = gF (¢)(E • 3) = gF [¢](E • 3) ≤ max{ gI([¢](E • 3) • [¢](Ä)), gF [¢](Ä) } ≤ max{ gF ([¢](E • 3 • Ä)), gF [¢](Ä) } ≤ max{ gF ((¢(E • 3 • Ä)), gF (¢(Ä)) } ≤ max{ gF (E • 3 • Ä), gF (Ä) }. 5. Direct Product of Neutrosophic Bi-ideal of INK sub-algebra Definition 5.1. Let g=(gT , gI , gF ) and h=(hT , hI , hF ) be two neutrosophic sets of INK algebras ¨̈U1 and ¨̈U2 respectively. Then the direct product of neutrosophic set g and h is given by g × h= (gT (g×h), gI (g×h), gF (g×h)) with (i) gT (g×h) (E1,31) = min {gT (g)(E1 ), gT (h)( 31)} (ii) gI (g×h) (E1,31) = max {gI (g)(E1 ), gI (h)( 31)} (iii) gF (g×h) (E1,31) = max {gF (g)(E1), gF (h)(31)} for all E1,31 ∈ ¨̈U1 × ¨̈U2. Definition 5.2. Let g × h= (gT (g×h), gI (g×h), gF (g×h)) be neutrosophic set in INK- algebra ¨̈U1 and ¨̈U2 respectively. Then the direct product of neutrosophic INK algebra of ¨̈U1 × ¨̈U2. (i) gT (g×h) (E1 ,31)• (E2 ,32)) ≥ min{ gT (g×h) (E1 ,31), gT (g×h) (E2 ,32)} (ii) gI (g×h) (E1 ,31)• (E2 ,32)) ≤ max{ gI (g×h) (E1 ,31), gI (g×h) (E2 ,32)} (iii) gF (g×h) (E1 ,31)• (E2 ,32)) ≤ max{ gF (g×h) (E1 ,31), gF (g×h) (E2 ,32)} for all (E1, E2, E3) and (31, 32, 33) ∈ ¨̈U1 × ¨̈U2. ¨̈U2. Theorem 5.1. Let g=(gT , gI , gF ) and h=(hT , hI , hF ) be two neutrosophic INK algebras ¨̈U1 and ¨̈U2 respectively. Then g × h = (gT (g×h), gI (g×h), gF (g×h)) is also neutrosphic sub algebra of INK-algebra of ¨̈U1 × ¨̈U2 . M. Remala, E. Tamma, Y. Bhargavi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6392 12 of 20 Proof . For any (E1, 31), (E2, 32) ∈ ¨̈U1 × ¨̈U2. Now, gT (g×h)(E1, 31) • (E2, 32) = gT (g×h)(E1 • E2), (31 • 32) = min{ gT (g)(E1 • E2), gT (h)(31 • 32) } = min{ gT (g)(E1 • E2), gT (h)(31 • 32) } ≥ min{min{gT (g)(E1), gT (g)(E2)}, min{gT (h)(31), gT (h)(32)} } ≥ min{ gT (g×h)(E1, E2), gT (g×h)(31, 32) }. Then, gI(g×h)(E1, 31) • (E2, 32) = gI(g×h)(E1 • E2), (31 • 32) = max{ gI(g)(E1 • E2), gI(h)(31 • 32) } = max{ gI(g)(E1 • E2), gI(h)(31 • 32) } ≤ max{max{gI(g)(E1), gI(g)(E2)}, max{gI(h)(31), gI(h)(32)} } ≤ max{ gI(g×h)(E1, E2), gI(g×h)(31, 32) }. Also, gF (g×h)(E1, 31) • (E2, 32) = gF (g×h)(E1 • E2), (31 • 32) = max{ gF (g)(E1 • E2), gF (h)(31 • 32) } = max{ gF (g)(E1 • E2), gF (h)(31 • 32) } ≤ max{max{gF (g)(E1), gF (h)(E2)}, max{gF (g)(31), gF (h)(32)} } ≤ max{ gF (g×h)(E1, E2), gF (g×h)(31, 32) }. Theorem 5.2. Let g = (gT , gI , gF ) and h = (hT , hI , hF ) be two neutrosophic INK algebras ¨̈U1 and ¨̈U2 respectively. Then (i) gT (g×h)(0, 0) = gT (g×h)(E1, 31), (ii) gI(g×h)(0, 0) = gI(g×h)(E1, 31), (iii) gI(g×h)(0, 0) = gI(g×h)(E1, 31) for all (E1, 31) ∈ ¨̈U1 × ¨̈U2. Proof . By definition, gT (g×h)(0, 0) = gT (g×h) ( (E1, 31) • (E1, 31) ) = min { gT (g×h) ( (E1, 31) • (E1, 31) ) , gT (g×h)(E1, 31) } ≥ gT (g×h)(E1, 31). gI(g×h)(0, 0) = gI(g×h) ( (E1, 31) • (E1, 31) ) = max { gI(g×h) ( (E1, 31) • (E1, 31) ) , gI(g×h)(E1, 31) } ≤ gI(g×h)(E1, 31). M. Remala, E. Tamma, Y. Bhargavi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6392 13 of 20 gF (g×h)(0, 0) = gF (g×h) ( (E1, 31) • (E1, 31) ) = max { gF (g×h) ( (E1, 31) • (E1, 31) ) , gF (g×h)(E1, 31) } ≤ gF (g×h)(E1, 31). Definition 5.3. Let g×h = ( gT (g×h), gI(g×h), gF (g×h) ) of ¨̈U1 and ¨̈U2 be the direct product of neutrosophic bi-ideals of ¨̈U1 × ¨̈U2 if (i) gT (g×h)(0, 0) ≥ gT (g×h)(E1, 31), (ii) gI(g×h)(0, 0) ≤ gI(g×h)(E1, 31), (iii) gF (g×h)(0, 0) ≤ gF (g×h)(E1, 31), (iv) gT (g×h) ( (E1, 31)•(E2, 32) ) ≥ min{ gT (g×h) ( (E1, 31)•(E2, 32)•(E3, 33) ) , gT (g×h)(E3, 33) }, (v) gI(g×h) ( (E1, 31)•(E2, 32) ) ≤ max{ gI(g×h) ( (E1, 31)•(E2, 32)•(E3, 33) ) , gI(g×h)(E3, 33) }, (vi) gF (g×h) ( (E1, 31)•(E2, 32) ) ≤ max{ gF (g×h) ( (E1, 31)•(E2, 32)•(E3, 33) ) , gF (g×h)(E3, 33) }. Theorem 5.3. Let g = (gT , gI , gF ) and h = (hT , hI , hF ) be two neutrosophic bi-ideals of INK algebras ¨̈U1 and ¨̈U2 respectively. Then the direct product of neutrosophic bi-ideals of INK-algebra g and h is given by g × h = ( gT (g×h), gI(g×h), gF (g×h) ) . Proof . For any (E1, E2, E3) and (31, 32, 33) ∈ g × h. Then gT (g×h)(0, 0) = min{ gT (g)(0), gT (h)(0) } ≥ { gT (g)(E1), gT (h)(31) } ≥ gT (g×h)(E1, 31). Now gI(g×h)(0, 0) = max{ gT (g)(0), gI(h)(0) } ≤ { gI(g)(E1), gI(h)(31) } ≤ gI(g×h)(E1, 31). Also gI(g×h)(0, 0) = max{ gT (g)(0), gI(h)(0) } ≤ { gI(g)(E1), gI(h)(31) } ≤ gI(g×h)(E1, 31). Now gT (g×h) ( (E1, 31) • (E2, 32) ) = gT (g×h) ( E1 • 31, E2 • 32 ) = min{ gT (g) ( E1 • 31, E2 • 32 ) } = min{min{gT (g)(E1 • E2 • E3), gT (g)(E3)}, min{gT (h)(31 • 32 • 33), gT (h)(33)} } ≥ min{ gT (g×h) ( (E1, 31) • (E2, 32) • (E3, 33) ) , gT (g×h)(E3, 33) }. M. Remala, E. Tamma, Y. Bhargavi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6392 14 of 20 Then, gI(g×h) ( (E1, 31) • (E2, 32) ) = gI(g×h) ( E1 • 31, E2 • 32 ) = max{ gI(g) ( E1 • 31, E2 • 32 ) } = max{max{gI(g)(E1 • E2 • E3), gI(g)(E3)}, max{gI(h)(31 • 32 • 33), gI(h)(33)} } ≤ max{ gI(g×h) ( (E1, 31) • (E2, 32) • (E3, 33) ) , gI(g×h)(E3, 33) }. Also, gF (g×h) ( (E1, 31) • (E2, 32) ) = gF (g×h) ( E1 • 31, E2 • 32 ) = max{ gF (g) ( E1 • 31, E2 • 32 ) } = max{max{gF (g)(E1 • E2 • E3), gF (g)(E3)}, max{gF (h)(31 • 32 • 33), gF (h)(33)} } ≤ max{ gF (g×h) ( (E1, 31) • (E2, 32) • (E3, 33) ) , gF (g×h)(E3, 33) }. Theorem 5.4. Let g × h = (gT (g×h), gI (g×h), gF (g×h)) and i × j = (g T (i×j), gI (i×j), gF (i×j)) is a neutrosophic bi-ideal of INK-algebra ¨̈U1 and ¨̈U2. Then (g × h) ∩ (i × j) = (gT (g×h) ∩ (i×j), gI (g×h) ∩ (i×j), gF (g×h) ∩ (i×j)) Proof . For any (E1, E2, E3) and (31, 32, 33) ∈ ¨̈U1 × ¨̈U2. Consider gT (g×h)(0, 0) ≥ min { gT (g×h)(E1, 31) } and gT (i×j)(0, 0) ≥ min { gT (i×j)(E1, 31) } . { gT (g×h)(0, 0), gT (i×j)(0, 0) } ≥ { gT (g×h)(E1, 31), gT (i×j)(E1, 31) }, min{ gT (g×h)(0, 0), gT (i×j)(0, 0) } ≥ min{ gT (i×j)(E1, 31), gT (i×j)(E1, 31) }. g T ( (g×h)∩(i×j) )(0, 0) ≥ g T ( (g×h)∩(i×j) )(E1, 31). gI(g×h)(0, 0) ≤ max { gI(g×h)(E1, 31) } , gI(i×j)(0, 0) ≤ max { gI(i×j)(E1, 31) } , { gI(g×h)(0, 0), gI(i×j)(0, 0) } ≤ { gI(g×h)(E1, 31), gI(i×j)(E1, 31) }, max{ gI(g×h)(0, 0), gI(i×j)(0, 0) } ≤ max{ gI(i×j)(E1, 31), gI(i×j)(E1, 31) }. g F ( (g×h)∩(i×j) )(0, 0) ≤ g F ( (g×h)∩(i×j) )(E1, 31). gF (g×h)(0, 0) ≤ max { gF (g×h)(E1, 31) } , gF (i×j)(0, 0) ≤ max { gF (i×j)(E1, 31) } , { gF (g×h)(0, 0), gF (i×j)(0, 0) } ≤ { gF (g×h)(E1, 31), gF (i×j)(E1, 31) }, max{ gF (g×h)(0, 0), gF (i×j)(0, 0) } ≤ max{ gF (i×j)(E1, 31), gF (i×j)(E1, 31) }. M. Remala, E. Tamma, Y. Bhargavi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6392 15 of 20 g F ( (g×h)∩(i×j) )(0, 0) ≤ g F ( (g×h)∩(i×j) )(E1, 31). Now (E1, 31,Ä1), (E2, 32,Ä2) ∈ X1 ×X2. Consider gT (g×h)(E1, 31) = min { gT (g×h) ( (E1, 31) • (E2, 32) • (E3, 33) ) , gT (g×h)(E3, 33) } , gT (i×j)(E1, 31) = min { gT (i×j) ( (E1, 31) • (E2, 32) • (E3, 33) ) , gT (i×j)(E3, 33) } . gT (g×h)(E1, 31), gT (i×j)(E1, 31) ≥ min { min{ gT (g×h)((E1, 31) • (E2, 32) • (E3, 33)), gT (g×h)(E3, 33) }, min{ gT (i×j)((E1, 31) • (E2, 32) • (E3, 33)), gT (i×j)(E3, 33) } } ≥ min { min{ gT (g×h)((E1, 31) • (E2, 32) • (E3, 33)), gT (i×j)((E1, 31) • (E2, 32) • (E3, 33)) }, min{ gT (g×h)(E3, 33), gT (i×j)(E3, 33) } } . g T ( (g×h)∩(i×j) )(E1, 31) ≥ { g T ( (g×h)∩(i×j) )((E1, 31)•(E2, 32)•(E3, 33) ) , g T ( (g×h)∩(i×j) )(E3, 33) } . Also for gI(g×h)(E1, 31) = max { gI(g×h) ( (E1, 31) • (E2, 32) • (E3, 33) ) , gI(g×h)(E3, 33) } , gI(i×j)(E1, 31) = max { gI(i×j) ( (E1, 31) • (E2, 32) • (E3, 33) ) , gI(i×j)(E3, 33) } , gI(g×h)(E1, 31), gI(i×j)(E1, 31) ≤ max { max{ gI(g×h)((E1, 31) • (E2, 32) • (E3, 33)), gI(g×h)(E3, 33) }, max{ gI(i×j)((E1, 31) • (E2, 32) • (E3, 33)), gI(i×j)(E3, 33) } } ≤ max { max{ gI(g×h)((E1, 31) • (E2, 32) • (E3, 33)), gI(i×j)((E1, 31) • (E2, 32) • (E3, 33)) }, max{ gI(g×h)(E3, 33), gI(i×j)(E3, 33) } } . g I ( (g×h)∩(i×j) )(E1, 31) ≤ { g I ( (g×h)∩(i×j) )((E1, 31)•(E2, 32)•(E3, 33) ) , g I ( (g×h)∩(i×j) )(E3, 33) } . Similarly for gF (g×h)(E1, 31) = max { gF (g×h) ( (E1, 31) • (E2, 32) • (E3, 33) ) , gF (g×h)(E3, 33) } , gF (i×j)(E1, 31) = max { gF (i×j) ( (E1, 31) • (E2, 32) • (E3, 33) ) , gF (i×j)(E3, 33) } , M. Remala, E. Tamma, Y. Bhargavi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6392 16 of 20 gF (g×h)(E1, 31), gF (i×j)(E1, 31) ≤ max { max{ gF (g×h)((E1, 31) • (E2, 32) • (E3, 33)), gF (g×h)(E3, 33) }, max{ gF (i×j)((E1, 31) • (E2, 32) • (E3, 33)), gF (i×j)(E3, 33) } } ≤ max { max{ gF (g×h)((E1, 31) • (E2, 32) • (E3, 33)), gF (i×j)((E1, 31) • (E2, 32) • (E3, 33)) }, max{ gF (g×h)(E3, 33), gF (i×j)(E3, 33) } } . g F ( (g×h)∩(i×j) )(E1, 31) ≤ { g F ( (g×h)∩(i×j) )((E1, 31)•(E2, 32)•(E3, 33) ) , g F ( (g×h)∩(i×j) )(E3, 33) } . This completes the required formulation for (g×h)∩(i×j) = ( gT (g×h)∩gT (i×j), gI(g×h)∩ gI(i×j), gF (g×h) ∩ gF (i×j) ) . 6. Application of Bi-ideal in INK-algebra Flowchart for Neutrosophic Bi-ideal in INK-algebra Figure 1: Flowchart for Neutrosophic Bi-ideal in INK-algebra 7. Selection of Research Guide/Supervisor by using Neutrosophic Bi-ideals in INK-algebra In academic research, selecting the right guide is necessary a critical decision that can be effect and shape scholars entire research experience. The choice or consideration M. Remala, E. Tamma, Y. Bhargavi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6392 17 of 20 of selection of guide depends on multiple parameters. To model this selection process logically, we apply the concept of neutrosophic Bi-ideal in INK-algebra, which allows us to evaluate whether removing a less important factor still results in a valid and trustworthy decision. We define a set of elements in our INK-algebra ¨̈U={0, A, G, C} where each and every parameter represents guide’s selection. Here, A= Alignment / P = Proposal idea, G=Guide’s Academic and research experience, C=Availability of communication with guide. The element “0” represents an ideal guide selection. In this context, the term E1 •31 interpreted as the how much mismatch is still there when choosing a guide based on those two factors in that order. This is taken for a better experience, a mismatch in any of them may not invalidate the overall decision if the core match is strong. To evaluate this system, we construct a cayley table of INK-algebra such as associa- tivity –like behaviour, and conditions of INK-algebras also we have to define membership values for each parameter. Table: 1 Cayley Table on ¨̈U • 0 A G C 0 0 A G C A A 0 C G G G C 0 A C C G A 0 Table: 2 Neutrosophic Membership Degrees • 0 A G C gT 1.0 0.9 0.9 0.6 gI 0.0 0.2 0.5 0.2 gF 0.0 0.3 0.1 0.4 {If E1 • 31 • Ä1 = E1 • 31, and one factor (eg = Ä1) is trusted, then does the core result E1 • 31 hold up on its own.} If the Researcher consider the criteria like Guide academic & research experience, fol- lowed by Alignment/Proposal idea and then Communication of avialbility of Guide. Let’s break down (i) gT (0) ≥ gT (G) ⇒ 1.0 > 0.9, (ii) gI(0) ≤ gI(G) ⇒ 0.0 < 0.5, (iii) gF (0) ≤ gF (G) ⇒ 0.0 < 0.1. M. Remala, E. Tamma, Y. Bhargavi / Eur. J. Pure Appl. Math, 18 (4) (2025), 6392 18 of 20 (iv) gT (G •A) ≥ min{gT (G •A • C), gT (C)}, gT (G) ≥ min{gT (G), gT (C)} 0.9 ≥ min{0.9, 0.8} 0.9 > 0.8. (v) gI(G •A) ≤ max{gI(G •A • C), gI(C)}, gI(G) ≤ max{gI(G), gT (C)} 0.5 ≤ max{0.5, 0.1} 0.5 = 0.5. (vi) gF (G •A) ≤ max{gF (G •A • C), gF (C)}, gF (G) ≤ max{gF (G), gF (C)} 0.1 ≤ max{0.1, 0.2} 0.1 < 0.2. Based on the result as long as starting with guide is experienced. Then, it is sufficient to make reliable descion. This allows us to ignore whether Alignment(A) is perfect or Communcation(C) is ideal. 8. Conclusion Our article investigates into the thought of neutrosophic bi-ideals within the frame- work of INK-algebras. We begin by starting neutrosophic bi-ideals of INK-algebras. We then explore their properties, including how intersection of neutrosophic bi-ideals works, and how containment relationships define them union of neutrosophic bi-ideals by one containing the other., Also, we investigate homomorphisms and epimorphisms of neutro- sophic bi-ideals. Subsequently, we discuss the direct product of neutrosophic sets and demonstrate that the intersection of direct products of neutrosophic bi-ideals remains a neutrosophic bi-ideal. Finally, we look over the application of bi-ideal in INK-algebra by considering parameters of selecting guide by a research scholar by using neutrosophic bi-ideals in INK-algebra. References [1] K. Iseki. On BCI-algebras. Math. Semin. Notes, Kobe Univ., 8:125–130, 1980. [2] K. Iseki and S. Tanaka. An introduction to the theory of BCK-algebras. Math. Japan, 23:1–26, 1978. [3] M. Kaviyarasu, K. Indhira, and V. M. Chandrasekaran. Fuzzy sub-algebras and fuzzy K-ideals in INK-algebras. International Journal of Pure and Applied Mathematics, 113(6):47–55, 2017. [4] M. Kaviyarasu, K. Indhira, V. M. Chandrasekaran, and K. Jacob. Interval valued fuzzy subalgebra and fuzzy INK-ideal in INK-algebra. In Advances in Algebra and Analysis, pages 19–25. 2018. M. Remala, E. Tamma, Y. Bhargavi / Eur. J. Pure Appl. 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