RATIO MATHEMATICA 29 (2015) 65-76 ISSN:1592-7415 Neutrosophic filters in BE-algebras Akbar Rezaei1, Arsham Borumand Saeid2, Florentin Smarandache3 1Department of Mathematics, Payame Noor University, P.O.BOX. 19395-3697, Tehran, Iran. rezaei@pnu.ac.ir 2Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran. arsham@uk.ac.ir 3Florentin Smarandache, University of New Mexico, Gallup, NM 87301, USA. smarand@unm.edu Abstract In this paper, we introduce the notion of (implicative) neutrosophic filters in BE-algebras. The relation between implicative neutrosophic filters and neutrosophic filters is investigated and we show that in self distributive BE- algebras these notions are equivalent. Keywords: BE-algebra, neutrosophic set, (implicative) neutrosophic filter. 2010 AMS subject classifications: 03B60, 06F35, 03G25. doi:10.23755/rm.v29i1.22 1 Introduction Neutrosophic set theory was introduced by Smarandache in 1998 ([10]). Neu- trosophic sets are a new mathematical tool for dealing with uncertainties which are free from many difficulties that have troubled the usual theoretical approaches. Research works on neutrosophic set theory for many applications such as infor- mation fussion, probability theory, control theory, decision making, measurement 65 A. Rezaei, A. Borumand Saeid and F. Smarandache theory, etc. Kandasamy and Smarandache introduced the concept of neutrosophic algebraic structures ([3, 4, 5]). Since then many researchers worked in this area and lots of literatures had been produced about the theory of neutrosophic set. In the neutrosophic set one can have elements which have paraconsistent information (sum of components > 1), others incomplete information (sum of components < 1), others consistent information (in the case when the sum of components =1) and others interval-valued components (with no restriction on their superior or inferior sums). H.S. Kim and Y.H. Kim introduced the notion of a BE-algebra as a generaliza- tion of a dual BCK-algebra ([6]). B.L. Meng give a procedure which generated a filter by a subset in a transitive BE-algebra ([7]). A. Walendziak introduced the no- tion of a normal filter in BE-algebras and showed that there is a bijection between congruence relations and filters in commutative BE-algebras ([11]). A. Borumand Saeid and et al. defined some types of filters in BE-algebras and showed the re- lationship between them ([1]). A. Rezaei and et al. discussed on the relationship between BE-algebras and Hilbert algebras ([9]). Recently, A. Rezaei and et al. introduced the notion of hesitant fuzzy (implicative) filters and get some results on BE-algebras ([8]). In this paper, we introduce the notion of (implicative) neutrosophic filters and study it in details. In fact, we show that in self distributive BE-algebras concepts of implicative neutrosophic filter and neutrosophic filter are equivalent. 2 Preliminaries In this section, we cite the fundamental definitions that will be used in the sequel: Definition 2.1. [6] By a BE-algebra we shall mean an algebra X = (X;∗, 1) of type (2, 0) satisfying the following axioms: (BE1) x∗x = 1, (BE2) x∗1 = 1, (BE3) 1∗x = x, (BE4) x∗ (y ∗z) = y ∗ (x∗z), for all x, y, z ∈ X. From now on, X is a BE-algebra, unless otherwise is stated. We introduce a relation “≤” on X by x ≤ y if and only if x∗y = 1. A BE-algebra X is said to be self distributive if x∗(y∗z) = (x∗y)∗(x∗z), for all x, y, z ∈ X. A BE-algebra X is said to be commutative if satisfies: 66 Neutrosophic filters in BE-algebras (x∗y)∗y = (y ∗x)∗x, for all x, y ∈ X. Proposition 2.1. [11] If X is a commutative BE-algebra, then for all x, y ∈ X, x∗y = 1 and y ∗x = 1 imply x = y. We note that “≤” is reflexive by (BE1). If X is self distributive then relation “≤” is a transitive ordered set on X, because if x ≤ y and y ≤ z, then x∗z = 1∗ (x∗z) = (x∗y)∗ (x∗z) = x∗ (y ∗z) = x∗1 = 1. Hence x ≤ z. If X is commutative then by Proposition 2.1, relation “≤” is anti- symmetric. Hence if X is a commutative self distributive BE-algebra, then relation “≤” is a partial ordered set on X. Proposition 2.2. [6] In a BE-algebra X, the following hold: (i) x∗ (y ∗x) = 1, (ii) y ∗ ((y ∗x)∗x) = 1, for all x, y ∈ X. A subset F of X is called a filter of X if it satisfies: (F1) 1 ∈ F, (F2) x ∈ F and x∗y ∈ F imply y ∈ F . Define A(x, y) = {z ∈ X : x∗ (y ∗z) = 1}, which is called an upper set of x and y. It is easy to see that 1, x, y ∈ A(x, y), for any x, y ∈ X. Every upper set A(x, y) need not be a filter of X in general. Definition 2.2. [1] A non-empty subset F of X is called an implicative filter if satisfies the following conditions: (IF1) 1 ∈ F , (IF2) x∗ (y ∗z) ∈ F and x∗y ∈ F imply that x∗z ∈ F , for all x, y, z ∈ X. If we replace x of the condition (IF2) by the element 1, then it can be easily observed that every implicative filter is a filter. However, every filter is not an implicative filter as shown in the following example. 67 A. Rezaei, A. Borumand Saeid and F. Smarandache Example 2.1. Let X = {1, a, b} be a BE-algebra with the following table: ∗ 1 a b 1 1 a b a 1 1 a b 1 a 1 Then F = {1, a} is a filter of X, but it is not an implicative filter, since 1∗ (a∗ b) = 1∗a = a ∈ F and 1∗a = a ∈ F but 1∗ b = b /∈ F . Definition 2.3. [10] Let X be a set. A neutrosophic subset A of X is a triple (TA, IA, FA) where TA : X → [0, 1] is the membership function, IA : X → [0, 1] is the indeterminacy function and FA : X → [0, 1] is the nonmembership function. Here for each x ∈ X, TA(x), IA(x) and FA(x) are all standard real numbers in [0, 1]. We note that 0 ≤ TA(x) + IA(x) + FA(x) ≤ 3, for all x ∈ X. The set of neutrosophic subset of X is denoted by NS(X). Definition 2.4. [10] Let A and B be two neutrosophic sets on X. Define A ≤ B if and only if TA(x) ≤ TB(x), IA(x) ≥ IB(x), FA(x) ≥ FB(x), for all x ∈ X. Definition 2.5. Let X1 = (X1;∗, 1) and X2 = (X2;◦, 1′) be two BE-algebras. Then a mapping f : X1 → X2 is called a homomorphism if, for all x1, x2 ∈ X1 f(x1 ∗ x2) = f(x1) ◦ f(x2). It is clear that if f : X1 → X2 is a homomorphism, then f(1) = 1′. 3 Neutrosophic Filters Definition 3.1. A neutrosophic set A of X is called a neutrosophic filter if satisfies the following conditions: (NF1) TA(x) ≤ TA(1), IA(x) ≥ IA(1) and FA(x) ≥ FA(1), (NF2) min{TA(x∗y), TA(x)}≤ TA(y), min{IA(x∗y), IA(x)}≥ IA(y) and min{FA(x∗y), FA(x)}≥ FA(y), for all x, y ∈ X. 68 Neutrosophic filters in BE-algebras The set of neutrosophic filter of X is denoted by NF(X). Example 3.1. In Example 2.1, put TA(1) = 0.9, TA(a) = TA(b) = 0.5, IA(1) = 0.2, IA(a) = IA(b) = 0.35 and FA(1) = 0.1, FA(a) = FA(b) = 0. Then A = (TA, IA, FA) is a neutrosophic filter. Proposition 3.1. Let A ∈ NF(X). Then (i) if x ≤ y, then TA(x) ≤ TA(y), IA(x) ≥ IA(y) and FA(x) ≥ FA(y), (ii) TA(x) ≤ TA(y ∗x), IA(x) ≥ IA(y ∗x) and FA(x) ≥ FA(y ∗x), (iii) min{TA(x), TA(y)}≤ TA(x∗y), min{IA(x), IA(y)}≥ IA(x∗y) and min{FA(x), FA(y)}≥ FA(x∗y), (iv) TA(x) ≤ TA((x∗y)∗y), IA(x) ≥ IA((x∗y)∗y) and FA(x) ≥ FA((x∗y)∗y), (v) min{TA(x), TA(y)}≤ TA((x∗ (y ∗z))∗z), min{IA(x), IA(y)}≥ IA((x∗ (y ∗z))∗z) and min{FA(x), FA(y)}≥ FA((x∗ (y ∗z))∗z), (vi) if min{TA(y), TA((x∗y)∗z)}≤ TA(z ∗x), then TA is order reversing and IA, FA are order (i.e. if x ≤ y, then TA(y) ≤ TA(x), IA(y) ≥ IA(x) and FA(y) ≥ FA(x)) (vii) if z ∈ A(x, y), then min{TA(x), TA(y)}≤ TA(z), min{IA(x), IA(y)}≥ IA(z) and min{FA(x), FA(y)}≥ FA(z) (viii) if n∏ i=1 ai ∗x = 1, then n∧ i=1 TA(ai) ≤ TA(x), n∧ i=1 IA(ai) ≥ IA(x) and n∧ i=1 FA(ai) ≥ FA(x) where n∏ i=1 ai ∗x = an ∗ (an−1 ∗ (. . . (a1 ∗x) . . . )). Proof. (i). Let x ≤ y. Then x∗y = 1 and so TA(x) = min{TA(x), TA(1)} = min{TA(x), TA(x∗y)}≤ TA(y), IA(x) = min{IA(x), IA(1)} = min{IA(x), IA(x∗y)}≥ IA(y), FA(x) = min{FA(x), FA(1)} = min{FA(x), FA(x∗y)}≥ FA(y). (ii). Since x ≤ y ∗x, by using (i) the proof is clear. 69 A. Rezaei, A. Borumand Saeid and F. Smarandache (iii). By using (ii) we have min{TA(x), TA(y)}≤ TA(y) ≤ TA(x∗y), min{IA(x), IA(y)}≥ IA(y) ≥ IA(x∗y), min{FA(x), FA(y)}≥ FA(y) ≥ FA(x∗y). (iv). It follows from Definition 3.1, TA(x) = min{TA(x), TA(1)} = min{TA(x), TA((x∗y)∗ (x∗y))} = min{TA(x), TA(x∗ ((x∗y)∗y))} ≤ TA((x∗y)∗y). Also, we have IA(x) = min{IA(x), IA(1)} = min{IA(x), IA((x∗y)∗ (x∗y))} = min{IA(x), IA(x∗ ((x∗y)∗y))} ≥ IA((x∗y)∗y) and FA(x) = min{FA(x), FA(1)} = min{FA(x), FA((x∗y)∗ (x∗y))} = min{FA(x), FA(x∗ ((x∗y)∗y))} ≥ FA((x∗y)∗y). (v). From (iv) we have min{TA(x), TA(y)} ≤ min{TA(x), TA((y ∗ (x∗z))∗ (x∗z))} = min{TA(x), TA((x∗ (y ∗z))∗ (x∗z))} = min{TA(x), TA(x∗ (x∗ (y ∗z))∗z))} ≤ TA((x∗ (y ∗z))∗z)), min{IA(x), IA(y)} ≥ min{IA(x), IA((y ∗ (x∗z))∗ (x∗z))} = min{IA(x), IA((x∗ (y ∗z))∗ (x∗z))} = min{IA(x), IA(x∗ (x∗ (y ∗z))∗z))} ≥ IA((x∗ (y ∗z))∗z)) 70 Neutrosophic filters in BE-algebras and min{FA(x), FA(y)} ≥ min{FA(x), FA((y ∗ (x∗z))∗ (x∗z))} = min{FA(x), FA((x∗ (y ∗z))∗ (x∗z))} = min{FA(x), FA(x∗ (x∗ (y ∗z))∗z))} ≥ FA((x∗ (y ∗z))∗z)). (vi). Let x ≤ y, that is, x∗y = 1. TA(y) = min{TA(y), TA(1∗1)} = min{TA(y), TA((x∗y)∗1)}≤ TA(1∗x) = TA(x), IA(y) = min{IA(y), IA(1∗1)} = min{IA(y), IA((x∗y)∗1)}≥ IA(1∗x) = IA(x), FA(y) = min{FA(y), FA(1∗1)} = min{FA(y), FA((x∗y)∗1)}≥ FA(1∗x) = FA(x). (vii). Let z ∈ A(x, y). Then x∗ (y ∗z) = 1. Hence min{TA(x), TA(y)} = min{TA(x), TA(y), TA(1)} = min{TA(x), TA(y), TA(x∗ (y ∗z))} ≤ min{TA(y), TA(y ∗z)} ≤ TA(z). Also, we have min{IA(x), IA(y)} = min{IA(x), IA(y), IA(1)} = min{IA(x), IA(y), IA(x∗ (y ∗z))} ≥ min{IA(y), IA(y ∗z)} ≥ IA(z), and min{FA(x), FA(y)} = min{FA(x), FA(y), FA(1)} = min{FA(x), FA(y), FA(x∗ (y ∗z))} ≥ min{FA(y), FA(y ∗z)} ≥ FA(z). (viii). The proof is by induction on n. By (vii) it is true for n = 1, 2. Assume that it satisfies for n = k, that is, 71 A. Rezaei, A. Borumand Saeid and F. Smarandache k∏ i=1 ai∗x = 1 ⇒ k∧ i=1 TA(ai) ≤ TA(x), k∧ i=1 IA(ai) ≥ IA(x) and k∧ i=1 FA(ai) ≥ FA(x) for all a1, . . . , ak, x ∈ X. Suppose that k+1∏ i=1 ai ∗x = 1, for all a1, . . . , ak, ak+1, x ∈ X. Then k+1∧ i=2 TA(ai) ≤ TA(a1 ∗x), k+1∧ i=2 IA(ai) ≥ IA(a1 ∗x), and k+1∧ i=2 FA(ai) ≥ FA(a1 ∗x). Since A is a neutrosophic filter of X, we have k+1∧ i=1 TA(ai) = min{( k+1∧ i=2 TA(ai)), TA(a1)}≤ min{TA(a1 ∗x), TA(a1)}≤ TA(x), k+1∧ i=1 IA(ai) = min{( k+1∧ i=2 IA(ai)), IA(a1)}≥ min{IA(a1 ∗x), IA(a1)}≥ IA(x) and k+1∧ i=1 FA(ai) = min{( k+1∧ i=2 FA(ai)), FA(a1)}≥ min{FA(a1 ∗x), FA(a1)}≥ FA(x). 2 Theorem 3.1. If {Ai}i∈I is a family of neutrosophic filters in X, then ⋂ i∈I Ai is too. Theorem 3.2. Let A ∈ NF(X). Then the sets (i) XTA = {x ∈ X : TA(x) = TA(1)}, (ii) XIA = {x ∈ X : IA(x) = IA(1)}, (iii) XFA = {x ∈ X : FA(x) = FA(1)}, are filters of X. Proof. (i). Obviously, 1 ∈ XhA. Let x, x∗y ∈ XTA . Then TA(x) = TA(x∗y) = TA(1). Now, by (NF1) and (NF2), we have TA(1) = min{TA(x), TA(x∗y)}≤ TA(y) ≤ TA(1). Hence TA(y) = TA(1). Therefore, y ∈ XTA. The proofs of (ii) and (iii) are similar to (i).2 72 Neutrosophic filters in BE-algebras Definition 3.2. A neutrosophic set A of X is called an implicative neutrosophic filter of X if satisfies the following conditions: (INF1) TA(1) ≥ TA(x), (INF2) TA(x∗z) ≥ min{TA(x∗ (y ∗z)), TA(x∗y)}, IA(x∗z) ≤ min{IA(x∗ (y ∗z)), IA(x∗y)} and FA(x∗z) ≤ min{FA(x∗ (y ∗z)), FA(x∗y)}, for all x, y, z ∈ X. The set of implicative neutrosophic filter of X is denoted by INF(X). If we replace x of the condition (INF2) by the element 1, then it can be easily observed that every implicative neutrosophic filter is a neutrosophic filter. However, ev- ery neutrosophic filter is not an implicative neutrosophic filter as shown in the following example. Example 3.2. Let X = {1, a, b, c, d} be a BE-algebra with the following table: ∗ 1 a b c d 1 1 a b c d a 1 1 b c b b 1 a 1 b a c 1 a 1 1 a d 1 1 1 b 1 Then X = (X;∗, 1) is a BE-algebra. Define a neutrosophic set A on X as follows: TA(x) = { 0.85 if x = 1, a 0.12 otherwise and IA(x) = FA(x) = 0.5, for all x ∈ X. Then clearly A = (TA, IA, FA) is a neutrosophic filter of X, but it is not an implicative neutrosophic filter of X, since TA(b∗ c) 6≥ min{TA(b∗ (d∗ c)), TA(b∗d)}. Theorem 3.3. Let X be a self distributive BE-algebra. Then every neutrosophic filter is an implicative neutrosophic filter. Proof. Let A ∈ NF(X) and x ∈ X. Obvious that TA(x) ≤ TA(1), IA(x) ≥ IA(1) and FA(x) ≥ FA(1). By self distributivity and (NF2), we have min{TA(x∗(y∗z)), TA(x∗y)} = min{TA((x∗y)∗(x∗z)), TA(x∗y)}≤ TA(x∗z), 73 A. Rezaei, A. Borumand Saeid and F. Smarandache min{IA(x∗(y∗z)), IA(x∗y)} = min{IA((x∗y)∗(x∗z)), IA(x∗y)}≥ IA(x∗z) and min{FA(x∗(y∗z)), FA(x∗y)} = min{FA((x∗y)∗(x∗z)), FA(x∗y)}≥ FA(x∗z). Therefore A ∈ INF(X).2 Let t ∈ [0, 1]. For a neutrosophic filter A of X, t-level subset which denoted by U(A; t) is defined as follows: U(A; t) := {x ∈ A : t ≤ TA(x), IA(x) ≤ t and FA(x) ≤ t} and strong t-level subset which denoted by U(A; t)> as U(A; t)> := {x ∈ A : t < TA(x), IA(x) < t and FA(x) < t}. Theorem 3.4. Let A ∈ NS(X). The following are equivalent: (i) A ∈ NF(X), (ii) (∀t ∈ [0, 1]) U(A; t) 6= ∅ imply U(A; t) is a filter of X. Proof. (i)⇒(ii). Let x, y ∈ X be such that x, x ∗ y ∈ U(A; t), for any t ∈ [0, 1]. Then t ≤ TA(x) and t ≤ TA(x∗y). Hence t ≤ min{TA(x), TA(x∗y)}≤ TA(y). Also, IA(x) ≤ t and IA(x ∗ y) ≤ t and so t ≥ min{IA(x), IA(x ∗ y)} ≥ IA(y). By a similar argument we have t ≥ min{FA(x), FA(x∗y)}≥ FA(y). Therefore, y ∈ U(A; t). (ii)⇒(i). Let U(A; t) be a filter of X, for any t ∈ [0, 1] with U(A; t) 6= ∅. Put TA(x) = IA(x) = FA(x) = t, for any x ∈ X. Then x ∈ U(A; t). Since U(A; t) is a filter of X, we have 1 ∈ U(A; t) and so TA(x) = t ≤ TA(1). Now, for any x, y ∈ X, let TA(x∗y) = IA(x∗y) = FA(x∗y) = tx∗y and TA(x) = IA(x) = FA(x) = tx. Put t = min{tx∗y, tx}. Then x, x ∗ y ∈ U(A; t), so y ∈ U(A; t). Hence t ≤ TA(y), t ≥ IA(y), t ≥ FA(y) and so min{TA(x∗y), TA(x)} = min{tx∗y, tx} = t ≤ TA(y), min{IA(x∗y), IA(x)} = min{tx∗y, tx} = t ≥ IA(y), and min{FA(x∗y), FA(x)} = min{tx∗y, tx} = t ≥ FA(y). Therefore, A ∈ NF(X).2 74 Neutrosophic filters in BE-algebras Theorem 3.5. Let A ∈ NF(X). Then we have (∀a, b ∈ X) (∀t ∈ [0, 1]) (a, b ∈ U(A; t) ⇒ A(a, b) ⊆ U(A; t)). Proof. Assume that A ∈ NF(X). Let a, b ∈ X be such that a, b ∈ U(A; t). Then t ≤ TA(a) and t ≤ TA(b). Let c ∈ A(a, b). Hence a ∗ (b ∗ c) = 1. Now, by Proposition 3.1(v) and (BE3), we have t ≤ min{TA(a), TA(b)}≤ TA((a∗ (b∗ c)∗ c)) = TA(1∗ c) = TA(c), t ≥ min{IA(a), IA(b)}≥ IA((a∗ (b∗ c)∗ c)) = IA(1∗ c) = IA(c) and t ≥ min{FA(a), FA(b)}≥ FA((a∗ (b∗ c)∗ c)) = FA(1∗ c) = FA(c). Then c ∈ U(A; t). Therefore, A(a, b) ⊆ U(A; t)).2 Corolary 3.1. Let A ∈ NF(X). Then (∀t ∈ [0, 1]) (U(A; t) 6= ∅ ⇒ U(A; t) = ⋃ a,b∈U(A;t) A(a, b)). Proof. It is sufficient prove that U(A; t) ⊆ ⋃ a,b∈U(A;t) A(a, b). For this, assume that x ∈ U(A; t). Since x∗ (1∗x) = 1, we have x ∈ A(x, 1). Hence U(A; t) ⊆ A(x, 1) ⊆ ⋃ x∈U(A;t) A(x, 1) ⊆ ⋃ x,y∈U(A;t) A(x, y). 2 Theorem 3.6. Let X be a self distributive BE-algebra and A ∈ NF(X). Then the following conditions are equivalent: (i) A ∈ INF(X), (ii) TA(y ∗ (y ∗x)) ≤ TA(y ∗x), IA(y ∗ (y ∗x)) ≥ IA(y ∗x) and FA(y ∗ (y ∗x)) ≥ FA(y ∗x), (iii) min{TA((z ∗ (y ∗ (y ∗x))), TA(z)}≤ TA(y ∗x), min{IA((z ∗ (y ∗ (y ∗x))), IA(z)}≥ IA(y ∗x) and min{FA((z ∗ (y ∗ (y ∗x))), FA(z)}≥ FA(y ∗x). 75 A. Rezaei, A. Borumand Saeid and F. Smarandache Proof. (i)⇒(ii). Let A ∈ NF(X). By (INF1) and (BE1) we have TA(y ∗ (y ∗x)) = min{TA(y ∗ (y ∗x)), TA(1)} = min{TA(y ∗ (y ∗x)), TA(y ∗y)} ≤ TA(y ∗x), IA(y ∗ (y ∗x)) = min{IA(y ∗ (y ∗x)), IA(1)} = min{IA(y ∗ (y ∗x)), IA(y ∗y)} ≥ IA(y ∗x) and FA(y ∗ (y ∗x)) = min{FA(y ∗ (y ∗x)), FA(1)} = min{FA(y ∗ (y ∗x)), FA(y ∗y)} ≥ FA(y ∗x). (ii)⇒(iii). Let A be a neutrosophic filter of X satisfying the condition (ii). By using (NF2) and (ii) we have min{TA(z ∗ (y ∗ (y ∗x))), TA(z)} ≤ TA(y ∗ (y ∗x)) ≤ TA(y ∗x), min{IA(z ∗ (y ∗ (y ∗x))), IA(z)} ≥ IA(y ∗ (y ∗x)) ≥ IA(y ∗x) and min{FA(z ∗ (y ∗ (y ∗x))), FA(z)} ≥ FA(y ∗ (y ∗x)) ≥ FA(y ∗x). (iii)⇒(i). Since x∗ (y ∗z) = y ∗ (x∗z) ≤ (x∗y)∗ (x∗ (x∗z)), we have TA(x∗ (y ∗z)) ≤ TA((x∗y)∗ (x∗ (x∗z))), IA(x∗ (y ∗z)) ≥ IA((x∗y)∗ (x∗ (x∗z))) and FA(x∗ (y ∗z)) ≥ FA((x∗y)∗ (x∗ (x∗z))), by Proposition 3.1(i). Thus min{TA(x∗ (y ∗z)), TA(x∗y)} ≤ min{TA((x∗y)∗ (x∗ (x∗z))), TA(x∗y)} ≤ TA(x∗z). 76 Neutrosophic filters in BE-algebras min{IA(x∗ (y ∗z)), IA(x∗y)} ≥ min{IA((x∗y)∗ (x∗ (x∗z))), IA(x∗y)} ≥ IA(x∗z) and min{FA(x∗ (y ∗z)), FA(x∗y)} ≥ min{FA((x∗y)∗ (x∗ (x∗z))), FA(x∗ y)} ≥ FA(x∗z). Therefore, A ∈ INF(X). Let f : X → Y be a homomorphism of BE-algebras and A ∈ NS(X). Define tree maps TAf : X → [0, 1] such that TAf (x) = TA(f(x)), IAf : X → [0, 1] such that IAf (x) = IA(f(x)) and FAf : X → [0, 1] such that FAf (x) = FA(f(x)), for all x ∈ X. Then TAf , IAf and FAf are well-define and Af = (TAf , IAf , FAf ) ∈ NS(X).2 Theorem 3.7. Let f : X → Y be an onto homomorphism of BE-algebras and A ∈ NS(Y). Then A ∈ NF(Y) (resp. A ∈ INF(Y)) if and only if Af ∈ NF(X) (resp. Af ∈ INF(X)). Proof. Assume that A ∈ NF(Y). For any x ∈ X, we have TAf (x) = TA(f(x)) ≤ TA(1Y ) = TA(f(1X)) = TAf (1X), IAf (x) = IA(f(x)) ≥ IA(1Y ) = IA(f(1X)) = IAf (1X) and FAf (x) = FA(f(x)) ≥ FA(1Y ) = FA(f(1X)) = FAf (1X). Hence (NF1) is valid. Now, let x, y ∈ X. By (NF1) we have min{TAf (x∗y), TAf (x)} = min{TA(f(x∗y)), TA(f(x))} = min{TA(f(x)∗f(y)), TA(f(x))} ≤ TA(f(y)) = TAf (y) Also, min{IAf (x∗y), IAf (x)} = min{IA(f(x∗y)), IA(f(x))} = min{IA(f(x)∗f(y)), IA(f(x))} ≥ IA(f(y)) = IAf (y). 77 A. Rezaei, A. Borumand Saeid and F. Smarandache By a similar argument we have min{FAf (x ∗ y), FAf (x)} ≥ FAf (y). Therefore, Af ∈ NF(X). Conversely, Assume that Af ∈ NF(X). Let y ∈ Y . Since f is onto, there exists x ∈ X such that f(x) = y. Then TA(y) = TA(f(x)) = TAf (x) ≤ TAf (1X) = TA(f(1X)) = TA(1Y ), IA(y) = IA(f(x)) = IAf (x) ≥ IAf (1X) = IA(f(1X)) = IA(1Y ) and FA(y) = FA(f(x)) = FAf (x) ≥ FAf (1X) = FA(f(1X)) = FA(1Y ), Now, let x, y ∈ Y . Then there exists a, b ∈ X such that f(a) = x and f(b) = y. Hence we have min{TA(x∗y), TA(x)} = min{TA(f(a)∗f(b)), TA(f(a))} = min{TA(f(a∗ b)), TA(f(a))} = min{TAf (a∗ b), TAf (a)} ≤ TAf (b) = TA(f(b)) = TA(y). Also, we have min{IA(x∗y), IA(x)} = min{IA(f(a)∗f(b)), IA(f(a))} = min{IA(f(a∗ b)), IA(f(a))} = min{IAf (a∗ b), IAf (a)} ≥ IAf (b) = IA(f(b)) = IA(y). By a similar argument we have min{FA(x∗y), FA(x)}≥ FA(y). Therefore, A ∈ NF(Y).2 4 Conclusion F. Smarandache as an extension of intuitionistic fuzzy logic introduced the concept of neutrosophic logic and then several researchers have studied of some neutrosophic algebraic structures. 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