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Annals of Fuzzy Mathematics and Informatics FMI @ Volume 14, No. 1, (July 2017), pp. 87–97 ISSN: 2093–9310 (print version) (cid:13)c Kyung Moon Sa Co. ISSN: 2287–6235 (electronic version) http://www.kyungmoon.com http://www.afmi.or.kr Neutrosophic subalgebras of BCK/BCI-algebras based on neutrosophic points A. Borumand Saeid, Young Bae Jun Received 27 March 2017; Revised 28 April 2017; Accepted 20 May 2017 Abstract. Properties on neutrosophic ∈∨q-subsets and neutrosophic q-subsets are investigated. Relations between an (∈, ∈∨q)-neutrosophic subalgebraanda(q,∈∨q)-neutrosophicsubalgebraareconsidered. Char- acterizationofan(∈,∈∨q)-neutrosophicsubalgebrabyusingneutrosophic ∈-subsets are discussed. Conditions for an (∈, ∈∨q)-neutrosophic subalgebra to be a (q, ∈∨q)-neutrosophic subalgebra are provided. 2010 AMS Classification: 06F35, 03B60, 03B52. Keywords: Neutrosophic set, neutrosophic ∈-subset, neutrosophic q-subset, neutrosophic ∈∨q-subset, neutrosophic T -point, neutrosophic I -point, neutrosophic Φ Φ F -point. Φ Corresponding Author: Y. B. Jun ([email protected]) 1. Introduction T he concept of neutrosophic set (NS) developed by Smarandache [17, 18, 19] is a more general platform which extends the concepts of the classic set and fuzzy set (see [20], [21]), intuitionistic fuzzy set (see [1]) and interval valued intuitionistic fuzzy set (see [2]). Neutrosophic set theory is applied to various part (see [4], [5], [8], [9], [10], [11], [12], [13], [15], [16]). For further particulars, we refer readers to the site http://fs.gallup.unm.edu/neutrosophy.htm. Barbhuiya [3] introduced and studied the concept of (∈, ∈ ∨q)-intuitionistic fuzzy ideals of BCK/BCI-algebras. Jun [7] introduced the notion of neutrosophic subalgebras in BCK/BCI-algebras with several types. He provided characterizations of an (∈,∈)-neutrosophic subalgebra and an (∈,∈ ∨q)-neutrosophic subalgebra. Given special sets, so called neutrosophic ∈-subsets, neutrosophic q-subsets and neutrosophic ∈∨q-subsets, he considered conditions for the neutrosophic ∈-subsets, neutrosophic q-subsets and neutrosophic ∈∨q-subsets to be subalgebras. He discussed conditions for a neutrosophic set to be a (q, ∈∨q)-neutrosophic subalgebra. A.BorumandSaeidetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,87–97 Inthispaper,wegiverelationsbetweenan(∈,∈∨q)-neutrosophicsubalgebraand a (q, ∈∨q)-neutrosophic subalgebra. We discuss characterization of an (∈, ∈∨q)- neutrosophic subalgebra by using neutrosophic ∈-subsets. We provide conditions foran(∈,∈∨q)-neutrosophicsubalgebratobea(q,∈∨q)-neutrosophicsubalgebra. We investigate properties on neutrosophic q-subsets and neutrosophic ∈∨q-subsets. 2. Preliminaries ByaBCI-algebrawemeananalgebra(X,∗,0)oftype(2,0)satisfyingtheaxioms: (a1) ((x∗y)∗(x∗z))∗(z∗y)=0, (a2) (x∗(x∗y))∗y =0, (a3) x∗x=0, (a4) x∗y =y∗x=0 ⇒ x=y, for all x,y,z ∈X. If a BCI-algebra X satisfies the axiom (a5) 0∗x=0 for all x∈X, thenwesaythatX isaBCK-algebra. AnonemptysubsetSofaBCK/BCI-algebra X is called a subalgebra of X if x∗y ∈S for all x,y ∈S. We refer the reader to the books [6] and [14] for further information regarding BCK/BCI-algebras. For any family {a |i∈Λ} of real numbers, we define i (cid:26) (cid:95) max{a |i∈Λ} if Λ is finite, {a |i∈Λ}:= i i sup{a |i∈Λ} otherwise. i (cid:26) (cid:94) min{a |i∈Λ} if Λ is finite, {a |i∈Λ}:= i i inf{a |i∈Λ} otherwise. i (cid:87) If Λ = {1,2}, we will also use a ∨a and a ∧a instead of {a | i ∈ Λ} and 1 2 1 2 i (cid:86) {a |i∈Λ}, respectively. i Let X be a non-empty set. A neutrosophic set (NS) in X (see [18]) is a structure of the form: A:={(cid:104)x;A (x),A (x),A (x)(cid:105)|x∈X} T I F where A : X → [0,1] is a truth membership function, A : X → [0,1] is an T I indeterminate membership function, and A : X → [0,1] is a false membership F function. For the sake of simplicity, we shall use the symbol A = (A ,A ,A ) for T I F the neutrosophic set A:={(cid:104)x;A (x),A (x),A (x)(cid:105)|x∈X}. T I F 88 A.BorumandSaeidetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,87–97 3. Neutrosophic subalgebras of several types GivenaneutrosophicsetA=(A ,A ,A )inasetX,α,β ∈(0,1]andγ ∈[0,1), T I F we consider the following sets: T (A;α):={x∈X |A (x)≥α}, ∈ T I (A;β):={x∈X |A (x)≥β}, ∈ I F (A;γ):={x∈X |A (x)≤γ}, ∈ F T (A;α):={x∈X |A (x)+α>1}, q T I (A;β):={x∈X |A (x)+β >1}, q I F (A;γ):={x∈X |A (x)+γ <1}, q F T (A;α):={x∈X |A (x)≥α or A (x)+α>1}, ∈∨q T T I (A;β):={x∈X |A (x)≥β or A (x)+β >1}, ∈∨q I I F (A;γ):={x∈X |A (x)≤γ or A (x)+γ <1}. ∈∨q F F WesayT (A;α),I (A;β)andF (A;γ)areneutrosophic∈-subsets;T (A;α),I (A;β) ∈ ∈ ∈ q q andF (A;γ)areneutrosophic q-subsets; andT (A;α), I (A;β)andF (A;γ) q ∈∨q ∈∨q ∈∨q are neutrosophic ∈∨q-subsets. For Φ∈{∈,q,∈∨q}, the element of T (A;α) (resp., Φ I (A;β) and F (A;γ)) is called a neutrosophic T -point (resp., neutrosophic I - Φ Φ Φ Φ point and neutrosophic F -point) with value α (resp., β and γ) (see [7]). Φ It is clear that (3.1) T (A;α)=T (A;α)∪T (A;α), ∈∨q ∈ q (3.2) I (A;β)=I (A;β)∪I (A;β), ∈∨q ∈ q (3.3) F (A;γ)=F (A;γ)∪F (A;γ). ∈∨q ∈ q Definition3.1([7]). GivenΦ,Ψ∈{∈,q,∈∨q},aneutrosophicsetA=(A ,A ,A ) T I F inaBCK/BCI-algebraX iscalleda(Φ, Ψ)-neutrosophic subalgebraofX ifthefol- lowing assertions are valid. x∈T (A;α ), y ∈T (A;α ) ⇒ x∗y ∈T (A;α ∧α ), Φ x Φ y Ψ x y (3.4) x∈I (A;β ), y ∈I (A;β ) ⇒ x∗y ∈I (A;β ∧β ), Φ x Φ y Ψ x y x∈F (A;γ ), y ∈F (A;γ ) ⇒ x∗y ∈F (A;γ ∨γ ) Φ x Φ y Ψ x y for all x,y ∈X, α ,α , β ,β ∈(0,1] and γ ,γ ∈[0,1). x y x y x y Lemma 3.2 ([7]). A neutrosophic set A = (A ,A ,A ) in a BCK/BCI-algebra T I F X is an (∈, ∈∨q)-neutrosophic subalgebra of X if and only if it satisfies:  (cid:86)  A (x∗y)≥ {A (x),A (y),0.5} T T T (3.5) (∀x,y ∈X) A (x∗y)≥(cid:86){A (x),A (y),0.5} .  I I I  (cid:87) A (x∗y)≤ {A (x),A (y),0.5} F F F Theorem 3.3. A neutrosophic set A=(A ,A ,A ) in a BCK/BCI-algebra X is T I F an (∈, ∈∨q)-neutrosophic subalgebra of X if and only if the neutrosophic ∈-subsets T (A;α), I (A;β) and F (A;γ) are subalgebras of X for all α,β ∈ (0,0.5] and ∈ ∈ ∈ γ ∈[0.5,1). 89 A.BorumandSaeidetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,87–97 Proof. Assume that A = (A ,A ,A ) is an (∈, ∈∨q)-neutrosophic subalgebra of T I F X. For any x,y ∈X, let α∈(0,0.5] be such that x,y ∈T (A;α). Then A (x)≥α ∈ T and A (y)≥α. It follows from (3.5) that T (cid:94) A (x∗y)≥ {A (x),A (y),0.5}≥α∧0.5=α T T T andsothatx∗y ∈T (A;α). ThusT (A;α)isasubalgebraofX forallα∈(0,0.5]. ∈ ∈ Similarly, I (A;β) is a subalgebra of X for all β ∈ (0,0.5]. Now, let γ ∈ [0.5,1) be ∈ such that x,y ∈F (A;γ). Then A (x)≤γ and A (y)≤γ. Hence ∈ F F (cid:95) A (x∗y)≤ {A x),A (y),0.5}≤γ∨0.5=γ F F F by (3.5), and so x ∗ y ∈ F (A;γ). Thus F (A;γ) is a subalgebra of X for all ∈ ∈ γ ∈[0.5,1). Conversely, letα,β ∈(0,0.5]andγ ∈[0.5,1)besuchthatT (A;α), I (A;β)and ∈ ∈ F (A;γ) are subalgebras of X. If there exist a,b∈X such that ∈ (cid:94) A (a∗b)< {A (a),A (b),0.5}, I I I then we can take β ∈(0,1) such that (cid:94) (3.6) A (a∗b)<β < {A (a),A (b),0.5}. I I I Thus a,b∈I (A;β) and β <0.5, and so a∗b∈I (A;β). But, the left inequality in ∈ ∈ (3.6) induces a∗b∈/ I (A;β), a contradiction. Hence ∈ (cid:94) A (x∗y)≥ {A (x),A (y),0.5} I I I for all x,y ∈X. Similarly, we can show that (cid:94) A (x∗y)≥ {A (x),A (y),0.5} T T T for all x,y ∈X. Now suppose that (cid:95) A (a∗b)> {A (a),A (b),0.5} F F F for some a,b∈X. Then there exists γ ∈(0,1) such that (cid:95) A (a∗b)>γ > {A (a),A (b),0.5}. F F F It follows that γ ∈(0.5,1) and a,b∈F (A;γ). Since F (A;γ) is a subalgebra of X, ∈ ∈ we have a∗b∈F (A;γ) and so A (a∗b)≤γ. This is a contradiction, and thus ∈ F (cid:95) A (x∗y)≤ {A (x),A (y),0.5} F F F for all x,y ∈ X. Using Lemma 3.2, A = (A ,A ,A ) is an (∈, ∈∨q)-neutrosophic T I F subalgebra of X. (cid:3) Using Theorem 3.3 and [7, Theorem 3.8], we have the following corollary. Corollary 3.4. For a neutrosophic set A = (A ,A ,A ) in a BCK/BCI-algebra T I F X,ifthenonemptyneutrosophic∈∨q-subsetsT (A;α),I (A;β)andF (A;γ) ∈∨q ∈∨q ∈∨q are subalgebras of X for all α,β ∈ (0,1] and γ ∈ [0,1), then the neutrosophic ∈- subsets T (A;α), I (A;β) and F (A;γ) are subalgebras of X for all α,β ∈ (0,0.5] ∈ ∈ ∈ and γ ∈[0.5,1). 90 A.BorumandSaeidetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,87–97 Theorem 3.5. Given neutrosophic set A = (A ,A ,A ) in a BCK/BCI-algebra T I F X, the nonempty neutrosophic ∈-subsets T (A;α), I (A;β) and F (A;γ) are sub- ∈ ∈ ∈ algebras of X for all α,β ∈ (0.5,1] and γ ∈ [0,0.5) if and only if the following assertion is valid.   A (x∗y)∨0.5≥A (x)∧A (y) T T T (3.7) (∀x,y ∈X) A (x∗y)∨0.5≥A (x)∧A (y) .  I I I  A (x∗y)∧0.5≤A (x)∨A (y) F F F Proof. Assume that the nonempty neutrosophic ∈-subsets T (A;α), I (A;β) and ∈ ∈ F (A;γ) are subalgebras of X for all α,β ∈ (0.5,1] and γ ∈ [0,0.5). Suppose ∈ that there are a,b ∈ X such that A (a∗b)∨0.5 < A (a)∧A (b) := α. Then T T T α∈(0.5,1] and a,b∈T (A;α). Since T (A;α) is a subalgebra of X, it follows that ∈ ∈ a∗b∈T (A;α), that is, A (a∗b)≥α which is a contradiction. Thus ∈ T A (x∗y)∨0.5≥A (x)∧A (y) T T T for all x,y ∈ X. Similarly, we know that A (x ∗ y) ∨ 0.5 ≥ A (x) ∧ A (y) for I I I all x,y ∈ X. Now, if A (x∗y)∧0.5 > A (x)∨A (y) for some x,y ∈ X, then F F F x,y ∈ F (A;γ) and γ ∈ [0,0.5) where γ = A (x)∨A (y). But, x∗y ∈/ F (A;γ) ∈ F F ∈ which is a contradiction. Hence A (x∗y)∧0.5≤A (x)∨A (y) for all x,y ∈X. F F F Conversely, let A = (A ,A ,A ) be a neutrosophic set in X satisfying the con- T I F dition (3.7). Let x,y,a,b ∈ X and α,β ∈ (0.5,1] be such that x,y ∈ T (A;α) and ∈ a,b∈I (A;β). Then ∈ A (x∗y)∨0.5≥A (x)∧A (y)≥α>0.5, T T T A (a∗b)∨0.5≥A (a)∧A (b)≥β >0.5. I I I It follows that A (x∗y) ≥ α and A (a∗b) ≥ β, that is, x∗y ∈ T (A;α) and T I ∈ a∗b ∈ I (A;β). Now, let x,y ∈ X and γ ∈ [0,0.5) be such that x,y ∈ F (A;γ). ∈ ∈ Then A (x ∗ y) ∧ 0.5 ≤ A (x) ∨ A (y) ≤ γ < 0.5 and so A (x ∗ y) ≤ γ, i.e., F F F F x∗y ∈F (A;γ). This completes the proof. (cid:3) ∈ We consider relations between a (q, ∈∨q)-neutrosophic subalgebra and an (∈, ∈∨q)-neutrosophic subalgebra. Theorem 3.6. In a BCK/BCI-algebra, every (q, ∈∨q)-neutrosophic subalgebra is an (∈, ∈∨q)-neutrosophic subalgebra. Proof. LetA=(A ,A ,A )bea(q,∈∨q)-neutrosophicsubalgebraofaBCK/BCI- T I F algebra X and let x,y ∈ X. Let α ,α ∈ (0,1] be such that x ∈ T (A;α ) and x y ∈ x y ∈T (A;α ). ThenA (x)≥α andA (y)≥α . Supposex∗y ∈/ T (A;α ∧α ). ∈ y T x T y ∈∨q x y Then (3.8) A (x∗y)<α ∧α , T x y (3.9) A (x∗y)+(α ∧α )≤1. T x y It follows that (3.10) A (x∗y)<0.5. T 91 A.BorumandSaeidetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,87–97 Combining (3.8) and (3.10), we have (cid:94) A (x∗y)< {α ,α ,0.5} T x y and so (cid:94) 1−A (x∗y)>1− {α ,α ,0.5} T x y (cid:95) = {1−α ,1−α ,0.5} x y (cid:95) ≥ {1−A (x),1−A (y),0.5}. T T Hence there exists α∈(0,1] such that (cid:95) (3.11) 1−A (x∗y)≥α> {1−A (x),1−A (y),0.5}. T T T The right inequality in (3.11) induces A (x)+α > 1 and A (y)+α > 1, that is, T T x,y ∈ T (A;α). Since A = (A ,A ,A ) is a (q, ∈∨q)-neutrosophic subalgebra q T I F of X, we have x∗y ∈ T (A;α). But, the left inequality in (3.11) implies that ∈∨q A (x∗y)+α≤1, i.e., x∗y ∈/ T (A;α), and A (x∗y)≤1−α<1−0.5=0.5<α, T q T i.e., x∗y ∈/ T (A;α). Hence x∗y ∈/ T (A;α), a contradiction. Thus x∗y ∈ ∈ ∈∨q T (A;α ∧α ). Similarly, we can show that if x ∈ I (A;β ) and y ∈ I (A;β ) ∈∨q x y ∈ x ∈ y for β ,β ∈ (0,1], then x∗y ∈ I (A;β ∧β ). Now, let γ ,γ ∈ [0,1) be such x y ∈∨q x y x y that x ∈ F (A;γ ) and y ∈ F (A;γ ). A (x) ≤ γ and A (y) ≤ γ . If x∗y ∈/ ∈ x ∈ y F x F y F (A;γ ∨γ ), then ∈∨q x y (3.12) A (x∗y)>γ ∨γ , F x y (3.13) A (x∗y)+(γ ∨γ )≥1. F x y It follows that (cid:95) A (x∗y)> {γ ,γ ,0.5} F x y and so that (cid:95) 1−A (x∗y)<1− {γ ,γ ,0.5} F x y (cid:94) = {1−γ ,1−γ ,0.5} x y (cid:94) ≤ {1−A (x),1−A (y),0.5}. F F Thus there exists γ ∈[0,1) such that (cid:94) (3.14) 1−A (x∗y)≤γ < {1−A (x),1−A (y),0.5}. F F F Itfollows fromtherightinequalityin(3.14)that A (x)+γ <1and A (y)+γ <1, F F that is, x,y ∈ F (A;γ), which implies that x ∗ y ∈ F (A;γ). But, we have q ∈∨q x∗y ∈/ F (A;γ) by the left inequality in (3.14). This is a contradiction, and so ∈∨q x∗y ∈F (A;γ ∨γ ). Therefore A=(A ,A ,A ) is an (∈, ∈∨q)-neutrosophic ∈∨q x y T I F subalgebra of X. (cid:3) The following example shows that the converse of Theorem 3.6 is not true. 92 A.BorumandSaeidetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,87–97 Table 1. Cayley table of the operation ∗ ∗ 0 1 2 3 4 0 0 0 0 0 0 1 1 0 0 1 1 2 2 1 0 2 2 3 3 3 3 0 3 4 4 4 4 4 0 X A (x) A (x) A (x) T I F 0 0.6 0.8 0.3 1 0.2 0.3 0.6 2 0.2 0.3 0.6 3 0.7 0.1 0.7 4 0.4 0.4 0.9 Example 3.7. Consider a BCK-algebra X ={0,1,2,3,4} with the following Cay- ley table. Let A=(A ,A ,A ) be a neutrosophic set in X defined by T I F Then  {0,3} if α∈(0.4,0.5],  T (A;α)= {0,3,4} if α∈(0.2,0.4], ∈  X if α∈(0,0.2],  {0} if β ∈(0.4,0.5],  {0,4} if β ∈(0.3,0.4], I (A;β)= ∈ {0,1,2,4} if β ∈(0.1,0.3],  X if β ∈(0,0.1],  X if γ ∈(0.9,1),  {0,1,2,3} if γ ∈[0.7,0.9), F (A;γ)= ∈ {0,1,2} if γ ∈[0.6,0.7),  {0} if γ ∈[0.5,0.6), which are subalgebras of X for all α,β ∈ (0,0.5] and γ ∈ [0.5,1). Using Theorem 3.3, A = (A ,A ,A ) is an (∈, ∈∨q)-neutrosophic subalgebra of X. But it is not T I F a (q, ∈∨q)-neutrosophic subalgebra of X since 2 ∈ T (A;0.83) and 3 ∈ T (A;0.4), q q but 2∗3=2∈/ T (A;0.4). ∈∨q Weprovideconditionsforan(∈,∈∨q)-neutrosophicsubalgebratobea(q,∈∨q)- neutrosophic subalgebra. Theorem 3.8. Assume that any neutrosophic T -point and neutrosophic I -point Φ Φ has the value α and β in (0,0.5], respectively, and any neutrosophic F -point has Φ the value γ in [0.5,1) for Φ ∈ {∈,q,∈ ∨q}. Then every (∈, ∈ ∨q)-neutrosophic subalgebra is a (q, ∈∨q)-neutrosophic subalgebra. Proof. Let X be a BCK/BCI-algebra and let A = (A ,A ,A ) be an (∈, ∈∨q)- T I F neutrosophic subalgebra of X. For x,y,a,b ∈ X, let α ,α ,β ,β ∈ (0,0.5] be x y a b 93 A.BorumandSaeidetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,87–97 such that x ∈ T (A;α ), y ∈ T (A;α ), a ∈ I (A;β ) and b ∈ T (A;β ). Then q x q y q a q b A (x)+α > 1, A (y)+α > 1, A (a)+β > 1 and A (b)+β > 1. Since T x T y I a I b α ,α ,β ,β ∈(0,0.5], it follows that A (x)>1−α ≥α , A (y)>1−α ≥α , x y a b T x x T y y A (a)>1−β ≥β andA (b)>1−β ≥β ,thatis,x∈T (A;α ),y ∈T (A;α ), I a a I b b ∈ x ∈ y a ∈ I (A;β ) and b ∈ I (A;β ). Also, let x ∈ F (A;γ ) and y ∈ F (A;γ ) for ∈ a ∈ b q x q y x,y ∈ X and γ ,γ ∈ [0.5,1). Then A (x)+γ < 1 and A (y)+γ < 1, and so x y F x F y A (x)<1−γ ≤γ andA (y)<1−γ ≤γ sinceγ ,γ ∈[0.5,1). Thisshowsthat F x x F y y x y x∈F (A;γ )andy ∈F (A;γ ). Itfollowsfrom(3.4)thatx∗y ∈T (A;α ∧α ), ∈ x ∈ y ∈∨q x y a∗b∈I (A;β ∧β ),andx∗y ∈F (A;γ ∨γ ). Consequently,A=(A ,A ,A ) ∈∨q a b ∈∨q x y T I F is a (q, ∈∨q)-neutrosophic subalgebra of X. (cid:3) Theorem 3.9. Both (∈, ∈)-neutrosophic subalgebra and (∈∨q, ∈∨q)-neutrosophic subalgebra are an (∈, ∈∨q)-neutrosophic subalgebra. Proof. It is clear that (∈, ∈)-neutrosophic subalgebra is an (∈, ∈∨q)-neutrosophic subalgebra. Let A = (A ,A ,A ) be an (∈∨q, ∈∨q)-neutrosophic subalgebra of T I F X. For any x,y,a,b ∈ X, let α , α , β , β ∈ (0,1] be such that x ∈ T (A;α ), x y a b ∈ x y ∈ T (A;α ), a ∈ I (A;β ) and b ∈ I (A;β ). Then x ∈ T (A;α ), y ∈ ∈ y ∈ a ∈ b ∈∨q x T (A;α ), a ∈ I (A;β ) and b ∈ I (A;β ) by (3.1) and (3.2). It follows ∈∨q y ∈∨q a ∈∨q b that x∗y ∈ T (A;α ∧α ) and a∗b ∈ I (A;β ∧β ). Now, let x,y ∈ X and ∈∨q x y ∈∨q a b γ ,γ ∈[0,1) be such that x∈F (A;γ ) and y ∈F (A;γ ). Then x∈F (A;γ ) x y ∈ x ∈ y ∈∨q x and y ∈ F (A;γ ) by (3.3). Hence x∗y ∈ F (A;γ ∨γ ). Therefore A = ∈∨q y ∈∨q x y (A ,A ,A ) is an (∈, ∈∨q)-neutrosophic subalgebra of X. (cid:3) T I F The converse of Theorem 3.9 is not true in general. In fact, the (∈, ∈ ∨q)- neutrosophic subalgebra A = (A ,A ,A ) in Example 3.7 is neither an (∈, ∈)- T I F neutrosophic subalgebra nor an (∈∨q, ∈∨q)-neutrosophic subalgebra. Theorem 3.10. For a neutrosophic set A=(A ,A ,A ) in a BCK/BCI-algebra T I F X, if the nonempty neutrosophic q-subsets T (A;α), I (A;β) and F (A;γ) are sub- q q q algebras of X for all α,β ∈(0.5,1] and γ ∈(0,0.5), then x∈T (A;α ), y ∈T (A;α ) ⇒ x∗y ∈T (A;α ∨α ), ∈ x ∈ y q x y (3.15) x∈I (A;β ), y ∈I (A;β ) ⇒ x∗y ∈I (A;β ∨β ), ∈ x ∈ y q x y x∈F (A;γ ), y ∈F (A;γ ) ⇒ x∗y ∈F (A;γ ∧γ ) ∈ x ∈ y q x y for all x,y ∈X, α ,α , β ,β ∈(0.5,1] and γ ,γ ∈(0,0.5). x y x y x y Proof. Let x,y,a,b,u,v ∈ X and α , α , β , β ∈ (0.5,1] and γ ,γ ∈ (0,0.5) be x y a b u v such that x∈T (A;α ), y ∈T (A;α ), a∈I (A;β ), b∈I (A;β ), u∈F (A;γ ) ∈ x ∈ y ∈ a ∈ b ∈ u and v ∈ F (A;γ ). Then A (x) ≥ α > 1−α , A (y) ≥ α > 1−α , A (a) ≥ ∈ v T x x T y y I β >1−β , A (b)≥β >1−β , A (u)≤γ <1−γ and A (v)≤γ <1−γ . a a I b b F u u F v v It follows that x,y ∈T (A;α ∨α ), a,b∈I (A;β ∨β ) and u,v ∈F (A;γ ∧γ ). q x y q a b q u v Sinceα ∨α ,β ∨β ∈(0.5,1]andγ ∧γ ∈(0,0.5),wehavex∗y ∈T (A;α ∨α ), x y a b u v q x y a∗b∈I (A;β ∨β ) and u∗v ∈F (A;γ ∧γ ) by hypothesis. This completes the q a b q u v proof. (cid:3) The following corollary is by Theorem 3.10 and [7, Theorem 3.7]. Corollary 3.11. Every (∈, ∈∨q)-neutrosophic subalgebra A = (A ,A ,A ) in a T I F BCK/BCI-algebra X satisfies the condition (3.15). 94 A.BorumandSaeidetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,87–97 Corollary 3.12. Every (q, ∈∨q)-neutrosophic subalgebra A = (A ,A ,A ) in a T I F BCK/BCI-algebra X satisfies the condition (3.15). Proof. It is by Theorem 3.6 and Corollary 3.11. (cid:3) Theorem 3.13. For a neutrosophic set A=(A ,A ,A ) in a BCK/BCI-algebra T I F X, if the nonempty neutrosophic q-subsets T (A;α), I (A;β) and F (A;γ) are sub- q q q algebras of X for all α,β ∈(0,0.5] and γ ∈(0.5,1), then x∈T (A;α ), y ∈T (A;α ) ⇒ x∗y ∈T (A;α ∨α ), q x q y ∈ x y (3.16) x∈I (A;β ), y ∈I (A;β ) ⇒ x∗y ∈I (A;β ∨β ), q x q y ∈ x y x∈F (A;γ ), y ∈F (A;γ ) ⇒ x∗y ∈F (A;γ ∧γ ) q x q y ∈ x y for all x,y ∈X, α ,α , β ,β ∈(0,0.5] and γ ,γ ∈(0.5,1). x y x y x y Proof. Let x,y,a,b,u,v ∈ X and α , α , β , β ∈ (0,0.5] and γ ,γ ∈ (0.5,1) x y a b u v be such that x ∈ T (A;α ), y ∈ T (A;α ), a ∈ I (A;β ), b ∈ I (A;β ), u ∈ q x q y q a q b F (A;γ ) and v ∈ F (A;γ ). Then x,y ∈ T (A;α ∨α ), a,b ∈ I (A;β ∨β ) and q u q v q x y q a b u,v ∈ F (A;γ ∧γ ). Since α ∨α ,β ∨β ∈ (0,0.5] and γ ∧γ ∈ (0.5,1), it q u v x y a b u v follows from the hypothesis that x∗y ∈ T (A;α ∨α ), a∗b ∈ I (A;β ∨β ) and q x y q a b u∗v ∈F (A;γ ∧γ ). Hence q u v A (x∗y)>1−(α ∨α )≥α ∨α , that is, x∗y ∈T (A;α ∨α ), T x y x y ∈ x y A (a∗b)>1−(β ∨β )≥β ∨β , that is, a∗b∈I (A;β ∨β ), I a b a b ∈ a b A (u∗v)<1−(γ ∧γ )≤γ ∧γ , that is, u∗v ∈F (A;γ ∧γ ). F u v u v ∈ u v Consequently, the condition (3.16) is valid for all x,y ∈ X, α ,α , β ,β ∈ (0,0.5] x y x y and γ ,γ ∈(0.5,1). (cid:3) x y Theorem 3.14. Given a neutrosophic set A = (A ,A ,A ) in a BCK/BCI- T I F algebra X, if the nonempty neutrosophic ∈∨q-subsets T (A;α), I (A;β) and ∈∨q ∈∨q F (A;γ) are subalgebras of X for all α,β ∈ (0,0.5] and γ ∈ [0.5,1), then the ∈∨q following assertions are valid. x∈T (A;α ), y ∈T (A;α ) ⇒ x∗y ∈T (A;α ∨α ), q x q y ∈∨q x y (3.17) x∈I (A;β ), y ∈I (A;β ) ⇒ x∗y ∈I (A;β ∨β ), q x q y ∈∨q x y x∈F (A;γ ), y ∈F (A;γ ) ⇒ x∗y ∈F (A;γ ∧γ ) q x q y ∈∨q x y for all x,y ∈X, α ,α , β ,β ∈(0,0.5] and γ ,γ ∈[0.5,1). x y x y x y Proof. Let x,y,a,b,u,v ∈ X and α , α , β , β ∈ (0,0.5] and γ ,γ ∈ [0.5,1) be x y a b u v such that x ∈ T (A;α ), y ∈ T (A;α ), a ∈ I (A;β ), b ∈ I (A;β ), u ∈ F (A;γ ) q x q y q a q b q u and v ∈ F (A;γ ). Then x ∈ T (A;α ), y ∈ T (A;α ), a ∈ I (A;β ), q v ∈∨q x ∈∨q y ∈∨q a b ∈ I (A;β ), u ∈ F (A;γ ) and v ∈ F (A;γ ). It follows that x,y ∈ ∈∨q b ∈∨q u ∈∨q v T (A;α ∨α ), a,b ∈ I (A;β ∨β ) and u,v ∈ F (A;γ ∧γ ) which imply ∈∨q x y ∈∨q a b ∈∨q u v from the hypothesis that x∗y ∈ T (A;α ∨α ), a∗b ∈ I (A;β ∨β ) and ∈∨q x y ∈∨q a b u∗v ∈F (A;γ ∧γ ). This completes the proof. (cid:3) ∈∨q u v Corollary 3.15. Every (∈, ∈∨q)-neutrosophic subalgebra A = (A ,A ,A ) of a T I F BCK/BCI-algebra X satisfies the condition (3.17). Proof. It is by Theorem 3.14 and [7, Theorem 3.9]. (cid:3) 95 A.BorumandSaeidetal./Ann. FuzzyMath. Inform. 14(2017),No. 1,87–97 Theorem 3.16. Given a neutrosophic set A = (A ,A ,A ) in a BCK/BCI- T I F algebra X, if the nonempty neutrosophic ∈∨q-subsets T (A;α), I (A;β) and ∈∨q ∈∨q F (A;γ) are subalgebras of X for all α,β ∈ (0.5,1] and γ ∈ [0,0.5), then the ∈∨q following assertions are valid. x∈T (A;α ), y ∈T (A;α ) ⇒ x∗y ∈T (A;α ∨α ), q x q y ∈∨q x y (3.18) x∈I (A;β ), y ∈I (A;β ) ⇒ x∗y ∈I (A;β ∨β ), q x q y ∈∨q x y x∈F (A;γ ), y ∈F (A;γ ) ⇒ x∗y ∈F (A;γ ∧γ ) q x q y ∈∨q x y for all x,y ∈X, α ,α , β ,β ∈(0.5,1] and γ ,γ ∈[0,0.5). x y x y x y Proof. It is similar to the proof Theorem 3.14. (cid:3) Corollary 3.17. Every (q, ∈∨q)-neutrosophic subalgebra A = (A ,A ,A ) of a T I F BCK/BCI-algebra X satisfies the condition (3.18). Proof. It is by Theorem 3.16 and [7, Theorem 3.10]. (cid:3) Combining Theorems 3.14 and 3.16, we have the following corollary. Corollary 3.18. Given a neutrosophic set A = (A ,A ,A ) in a BCK/BCI- T I F algebra X, if the nonempty neutrosophic ∈∨q-subsets T (A;α), I (A;β) and ∈∨q ∈∨q F (A;γ) are subalgebras of X for all α,β ∈(0,1] and γ ∈[0,1), then the following ∈∨q assertions are valid. x∈T (A;α ), y ∈T (A;α ) ⇒ x∗y ∈T (A;α ∨α ), q x q y ∈∨q x y x∈I (A;β ), y ∈I (A;β ) ⇒ x∗y ∈I (A;β ∨β ), q x q y ∈∨q x y x∈F (A;γ ), y ∈F (A;γ ) ⇒ x∗y ∈F (A;γ ∧γ ) q x q y ∈∨q x y for all x,y ∈X, α ,α , β ,β ∈(0,1] and γ ,γ ∈[0,1). x y x y x y Conclusions We have considered relations between an (∈, ∈∨q)-neutrosophic subalgebra and a (q, ∈∨q)-neutrosophic subalgebra. We have discussed characterization of an (∈, ∈∨q)-neutrosophic subalgebra by using neutrosophic ∈-subsets, and have provided conditions for an (∈, ∈∨q)-neutrosophic subalgebra to be a (q, ∈∨q)-neutrosophic subalgebra. We have investigated properties on neutrosophic q-subsets and neutrosophic ∈∨q-subsets. 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