EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 14, No. 3, 2021, 895-904 ISSN 1307-5543 – ejpam.com Published by New York Business Global Introduction to Neutrosophic B-algebras Danilo O. Jacobe1,∗, Jocelyn P. Vilela2 1 Institute of Computing and Engineering, Davao Oriental State University, Dahican, Mati City, 8200 Davao Oriental, Philippines 2 Department of Mathematics and Statistics, College of Science and Mathematics, Center of Graph Theory, Algebra and Analysis, Premier Research of Institute of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, Tibanga, Iligan City, 9200 Lanao del Norte, Philippines Abstract. This paper introduces the notion of neutrosophic B-algebra. Several results on prop- erties of neutrosophic B-algebras and neutrosophic subalgebras are presented and proved. 2020 Mathematics Subject Classifications: 03G25, 06F35 Key Words and Phrases: Neutrosophic B-algebra, neutrosophic subalgebra, neutrosophic al- gebraic structures 1. Introduction In 1995, Smarandache [16] introduced the concept of neutrosophic logic as an ex- tension of fuzzy logic in which indeterminacy is included. Indeterminacy means degrees of uncertainty, vagueness, imprecision, undefined, unknown, inconsistency or redundancy, for example, in tossing a die on irregular surface one can get {1, 2, 3, 4, 5, 6, indeterminacy}. In the neutrosophic logic, each proposition is estimated to have the percentage of truth in a subset T, the percentage of indeterminacy in a subset I, and the percentage of falsity in a subset F. Using neutrosophic theory, Vasantha Kandasamy and Florentin Smarandache [11] in 2003 introduced a neutrosophic structure based on indeterminacy “I” only, which they called I-Neutrosophic Algebraic Structures, an algebraic structure based on sets of neutro- sophic numbers of the form N = a + bI, where a, b are real (or complex) numbers, and I is called literal indeterminacy, which stands for unknown or non-determinate such that I2 = I. Here, a is called the determinate part of N and bI is called the indeterminate part of N , with mI + nI = (m + n)I, 0 · I = 0. The indeterminacy I is different from the imaginary i = √ −1. In general, one has In = I if n > 0, and is undefined for n ≤ 0. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v14i3.3996 Email addresses: danilo.jacobe@g.msuiit.edu.ph (D.O. Jacobe), jocelyn.vilela@g.msuiit.edu.ph (J.P. Vilela) http://www.ejpam.com 895 © 2021 EJPAM All rights reserved. D.O. Jacobe, J.P. Vilela / Eur. J. Pure Appl. Math, 14 (3) (2021), 895-904 896 In 2006, they introduced some neutrosophic algebraic structures like neutrosophic fields, neutrosophic vector spaces, neutrosophic groups, neutrosophic bigroups, neutrosophic N - groups, neutrosophic semigroups, neutrosophic bisemigroups, neutrosophic N-semigroup, neutrosophic loops, neutrosophic biloops, neutrosophic N-loop, neutrosophic groupoids, neutrosophic bigroupoids, and neutrosophic rings [12, 17]. In 2015, A.A.A Agboola and B. Davvaz [1] introduced the concept of neutrosophic BCI/BCK. In 2002, J. Neggers and H.S. Kim [14, 15] introduced the concept of B -algebra and established some properties of B -homomorphism[14]. From then on, several characteri- zations as to commutativity and center, cyclicity, isomorphism, direct product, Lagrange and Cauchy’s Theorems, B -action and the Sylow Theorems for B -algebras as exemplified by the following literatures [2–5, 8–10, 13]. In this paper, we introduce the concepts of neutrosophic B -algebra and neutrosophic subalgebra. Some properties of neutrosophic B -algebras and neutrosophic subalgebras are presented and proved. 2. Preliminaries For convenience, we view the neutrosophic number N = a+ bI as an ordered pair (a, bI). Definition 2.1. [7] Let X be a nonempty set and let I be an indeterminate. A set X(I) = 〈X, I〉 = {(x, yI) : x, y ∈ X} is called a neutrosophic set generated by X and I. A type (2, 0) algebra is an algebra formed from a nonempty set X together with 2-ary operation ∗ and a 0-ary operation(with constant element 0). Definition 2.2. [15] Let X be a nonempty set with a binary operation “ ∗ ” on X and a constant 0. Then the algebra (X; ∗, 0) of type (2, 0) is called a B-algebra if it satisfies the following axioms: for all x, y, z ∈ X, (B1) x ∗ x = 0; (B2) x ∗ 0 = x; (B3) (x ∗ y) ∗ z = x ∗ (z ∗ (0 ∗ y)). Example 2.3. The following are examples of B -algebra. (i) [15] Let X = {0, 1, 2}. Define the operation “ ∗ ” by the Cayley table shown below. ∗ 0 1 2 0 0 2 1 1 1 0 2 2 2 1 0 (ii) [15] Let X = {0, 1, 2, 3, 4, 5}. De- fine the operation “ ∗ ” by the Cayley table shown below. ∗ 0 1 2 3 4 5 0 0 2 1 3 4 5 1 1 0 2 4 5 3 2 2 1 0 5 3 4 3 3 4 5 0 2 1 4 4 5 3 1 0 2 5 5 3 4 2 1 0 D.O. Jacobe, J.P. Vilela / Eur. J. Pure Appl. Math, 14 (3) (2021), 895-904 897 The following properties of B-algebra can be found in [15] and [18]. Let (X; ∗, 0) be a B -algebra. Then for any x, y, z ∈ X, (P1): 0 ∗ (0 ∗ x) = x, (P2): 0 ∗ (x ∗ y) = y ∗ x, (P3): (x ∗ z) ∗ (y ∗ z) = x ∗ y, and (P4): x ∗ y = 0 =⇒ x = y. Lemma 2.4. Let X be a B-algebra. Then for any x, y, z ∈ X, (i) [6] (left cancellation law) x ∗ y = x ∗ z implies y = z. (ii) [15] x ∗ (y ∗ z) = (x ∗ (0 ∗ z)) ∗ y. Definition 2.5. [15] A B-algebra (X; ∗, 0) is said to be commutative if a ∗ (0 ∗ b) = b ∗ (0 ∗ a) for any a, b ∈ X. Lemma 2.6. [15] Let X be a commutative B-algebra. Then for any x, y, z ∈ X, x∗(x∗y) = y. Definition 2.7. [14] Let (X; ∗, 0) be a B -algebra. A nonempty subset N of X is said to be a subalgebra of X if a ∗ b ∈ N for all a, b ∈ N . N is said to be normal if for any x ∗ y, a ∗ b ∈ N implies (x ∗ a) ∗ (y ∗ b) ∈ N . Lemma 2.8. [9] Let X be a B-algebra. If {Nα : α ∈ A } is any nonempty collection of subalgebras (resp., normal subalgebras) of X, then ⋂ α∈A Nα is a subalgebra (resp., normal subalgebra) of X. 3. Some Properties of Neutrosophic B-algebras and Neutrosophic Subalgebras Definition 3.1. Let (X; ∗, 0) be any B -algebra. The set X(I) = {(x, yI) : x, y ∈ X} is the neutrosophic set determined by X and I. Moreover, (a, bI) = (c, dI) in X(I) if and only if a = c and b = d. Definition 3.2. Let (X; ∗, 0) be any B -algebra. For any x, y ∈ X, we denote x ∧ y = x ∗ (x ∗ y). Lemma 3.3. The mapping λ : X(I) × X(I) → X(I) defined by λ((a, bI), (c, dI)) = (a, bI) · (c, dI) = ( a∗c, ((a∗d∧b∗c)∧b∗d)I ) for any (a, bI), (c, dI) ∈ X(I) is well-defined. Proof : Let (a, bI), (c, dI), (x, yI), (u, vI) ∈ X(I) such that (a, bI) = (x, yI) and (c, dI) = (u, vI). Then λ ( (a, bI), (c, dI) ) = (a, bI) · (c, dI) = ( a ∗ c, ((a ∗ d ∧ b ∗ c) ∧ b ∗ d)I ) =( x ∗ u, ((x ∗ v ∧ y ∗ u) ∧ y ∗ v)I ) = (x, yI) · (u, vI) = λ ( (x, yI), (u, vI) ) . Hence, the λ is well-defined. � Definition 3.4. The triple (X(I); ·, (0, 0I)) is called a neutrosophic B-algebra determined by X and I with the binary operation · defined in Lemma 3.3 and (0, 0I) as its constant element. Remark 3.5. Every nonzero neutrosophic B-algebra X(I) always contains the B-algebra X ′ = {(x, 0I) : x ∈ X} as a proper subset. D.O. Jacobe, J.P. Vilela / Eur. J. Pure Appl. Math, 14 (3) (2021), 895-904 898 Let X(I) stand for a neutrosophic B-algebra (X(I); ·, (0, 0I)), unless otherwise stated. Example 3.6. Consider the commutative B-algebra X = {0, 1, 2} in Example 2.3(i). Then the neutrosophicB-algebra determined byX and I is given byX(I) = {(0, 0I), (0, I), (0, 2I), (1, 0I), (1, I), (1, 2I), (2, 0I), (2, I), (2, 2I)}. Lemma 3.7. Let X be a B-algebra. Then for any x, y ∈ X, (i) x ∧ x = x, (iv) x ∗ y ∧ y ∗ x = y ∗ x, (ii) x ∧ 0 = 0, (v) x ∧ y = 0 if and only if y = 0. (iii) 0 ∧ x = x, Proof : (i) By (B1) and (B2), x ∧ x = x ∗ (x ∗ x) = x ∗ 0 = x; (ii) By (B2) and (B1), x∧ 0 = x ∗ (x ∗ 0) = x ∗x = 0; (iii) By (P1), 0∧x = 0 ∗ (0 ∗x) = x; (iv) By Lemma 2.4(ii), (B1) and (P3), x ∗ y ∧ y ∗ x = (x ∗ y) ∗ [(x ∗ y) ∗ (y ∗ x)] = [(x ∗ y) ∗ [0 ∗ (y ∗ x)]] ∗ (x ∗ y) = [(x ∗ y) ∗ (x ∗ y)] ∗ (x ∗ y) = 0 ∗ (x ∗ y) = y ∗ x; (v) x ∧ y = 0 implies that x ∗ (x ∗ y) = 0. By (P4), x = x ∗ y which can be written as x ∗ 0 = x ∗ y. Hence, by Lemma 2.4(i), y = 0. The converse follows directly from (ii). � Lemma 3.8. If X(I) is a neutrosophic B-algebra, then for any (a, bI), (c, dI) ∈ X(I), (i) (a, bI) · (0, 0I) = ( a, ((a ∧ b) ∧ b)I ) , (ii) (a, cI) · (b, cI) = (a ∗ b, 0I), (iii) (a, aI) · (b, bI) = (a ∗ b, (a ∗ b)I), (iv) (a, bI) · (c, dI) = (0, 0I) if and only if (a, bI) = (c, dI). Proof : (i) By Definition 3.4 and (B2), (a, bI) · (0, 0I) = (a ∗ 0, ((a ∗ 0 ∧ b ∗ 0) ∧ b ∗ 0)I) = (a, ((a ∧ b) ∧ b)I); (ii) By Definition 3.4, (B1), (B2) and Lemma 3.7(ii), (a, cI) · (b, cI) = (a∗b, ((a∗c∧c∗b)∧c∗c)I) = (a∗b, ((a∗c∧c∗b)∧0)I) = (a∗b, 0I); (iii) Follows directly from Definition 3.4 and Lemma 3.7(i); (iv) By Definition 3.4, (a, bI) · (c, dI) = (0, 0I) implies that ( a∗c, ((a∗d∧b∗c)∧b∗d)I ) = (0, 0I). That is, a∗c = 0 and (a∗d∧b∗c)∧b∗d = 0. By (P4), a = c and by Lemma 3.7(v), b∗d = 0. Thus, by (P4), b = d. Hence, (a, bI) = (c, dI). Conversely, let (a, bI) = (c, dI). Then a = c and b = d. Thus, by Definition 3.4, (B1) and Lemma 3.7(ii and iv), (a, bI) · (c, dI) = (a ∗ c, ((a ∗ d∧ b ∗ c)∧ b ∗ d)I) = (a ∗ a, ((a ∗ b∧ b ∗ a) ∧ b ∗ b)I) = (0, ((b ∗ a) ∧ 0)I) = (0, 0I). � Lemma 3.9. If X is a commutative B-algebra, then for any x, y, z ∈ X, (i) x ∧ y = y, (ii) (x ∧ y) ∧ z = x ∧ (y ∧ z) = z. Proof : Let x, y, z ∈ X. (i) By Definition 3.4 and Lemma 2.6, x ∧ y = x ∗ (x ∗ y) = y; (ii) By (i), (x ∧ y) ∧ z = y ∧ z = z = x ∧ z = x ∧ (y ∧ z). � Theorem 3.10. If X is commutative, then X(I) is a B-algebra. Proof : Let (a, bI), (c, dI) ∈ X(I). By Lemma 3.9(ii), the binary operation in X(I) is (a, bI) · (c, dI) = ( a ∗ c, ((a ∗ d∧ b ∗ c)∧ b ∗ d)I ) = ( a ∗ c, (b ∗ d)I ) . This coincides with the binary operation of X ×X as a B-algebra. Therefore, X(I) is a B-algebra. � D.O. Jacobe, J.P. Vilela / Eur. J. Pure Appl. Math, 14 (3) (2021), 895-904 899 Remark 3.11. By Theorem 3.10, the neutrosophic B-algebra X(I) in Example 3.6 is a B-algebra. However, a neutrosophic B-algebra is not a B-algebra in general as shown in the following example. Example 3.12. Consider the non-commutative B-algebraX = {0, 1, 2, 3, 4, 5} in Example 2.3(ii). Then the setX(I) = {(0, 0I), (1, 0I), ..., (5, 0I), (0, I), (1, I), ..., (5, I), (0, 2I), (1, 2I), ..., (5, 2I), (0, 3I), (1, 3I), ..., (5, 3I), (0, 4I), (1, 4I), ..., (5, 4I), (0, 5I), (1, 5I), ..., (5, 5I)} is the neutrosophic B -algebra determined by X and I. X(I) is not a B-algebra since by Lemma 3.8(i), (3, 4I) · (0, 0I) = (3, ((3 ∧ 4) ∧ 4)I) = (3, (5 ∧ 4)I) = (3, 3I) 6= (3, 4I). Definition 3.13. A neutrosophic B-algebra X(I) is said to be commutative if (a, bI) · [(0, 0I) · (c, dI)] = (c, dI) · [(0, 0I) · (a, bI)] for any (a, bI), (c, dI) ∈ X(I). Remark 3.14. If X(I) is a commutative neutrosophic B-algebra, then X ′ = {(x, 0I) : x ∈ X} is a commutative B-algebra. In Example 3.12, X(I) is not commutative since (2, 2I), (5, 5I) ∈ X(I) but by Lemma 3.8(iii), (2, 2I) · [(0, 0I) · (5, 5I)] = (4, 4I) 6= (3, 3I) = (5, 5I) · [(0, 0I) · (2, 2I)]. In Example 3.6, X is a commutative B-algebra and it can be verified that X(I) is also commutative. This observation is generalized in the following result. Theorem 3.15. X is commutative if and only if X(I) is commutative. Proof : Let (a, bI), (c, dI) ∈ X(I). Suppose that X is commutative. Then by Definition 3.4 and Lemma 3.9 (ii), (a, bI) · [(0, 0I) · (c, dI)] = (a, bI) · ( 0 ∗ c, ((0 ∗ d ∧ 0 ∗ c) ∧ 0 ∗ d)I ) = (a, bI) · ( 0 ∗ c, (0 ∗ d)I ) = ( a ∗ (0 ∗ c), [( a ∗ (0 ∗ d) ∧ b ∗ (0 ∗ c) ) ∧ b ∗ (0 ∗ d) ] I ) = ( a ∗ (0 ∗ c), [b ∗ (0 ∗ d)]I ) = ( c ∗ (0 ∗ a), [d ∗ (0 ∗ b)]I ) = ( c ∗ (0 ∗ a), [( c ∗ (0 ∗ b) ∧ d ∗ (0 ∗ a) ) ∧ d ∗ (0 ∗ b) ] I ) = (c, dI) · ( 0 ∗ a, (0 ∗ b)I ) = (c, dI) · ( 0 ∗ a, ((0 ∗ b ∧ 0 ∗ a) ∧ 0 ∗ b)I ) = (c, dI) · [(0, 0I) · (a, bI)]. Therefore, X(I) is commutative. Conversely, suppose that X(I) is commutative. Then by Remark 3.14, X ′ is commutative so that for any (x, 0I), (y, 0I) ∈ X ′, x∗ (0∗y) = y ∗ (0∗x) for any x, y ∈ X. Hence, by Definition 2.5, X is commutative. � Corollary 3.16. If X(I) is commutative, then X(I) is a B-algebra. The notion of neutrosophic subalgebra of a neutrosophic B-algebra will now be intro- duced. D.O. Jacobe, J.P. Vilela / Eur. J. Pure Appl. Math, 14 (3) (2021), 895-904 900 Definition 3.17. Let X(I) be a neutrosophic B-algebra. A subset P (I) of X(I) is said to be a proper subset of X(I) if P (I) 6= X(I). Definition 3.18. A nonempty subset S(I) of a neutrosophic B-algebra X(I) is said to be a neutrosophic subalgebra of X(I) if the following conditions hold: (i) (a, bI) · (c, dI) ∈ S(I) for all (a, bI), (c, dI) ∈ S(I), and (ii) S(I) contains a proper subset which is a B-algebra. In view of Lemma 3.8(iv), (0, 0I) is an element of any neutrosophic subalgebra S(I) of X(I). Remark 3.19. Let X(I) be a nonzero neutrosophic B-algebra. (i) Then X(I) is a neutrosophic subalgebra of itself. However, {(0, 0I)} is not a neutro- sophic subalgebra of X(I) since it does contain a proper subset which is a B-algebra but {(0, 0I)} is a B-algebra. (ii) If S(I) is a neutrosophic subalgebra of X(I), then S(I) is a neutrosophic B-algebra in its own right. Theorem 3.20. Let X(I) be a nonzero neutrosophic B-algebra. Then (i) X ′ = {(x, 0I) : x ∈ X} is a neutrosophic subalgebra of X(I). (ii) X ′′ = {(0, xI) : x ∈ X} is a neutrosophic subalgebra of X(I). (iii) Xω(I) = {(a, aI) : a ∈ X} is a neutrosophic subalgebra of X(I). Proof : (i) Clearly, (0, 0I) ∈ X ′. Suppose that(a, 0I), (b, 0I) ∈ X ′. Then by Remark 3.5, (a, 0I) · (b, 0I) ∈ X ′. Moreover, {(0, 0I)} ( X ′ is a B-algebra. Hence, X ′ is a neutrosophic subalgebra of X(I). (ii) Clearly, (0, 0I) ∈ X ′′. Suppose that(0, aI), (0, bI) ∈ X ′′. Then a, b ∈ X and (0, aI) · (0, bI) = (0, ((0 ∗ b ∧ a ∗ 0) ∧ a ∗ b)I). Since X is a B-algebra, (0 ∗ b ∧ a ∗ 0) ∧ a ∗ b ∈ X and so (0, aI) · (0, bI) = (0, ((0 ∗ b ∧ a ∗ 0) ∧ a ∗ b)I) ∈ X ′′. Moreover, {(0, 0I)} ( X ′′ is a B-algebra. Therefore, X ′′ is a neutrosophic subalgebra of X(I). (iii) Clearly, (0, 0I) ∈ Xω(I) and {(0, 0I)} ( Xω(I). Let (a, aI), (b, bI) ∈ Xω(I). Then by Lemma 3.8(iii), (a, aI) · (b, bI) = (a ∗ b, (a ∗ b)I) ∈ Xω(I). Therefore, Xω(I) is a neutrosophic subalgebra of X(I). � Definition 3.21. A neutrosophic subalgebra N(I) of X(I) is normal if for any (a, bI) · (c, dI), (x, yI) · (u, vI) ∈ N(I), [(a, bI) · (x, yI)] · [(c, dI) · (u, vI)] ∈ N(I). Example 3.22. Consider the neutrosophic B-algebra X(I) in Example 3.12 and its sub- set N(I) = {(0, 0I), (0, I), (0, 2I), (1, 0I), (1, I), (1, 2I), (2, 0I), (2, I), (2, 2I)} determined by N = {0, 1, 2} and I. It can be verified that N(I) is a normal neutrosophic subal- gebra of X(I) with {(0, 0I), (1, 0I), (2, 0I)} as its proper subset which is a B-algebra.. It can also be verified that N is a normal subalgebra of X. D.O. Jacobe, J.P. Vilela / Eur. J. Pure Appl. Math, 14 (3) (2021), 895-904 901 The observation in the preceding example are generalized in the next theorem. Theorem 3.23. Let S(I) = {(a, bI) ∈ X(I) : a, b ∈ S, S ⊆ X}. Then S(I) is a neutro- sophic subalgebra of X(I) if and only if S is a nonzero subalgebra of X. Moreover, if S(I) is normal in X(I), then S is normal in X. Proof : Let S be a nonzero subalgebra of X. Let (a, bI), (c, dI) ∈ S(I). Then a, b, c, d ∈ S and (a, bI) · (c, dI) = ( a∗c, ((a∗d∧b∗c)∧b∗d)I ) . Since S is a subalgebra of X, a∗c, a∗d, b ∗ c, b ∗ d ∈ S so that (a ∗ d∧ b ∗ c)∧ b ∗ d ∈ S. Thus, (a, bI) · (c, dI) ∈ S(I). By Definition 3.4, S(I) is a neutrosophic B-algebra determined by S and I. Hence, by Remark 3.5, S(I) contains the B-algebra S′ = {(x, 0I) : x ∈ S} as a proper subset. Therefore, S(I) is a neutrosophic subalgebra of X(I). Conversely, let S(I) be a neutrosophic subalgebra of X(I). Then S 6= ∅ and S 6= {0}. By Remark 3.19(ii), S(I) is a neutrosophic B-algebra. Thus, S is a B-algebra by Definition 3.4. Hence, S is a nonzero subalgebra of X. Moreover, let a ∗ b, c ∗ d ∈ S. By definition of S(I) and Lemma 3.8(ii), (a ∗ b, 0I) = (a, 0I)·(b, 0I), (c∗d, 0I) = (c, 0I)·(d, 0I) ∈ S(I). By normality of S(I) and Lemma 3.8(ii), [(a, 0I) · (c, 0I)] · [(b, 0I) · (d, 0I)] = [ (a ∗ c, 0I) · (b ∗ d, 0I) ] = ( [(a ∗ c) ∗ (b ∗ d)], 0I ) ∈ S(I). By definition of S(I), [(a ∗ c) ∗ (b ∗ d)] ∈ S and hence S is normal in X. � Since a neutrosophic subalgebra is also a neutrosophic B-algebra contained in a given neutrosophic B-algebra, the following remark follows. Remark 3.24. If N(I) is a normal neutrosophic subalgebra of X(I), then N(I) is normal in every neutrosophic subalgebra of X(I) containing N(I). Example 3.25. In the neutrosophic B-algebra X(I) in Example 3.6, X(I) has three neutrosophic subalgebras: {(0, 0I), (0, I), (0, 2I)}, {(0, 0I), (1, 0I), (2, 0I)} and itself. The first two have {(0, 0I)} as their proper subset which is a B-algebra and the latter has {(0, 0I), (1, 0I), (2, 0I)}. In the neutrosophic B-algebra X(I) in Example 3.12, the follow- ing are some of its neutrosophic subalgebras: S(I)1 =X(I) = {(0, 0I), (1, 0I), (2, 0I), (3, 0I), (4, 0I), (5, 0I), (0, I), (1, I), (2, I), (3, I), (4, I), (5, I), (0, 2I), (1, 2I), (2, 2I), (3, 2I), (4, 2I), (5, 2I), (0, 3I), (1, 3I), (2, 3I), (3, 3I), (4, 3I), (5, 3I), (0, 4I), (1, 4I), (2, 4I), (3, 4I), (4, 4I), (5, 4I), (0, 5I), (1, 5I), (2, 5I), (3, 5I), (4, 5I), (5, 5I)} S(I)2 ={(0, 0I), (0, I), (0, 2I), (0, 3I), (0, 4I), (0, 5I), (1, 0I), (1, I), (1, 2I), (1, 3I), (1, 4I), (1, 5I), (2, 0I), (2, I), (2, 2I), (2, 3I), (2, 4I), (2, 5I)}, S(I)3 ={(0, 0I), (0, I), (0, 2I), (0, 3I), (0, 4I), (0, 5I), (3, 0I), (3, I), (3, 2I), (3, 3I), (3, 4I), (3, 5I)}, S(I)4 ={(0, 0I), (0, I), (0, 2I), (0, 3I), (0, 4I), (0, 5I), (4, 0I), (4, I), (4, 2I), (4, 3I), (4, 4I), (4, 5I)}, D.O. Jacobe, J.P. Vilela / Eur. J. Pure Appl. Math, 14 (3) (2021), 895-904 902 S(I)5 ={(0, 0I), (0, I), (0, 2I), (0, 3I), (0, 4I), (0, 5I), (5, 0I), (5, I), (5, 2I), (5, 3I), (5, 4I), (5, 5I)} S(I)6 ={(0, 0I), (1, I), (2, 2I), (3, 3I), (4, 4I), (5, 5I)} Aside from {(0, 0I)}, setsX ′, {(0, 0I), (1, 0I), (2, 0I)}, {(0, 0I), (3, 3I)}, {(0, 0I), (4, 4I)}, {(0, 0I), (5, 5I)} and {(0, 0I), (1, I), (2, 2I)} are proper subsets of S(I)i, i = 1, 2, 3, 4, 5, 6, respectively, which are B-algbras. Notice that the neutrosophic B-algebra in Example 3.6 is a neutrosophic subalgebra of the neutrosophic B-algebra in Example 3.12. In a B-algebra, the intersection of any collection of subalgebras is also a subalgebra by Lemma 2.8. However, this is not always the case for neutrosophic B-algebra. Consider the following example. Example 3.26. Consider the neutrosophic B-algebra in Example 3.12 and a collection of its neutrosophic subalgebras in Example 3.25. Clearly, 6⋂ i=1 S(I)i = {(0, 0I)} is not a neutro- sophic subalgebra of X(I) by Remark 3.19. However, 5⋂ i=1 S(I)i = {(0, 0I), (0, I), (0, 2I)} is a neutrosophic subalgebra of X(I) with {(0, 0I)} as its proper subset which is a B-algebra. The following theorem provides a necessary and sufficient condition for the intersection of neutrosophic subalgebras to be a neutrosophic subalgebra. Theorem 3.27. Let {S(I)α : α ∈ A } be any nonempty collection of neutrosophic sub- algebras (resp., normal neutrosophic subalgebras) of a neutrosophic B-algebra X(I). If⋂ α∈A S(I)α 6= {(0, 0I)}, then ⋂ α∈A S(I)α is a neutrosophic subalgebra (resp., normal neu- trosophic subalgebra) of X(I). Proof : Since (0, 0I) ∈ S(I)α for every α ∈ A , (0, 0I) ∈ ⋂ α∈A S(I)α so that ⋂ α∈A S(I)α 6= ∅. Since ⋂ α∈A S(I)α 6= {(0, 0I)}, there exists (a, bI) ∈ ⋂ α∈A S(I)α such that (a, bI) 6= (0, 0I). Thus, {(0, 0I)} ( ⋂ α∈A S(I)α which is a B-algebra. Let (a, bI), (c, dI) ∈ ⋂ α∈A S(I)α. Then (a, bI), (c, dI) ∈ S(I)α for every α ∈ A . Since for every α ∈ A , S(I)α is a neutrosophic subalgebra of X(I), (a, bI) · (c, dI) ∈ S(I)α for every α ∈ A . Thus, (a, bI) · (c, dI) ∈⋂ α∈A S(I)α. Hence, ⋂ α∈A S(I)α is a neutrosophic subalgebra of X(I). Moreover, let {S(I)α : α ∈ A } be any nonempty collection of normal neutrosophic subalgebras of a X(I) and (a, bI)·(c, dI), (x, yI)·(u, vI) ∈ ⋂ α∈A S(I)α. Then (a, bI)·(c, dI), (x, yI)·(u, vI) ∈ S(I)α for every α ∈ A . Since S(I)α is normal for every α ∈ A , [(a, bI) · (x, yI)] · [(c, dI) · (u, vI)] ∈ REFERENCES 903 S(I)α for every α ∈ A . Hence, [(a, bI) · (x, yI)] · [(c, dI) · (u, vI)] ∈ ⋂ α∈A S(I)α. Therefore,⋂ α∈A S(I)α is a normal neutrosophic subalgebra of X(I). � Example 3.28. Consider the neutrosophic B-algebra in Example 3.12 and a collection of its neutrosophic subalgebras in Example 3.25. The union 6⋃ i=4 S(I)i is not a neutrosophic subalgebra of X(I) since (1, I), (0, 5I) ∈ 6⋃ i=4 S(I)i but (1, I) · (0, 5I) = (1, 4I) /∈ 6⋃ i=4 S(I)i. Theorem 3.29. Let {S(I)i : i ∈ I } be any nonempty collection of neutrosophic subal- gebras of a neutrosophic B-algebra X(I) such that S(I)1 ⊆ S(I)2 ⊆ S(I)3 ⊆ · · · . Then⋃ i∈I S(I)i is a neutrosophic subalgebra of X(I). Proof : Clearly, ⋃ i∈I S(I)i 6= ∅. Let (a, bI), (c, dI) ∈ ⋃ i∈I S(I)i. Then for some i ∈ I , (a, bI), (c, dI) ∈ S(I)i and (a, bI) · (c, dI) ∈ S(I)i. Thus, (a, bI) · (c, dI) ∈ ⋃ i∈I S(I)i. Let P (I)i be a proper subset of S(I)i, for every i ∈ I which is a B-algebra. Then for any i ∈ I , P (I)i ( ⋃ i∈I S(I)i. Therefore, ⋃ i∈I S(I)i is a neutrosophic subalgebra of X(I). � References [1] Adesina Abdul Akeem Agboola and Bijan Davvaz. Introduction to neutrosophic BCI/BCK-algebras. International Journal of Mathematics and Mathematical Sci- ences, 2015, 2015. [2] PJ Allen, J Neggers, and Hee Sik Kim. B-algebras and groups. 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