Generalizations of Neutrosophic Subalgebras in BCK/BCI-Algebras Based on Neutrosophic Points by Ioan Degău

Neutrosophic Sets and Systems, Vol. 20, 2018

University of New Mexico

Generalizations of Neutrosophic Subalgebras in BCK/BCI-Algebras Based on Neutrosophic Points Seon Jeong Kim1 , Seok-Zun Song2 and Young Bae Jun1 1 Department

of Mathematics, Natural Science of College, Gyeongsang National University, Jinju 52828, Korea. E-mail: skim@gnu.ac.kr 2 Department

3 Department

of Mathematics, Jeju National University, Jeju 63243, Korea. E-mail: szsong@jejunu.ac.kr

of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea. E-mail: skywine@gmail.com ∗ Correspondence:

Y.B. Jun (skywine@gmail.com)

Abstract: Saeid and Jun introduced the notion of neutrosophic points, and studied neutrosophic subalgebras of several types in BCK/BCI-algebras by using the notion of neutrosophic points (see [4] and [6]). More general form of neutrosophic points is considered in this paper, and generalizations of Saeid and Jun’s results are

discussed. The concepts of (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra, (q(kT ,kI ,kF ) , ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra and (∈, q(kT ,kI ,kF ) )-neutrosophic subalgebra are introduced, and several properties are investigated. Characterizations of (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra are discussed.

Keywords: (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra; (q(kT ,kI ,kF ) , ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra; (∈, q(kT ,kI ,kF ) )-neutrosophic subalgebra.

Introduction

terizations of (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra. We consider relations between (∈, ∈)-neutrosophic subalgebra, (∈, As a generalization of fuzzy sets, Atanassov [1] introduced the q(kT ,kI ,kF ) )-neutrosophic subalgebra and (∈, ∈ ∨q(kT ,kI ,kF ) )degree of nonmembership/falsehood (f) in 1986 and defined the neutrosophic subalgebra. intuitionistic fuzzy set. As a more general platform which extends the notions of the classic set and fuzzy set, intuitionistic fuzzy set and interval valued (intuitionistic) fuzzy set, Smaran- 2 Preliminaries dache introduced the notion of neutrosophic sets (see [7, 8]), which is useful mathematical tool for dealing with incomplete, By a BCI-algebra, we mean a set X with a binary operation ∗ inconsistent and indeterminate information. For further particu- and the special element 0 satisfying the conditions (see [3, 5]): lars on neutrosophic set theory, we refer the readers to the site (a1) (∀x, y, z ∈ X)(((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0), http://fs.gallup.unm.edu/FlorentinSmarandache.htm Jun [4] introduced the notion of (Φ, Ψ)-neutrosophic subalgebra of a BCK/BCI-algebra X for Φ, Ψ ∈ {∈, q, ∈ ∨ q}, and investigated related properties. He provided characterizations of an (∈, ∈)-neutrosophic subalgebra and an (∈, ∈ ∨ q)-neutrosophic subalgebra, and considered conditions for a neutrosophic set to be a (q, ∈ ∨ q)-neutrosophic subalgebra. Saeid and Jun [6] gave relations between an (∈, ∈ ∨ q)-neutrosophic subalgebra and a (q, ∈ ∨ q)-neutrosophic subalgebra, and investigated properties on neutrosophic q-subsets and neutrosophic ∈ ∨ q-subsets. The purpose of this article is to give an algebraic tool of neutrosophic set theory which can be used in applied sciences, for example, decision making problems, medical sciences etc. We consider a general form of neutrosophic points, and then we discuss generalizations of the papers [4] and [6]. As a generalization of (∈, ∈ ∨ q)-neutrosophic subalgebras, we introduce the notions of (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra, and (∈, q(kT ,kI ,kF ) )-neutrosophic subalgebra in BCK/BCIalgebras, and investigate several properties. We discuss charac-

(a2) (∀x, y ∈ X)((x ∗ (x ∗ y)) ∗ y = 0), (a3) (∀x ∈ X)(x ∗ x = 0), (a4) (∀x, y ∈ X)(x ∗ y = y ∗ x = 0 ⇒ x = y). If a BCI-algebra X satisfies the axiom (a5) 0 ∗ x = 0 for all x ∈ X, then we say that X is a BCK-algebra (see [3, 5]). A nonempty subset S of a BCK/BCI-algebra X is called a subalgebra of X (see [3, 5]) if x ∗ y ∈ S for all x, y ∈ S. The collection of all BCK-algebras and all BCI-algebras are denoted by BK (X) and BI (X), respectively. Also B(X) := BK (X) ∪ BI (X). We refer the reader to the books [3] and [5] for further information regarding BCK/BCI-algebras. Let X be a non-empty set. A neutrosophic set (NS) in X (see [7]) is a structure of the form: A := {hx; AT (x), AI (x), AF (x)i | x ∈ X}

(2.1)

S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points

Neutrosophic Sets and Systems, Vol. 20, 2018

where AT , AI and AF are a truth membership function, an inde- (0, 1] and γ ∈ [0, 1), we consider the following sets: terminate membership function and a false membership function, TqkT (A; α) := {x ∈ X | AT (x) + α + kT > 1}, respectively, from X into the unit interval [0, 1]. The neutroIqkI (A; β) := {x ∈ X | AI (x) + β + kI > 1}, sophic set (2.1) will be denoted by A = (AT , AI , AF ). FqkF (A; γ) := {x ∈ X | AF (x) + γ + kF < 1}, Given a neutrosophic set A = (AT , AI , AF ) in a set X, α, β ∈ T∈∨ qkT (A; α) := {x ∈ X | AT (x) ≥ α or (0, 1] and γ ∈ [0, 1), we consider the following sets (see [4]): AT (x) + α + kT > 1}, I (A; β) := {x ∈ X | AI (x) ≥ β or ∈∨ q kI T∈ (A; α) := {x ∈ X | AT (x) ≥ α}, AI (x) + β + kI > 1}, I∈ (A; β) := {x ∈ X | AI (x) ≥ β}, F∈∨ qkF (A; γ) := {x ∈ X | AF (x) ≤ γ or F∈ (A; γ) := {x ∈ X | AF (x) ≤ γ}, AF (x) + γ + kF < 1}. Tq (A; α) := {x ∈ X | AT (x) + α > 1}, We say TqkT (A; α), IqkI (A; β) and FqkF (A; γ) are neuIq (A; β) := {x ∈ X | AI (x) + β > 1}, trosophic qk -subsets; and T∈∨ qkT (A; α), I∈∨ qkI (A; β) and Fq (A; γ) := {x ∈ X | AF (x) + γ < 1}, F∈∨ qkF (A; γ) are neutrosophic (∈ ∨ qk )-subsets. For Φ ∈ {∈, T∈∨ q (A; α) := {x ∈ X | AT (x) ≥ α or AT (x) + α > 1}, q, qk , qkT , qkI , qkF , ∈ ∨ q, ∈ ∨ qk , ∈ ∨ qkT , ∈ ∨ qkI , ∈ ∨ qkF }, I∈∨ q (A; β) := {x ∈ X | AI (x) ≥ β or AI (x) + β > 1}, the element of TΦ (A; α) (resp., IΦ (A; β) and FΦ (A; γ)) is called a neutrosophic TΦ -point (resp., neutrosophic IΦ -point and neuF∈∨ q (A; γ) := {x ∈ X | AF (x) ≤ γ or AF (x) + γ < 1}. trosophic FΦ -point) with value α (resp., β and γ). We say T∈ (A; α), I∈ (A; β) and F∈ (A; γ) are neutrosophic ∈-subsets; Tq (A; α), Iq (A; β) and Fq (A; γ) are neutrosophic qsubsets; and T∈∨ q (A; α), I∈∨ q (A; β) and F∈∨ q (A; γ) are neutrosophic ∈ ∨ q-subsets. It is clear that T∈∨ q (A; α) = T∈ (A; α) ∪ Tq (A; α), I∈∨ q (A; β) = I∈ (A; β) ∪ Iq (A; β), F∈∨ q (A; γ) = F∈ (A; γ) ∪ Fq (A; γ).

T∈∨ qkT (A; α) = T∈ (A; α) ∪ TqkT (A; α),

(3.1)

I∈∨ qkI (A; β) = I∈ (A; β) ∪ IqkI (A; β),

(3.2)

F∈∨ qkF (A; γ) = F∈ (A; γ) ∪ FqkF (A; γ). (3.3) (2.2) (2.3) Given a neutrosophic set A = (AT , AI , AF ) in a set X, α, β ∈ (2.4) (0, 1] and γ ∈ [0, 1), we consider the following sets:

Given Φ, Ψ ∈ {∈, q, ∈ ∨ q}, a neutrosophic set A = (AT , AI , AF ) in X ∈ B(X) is called a (Φ, Ψ)-neutrosophic subalgebra of X (see [4]) if the following assertions are valid. x ∈ TΦ (A; αx ), y ∈ TΦ (A; αy ) ⇒ x ∗ y ∈ TΨ (A; αx ∧ αy ), x ∈ IΦ (A; βx ), y ∈ IΦ (A; βy ) ⇒ x ∗ y ∈ IΨ (A; βx ∧ βy ), x ∈ FΦ (A; γx ), y ∈ FΦ (A; γy ) ⇒ x ∗ y ∈ FΨ (A; γx ∨ γy )

It is clear that

T∈∗ (A; α) := {x ∈ X | AT (x) > α}, I∈∗ (A; β) := {x ∈ X | AI (x) > β}, F∈∗ (A; γ) := {x ∈ X | AF (x) < γ}.

(3.4) (3.5) (3.6)

Proposition 3.1. For any neutrosophic set A = (AT , AI , AF ) in a set X, α, β ∈ (0, 1] and γ ∈ [0, 1), we have (2.5)

for all x, y ∈ X, αx , αy , βx , βy ∈ (0, 1] and γx , γy ∈ [0, 1).

α≤ β≤ γ≥ α> β> γ<

1−k 2 1−k 2 1−k 2 1−k 2 1−k 2 1−k 2

⇒ Tqk (A; α) ⊆ T∈∗ (A; α),

(3.7)

Iqk (A; β) ⊆ I∈∗ (A; β), Fqk (A; γ) ⊆ F∈∗ (A; γ),

(3.8) (3.9)

⇒ T∈ (A; α) ⊆ Tqk (A; α),

(3.10)

⇒ I∈ (A; β) ⊆ Iqk (A; β),

(3.11)

⇒ F∈ (A; γ) ⊆ Fqk (A; γ).

(3.12)

⇒ ⇒

Generalizations of (∈, ∈ ∨q)-neutrosophic subalgebras

1+k Proof. If α ≤ 1−k 2 , then 1 − α ≥ 2 and α ≤ 1 − α. Assume that x ∈ Tqk (A; α). Then AT (x) + k > 1 − α ≥ 1+k 2 , and 1+k 1−k ∗ so AT (x) > 2 − k = 2 ≥ α. Hence x ∈ T∈ (A; α). Similarly, we have the result (3.8). Suppose that γ ≥ 1−k 2 and let x ∈ F (A; γ). Then A (x) + γ + k < 1, and thus q F k In what follows, let kT , kI and kF denote arbitrary elements of [0, 1) unless otherwise specified. If kT , kI and kF are the same 1−k AF (x) < 1 − γ − k ≤ 1 − 1−k 2 − k = 2 ≤ γ. number in [0, 1), then it is denoted by k, i.e., k = kT = kI = kF .

Given a neutrosophic set A = (AT , AI , AF ) in a set X, α, β ∈ Hence x ∈ F∈∗ (A; γ). Suppose that α >

1−k 2 .

If x ∈ T∈ (A; α),

S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points

Neutrosophic Sets and Systems, Vol. 20, 2018

Corollary 3.5. If A = (AT , AI , AF ) is an (∈, ∈)-neutrosophic subalgebra of X ∈ B(X), then neutrosophic qk -subsets Tqk (A; α), Iqk (A; β) and Fqk (A; γ) are subalgebras of X for all α, β ∈ (0, 1] and γ ∈ [0, 1) whenever they are nonempty.

then AT (x) + α + k ≥ 2α + k > 2 ·

1−k 2

+k =1

and so x ∈ Tqk (A; α). Hence T∈ (A; α) ⊆ Tqk (A; α). Similarly, we can verify that if β > 1−k If we take kT = kI = kF = 0 in Theorem 3.4, then we have 2 , then I∈ (A; β) ⊆ Iqk (A; β). Supthe following corollary. pose that γ < 1−k . If x ∈ F (A; γ), then A (x) ≤ γ, and ∈ F 2 thus Corollary 3.6 ([4]). If A = (AT , AI , AF ) is an (∈, ∈)1−k neutrosophic subalgebra of X ∈ B(X), then neutrosophic qAF (x) + γ + k ≤ 2γ + k < 2 · 2 + k = 1, subsets Tq (A; α), Iq (A; β) and Fq (A; γ) are subalgebras of X for all α, β ∈ (0, 1] and γ ∈ [0, 1) whenever they are nonempty. that is, x ∈ Fqk (A; γ). Hence F∈ (A; γ) ⊆ Fqk (A; γ).

Definition 3.2. A neutrosophic set A = (AT , AI , AF ) in X ∈ Definition 3.7. A neutrosophic set A = (AT , AI , AF ) in X ∈ B(X) is called an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra B(X) is called a (q(kT ,kI ,kF ) , ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subof X if algebra of X if x ∈ T∈ (A; αx ), y ∈ T∈ (A; αy ) ⇒ x ∗ y ∈ T∈∨qkT (A; αx ∧ αy ), x ∈ I∈ (A; βx ), y ∈ I∈ (A; βy ) ⇒ x ∗ y ∈ I∈∨qkI (A; βx ∧ βy ), x ∈ F∈ (A; γx ), y ∈ F∈ (A; γy ) ⇒ x ∗ y ∈ F∈∨qkF (A; γx ∨ γy )

(3.13)

for all x, y ∈ X, αx , αy , βx , βy ∈ (0, 1] and γx , γy ∈ [0, 1).

x ∈ TqkT (A; αx ), y ∈ TqkT (A; αy ) ⇒ x ∗ y ∈ T∈∨qkT (A; αx ∧ αy ), x ∈ IqkI (A; βx ), y ∈ IqkI (A; βy ) ⇒ x ∗ y ∈ I∈∨qkI (A; βx ∧ βy ), x ∈ FqkF (A; γx ), y ∈ FqkF (A; γy ) ⇒ x ∗ y ∈ F∈∨qkF (A; γx ∨ γy )

(3.15)

for all x, y ∈ X, αx , αy , βx , βy ∈ (0, 1] and γx , γy ∈ [0, 1).

An (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra with kT = kI = kF = k is called an (∈, ∈ ∨qk )-neutrosophic subalgebra.

A (q(kT ,kI ,kF ) , ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra with kT = kI = kF = k is called a (qk , ∈ ∨qk )-neutrosophic subalLemma 3.3 ([4]). A neutrosophic set A = (AT , AI , AF ) in gebra. X ∈ B(X) is an (∈, ∈)-neutrosophic subalgebra of X if and Theorem 3.8. If A = (AT , AI , AF ) is a (q(kT ,kI ,kF ) , ∈ only if it satisfies: ∨q (kT ,kI ,kF ) )-neutrosophic subalgebra of X ∈ B(X), then neu  AT (x ∗ y) ≥ AT (x) ∧ AT (y) trosophic qk -subsets TqkT (A; α), IqkI (A; β) and FqkF (A; γ) are   1−kI T (∀x, y ∈ X)  AI (x ∗ y) ≥ AI (x) ∧ AI (y)  . (3.14) subalgebras of X for all α ∈ ( 1−k 2 , 1], β ∈ ( 2 , 1] and 1−kF γ ∈ [0, 2 ) whenever they are nonempty. AF (x ∗ y) ≤ AF (x) ∨ AF (y)

T Proof. Let x, y ∈ TqkT (A; α) for α ∈ ( 1−k 2 , 1]. Then x ∗ y ∈ T∈∨ qkT (A; α), that is, x ∗ y ∈ T∈ (A; α) or x ∗ y ∈ TqkT (A; α). If x ∗ y ∈ T∈ (A; α), then x ∗ y ∈ TqkT (A; α) by (3.10). Therefore TqkT (A; α) is a subalgebra of X. Similarly, we prove that IqkI (A; β) is a subalgebra of X. Let x, y ∈ FqkF (A; γ) Proof. Let x, y ∈ TqkT (A; α). Then AT (x) + α + kT > 1 and F for γ ∈ [0, 1−k 2 ). Then x ∗ y ∈ F∈∨ qkF (A; γ), and so AT (y) + α + kT > 1. It follows from Lemma 3.3 that x∗y ∈ F∈ (A; γ) or x∗y ∈ FqkF (A; γ). If x∗y ∈ F∈ (A; γ), then x ∗ y ∈ FqkF (A; γ) by (3.12). Hence FqkF (A; γ) is a subalgebra AT (x ∗ y) + α + kT ≥ (AT (x) ∧ AT (y)) + α + kT of X. = (AT (x) + α + kT ) ∧ (AT (y) + α + kT ) > 1

Theorem 3.4. If A = (AT , AI , AF ) is an (∈, ∈)-neutrosophic subalgebra of X ∈ B(X), then neutrosophic qk -subsets TqkT (A; α), IqkI (A; β) and FqkF (A; γ) are subalgebras of X for all α, β ∈ (0, 1] and γ ∈ [0, 1) whenever they are nonempty.

Taking kT = kI = kF = 0 in Theorem 3.8 induces the foland so that x∗y ∈ TqkT (A; α). Hence TqkT (A; α) is a subalgebra lowing corollary. of X. Similarly, we can prove that IqkI (A; β) is a subalgebra of X. Now let x, y ∈ FqkF (A; γ). Then AF (x) + γ + kF < 1 and Corollary 3.9 ([4]). If A = (AT , AI , AF ) is a (q, ∈ ∨ q)AF (y) + γ + kF < 1, which imply from Lemma 3.3 that neutrosophic subalgebra of X ∈ B(X), then neutrosophic qsubsets Tq (A; α), Iq (A; β) and Fq (A; γ) are subalgebras of X AF (x ∗ y) + γ + kF ≤ (AF (x) ∨ AF (y)) + γ + kF for all α, β ∈ (0.5, 1] and γ ∈ [0, 0, 5) whenever they are = (AF (x) + γ + kF ) ∨ (AF (y) + γ + kF ) < 1. nonempty. Hence x ∗ y ∈ FqkF (A; γ) and so FqkF (A; γ) is a subalgebra of We provide characterizations of an (∈, ∈ ∨q(kT ,kI ,kF ) )-neuX. trosophic subalgebra. S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points

Neutrosophic Sets and Systems, Vol. 20, 2018

Theorem 3.10. Given a neutrosophic set A = (AT , AI , AF ) in that is, a ∗ b ∈ / FqkF (A; γF ). Thus a ∗ b ∈ / F∈∨ qkF (A; γF ), F X ∈ B(X), the following are equivalent. which is a contradiction. If AF (a) ∨ AF (b) < 1−k 2 , then a, b ∈ 1−kF 1−kF / F∈ (A; 2 ). Also, (1) A = (AT , AI , AF ) is an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic F∈ (A; 2 ) and a ∗ b ∈ subalgebra of X. F F F AF (a ∗ b) + 1−k > 1−k + 1−k = 1 − kF 2 2 2 (2) A = (AT , AI , AF ) satisfies the following assertion. F F and so a∗b ∈ / FqkF (A; 1−k / F∈∨ qkF (A; 1−k V 2 ). Hence a∗b ∈ 2 ), T AT (x ∗ y) ≥ {AT (x), AT (y), 1−k 2 } a contradiction. Therefore V I (3.16) AI (x ∗ y) ≥ {AI (x), AI (y), 1−k _ 2 } F W AF (x ∗ y) ≤ {AF (x), AF (y), 1−k F 2 } AF (x ∗ y) ≤ {AF (x), AF (y), 1−k } 2 for all x, y ∈ X. Conversely, let A = (AT , AI , AF ) be a neutrosophic set in X Proof. Let A = (AT , AI , AF ) be an (∈, ∈ ∨q(kT ,kI ,kF ) )- which satisfies the condition (3.16). Let x, y ∈ X and β , β ∈ x y neutrosophic subalgebra of X. Assume that there exist a, b ∈ X (0, 1] be such that x ∈ I (A; β ) and y ∈ I (A; β ). Then ∈ x ∈ y such that ^ ^ ^ I I AI (x ∗ y) ≥ {AI (x), AI (y), 1−k } ≥ {βx , βy , 1−k 1−kT 2 2 }. AT (a ∗ b) < {AT (a), AT (b), 2 }. for all x, y ∈ X.

If AT (a) ∧ AT (b) < Hence

1−kT 2

, then AT (a ∗ b) < AT (a) ∧ AT (b).

AT (a ∗ b) < αt ≤ AT (a) ∧ AT (b)

I I or βy ≤ 1−k Suppose that βx ≤ 1−k 2 2 . Then AI (x ∗ y) ≥ βx ∧ βy , and so x ∗ y ∈ I∈ (A; βx ∧ βy ). Now, assume that 1−kI I I and βy > 1−k βx > 1−k 2 2 . Then AI (x ∗ y) ≥ 2 , and so

AI (x ∗ y) + βx ∧ βy >

1−kI 2

= 1 − kI ,

for some αt ∈ (0, 1]. It follows that a ∈ T∈ (A; αt ) and b ∈ that is, x ∗ y ∈ IqkI (A; βx ∧ βy ). Hence T∈ (A; αt ) but a ∗ b ∈ / T∈ (A; αt ). Moreover, AT (a ∗ b) + αt < 2αt < 1 − kT ,

x ∗ y ∈ I∈∨ qkI (A; βx ∧ βy ).

and so a ∗ b ∈ / TqkT (A; αt ). Thus a ∗ b ∈ / T∈∨ qkT (A; αt ), a conT T tradiction. If AT (a) ∧ AT (b) ≥ 1−k , then a ∈ T∈ (A; 1−k 2 2 ), 1−kT 1−kT / T∈ (A; 2 ). Also, b ∈ T∈ (A; 2 ) and a ∗ b ∈

Similarly, we can verify that if x ∈ T∈ (A; αx ) and y ∈ T∈ (A; αy ), then x ∗ y ∈ T∈∨ qkT (A; αx ∧ αy ). Finally, let x, y ∈ X and γx , γy ∈ [0, 1) be such that x ∈ F∈ (A; γx ) and y ∈ F∈ (A; γy ). Then _ _ F F {γx , γy , 1−k AF (x ∗ y) ≤ {AF (x), AF (y), 1−k 2 }≤ 2 }.

AT (a ∗ b) +

1−kT 2

= 1 − kT ,

T T i.e., a ∗ b ∈ / TqkT (A; 1−k / T∈∨ qkT (A; 1−k 2 ). Hence a ∗ b ∈ 2 ), a contradiction. Consequently, ^ T AT (x ∗ y) ≥ {AT (x), AT (y), 1−k 2 }

for all x, y ∈ X. Similarly, we know that ^ I AI (x ∗ y) ≥ {AI (x), AI (y), 1−k 2 } for all x, y ∈ X. Suppose that there exist a, b ∈ X such that _ F AF (a ∗ b) > {AF (a), AF (b), 1−k 2 }.

F F If γx ≥ 1−k or γy ≥ 1−k 2 2 , then AF (x ∗ y) ≤ γx ∨ γy and thus F F x ∗ y ∈ F∈ (A; γx ∨ γy ). If γx < 1−k and γy < 1−k 2 2 , then 1−kF AF (x ∗ y) ≤ 2 . Hence

AF (x ∗ y) + γx ∨ γy <

1−kF 2

= 1 − kF ,

that is, x ∗ y ∈ FqkF (A; γx ∨ γy ). Thus x ∗ y ∈ F∈∨ qkF (A; γx ∨ γy ). Therefore A = (AT , AI , AF ) is an (∈, ∈ ∨ qkF )-neutrosophic subalgebra of X.

W F Then AF (a ∗ b) > γF ≥ {AF (a), AF (b), 1−k 2 } for some Corollary 3.11 ([4]). A neutrosophic set A = (AT , AI , AF ) in 1−kF γF ∈ [0, 1). If AF (a) ∨ AF (b) ≥ 2 , then X ∈ B(X) is an (∈, ∈ ∨ q)-neutrosophic subalgebra of X if and only if it satisfies: AF (a ∗ b) > γF ≥ AF (a) ∨ AF (b)   V AT (x ∗ y) ≥ {AT (x), AT (y), 0.5} V which implies that a, b ∈ F∈ (A; γF ) and a ∗ b ∈ / F∈ (A; γF ).   (∀x, y ∈ X)  AI (x ∗ y) ≥ {AI (x), AI (y).0.5}  . Also, W AF (x ∗ y) ≤ {AF (x), AF (y), 0.5} A (a ∗ b) + γ > 2γ ≥ 1 − k , F

S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points

Neutrosophic Sets and Systems, Vol. 20, 2018

Proof. It follows from taking kT = kI = kF = 0 in Theorem AF (a ∗ b) ≤ γF which is a contradiction. Thus 3.10. _ F AF (x ∗ y) ≤ {AF (x), AF (y), 1−k 2 } Theorem 3.12. Let A = (AT , AI , AF ) be a neutrosophic set in X ∈ B(X). Then A = (AT , AI , AF ) is an (∈, ∈ ∨q(kT ,kI ,kF ) )- for all x, y ∈ X. Therefore A = (AT , AI , AF ) is an (∈, ∈ neutrosophic subalgebra of X if and only if neutrosophic ∈- ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra of X by Theorem 3.10. subsets T∈ (A; α), I∈ (A; β) and F∈ (A; γ) are subalgebras of X 1−kI 1−kF T for all α ∈ (0, 1−k 2 ], β ∈ (0, 2 ] and γ ∈ [ 2 , 1) when- Corollary 3.13. Let A = (A , A , A ) be a neutrosophic set T I F ever they are nonempty. in X ∈ B(X). Then A = (AT , AI , AF ) is an (∈, ∈ ∨ q)neutrosophic subalgebra of X if and only if neutrosophic ∈Proof. Assume that A = (AT , AI , AF ) is an (∈, ∈ subsets T∈ (A; α), I∈ (A; β) and F∈ (A; γ) are subalgebras of 1−kI ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra of X. Let β ∈ (0, 2 ] X for all α, β ∈ (0, 0.5] and γ ∈ [0.5, 1) whenever they are and x, y ∈ I∈ (A; β). Then AI (x) ≥ β and AI (y) ≥ β. It nonempty. follows from Theorem 3.10 that ^ Proof. It follows from taking kT = kI = kF = 0 in Theorem 1−kI I AI (x ∗ y) ≥ {AI (x), AI (y), 1−k =β 2 }≥β∧ 2 3.12. and so that x ∗ y ∈ I∈ (A; β). Hence I∈ (A; β) is a subalgebra I of X for all β ∈ (0, 1−k 2 ]. Similarly, we know that T∈ (A; α) 1−kF T is a subalgebra of X for all α ∈ (0, 1−k 2 ]. Let γ ∈ [ 2 , 1) and x, y ∈ F∈ (A; γ). Then AF (x) ≤ γ and AF (y) ≤ γ. Using Theorem 3.10 implies that _ 1−kF F = γ. AF (x ∗ y) ≤ {AF (x), AF (y), 1−k 2 }≤γ∨ 2

Theorem 3.14. Every (∈, ∈)-neutrosophic subalgebra is an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra. Proof. Straightforward. The converse of Theorem 3.14 is not true as seen in the following example.

Example 3.15. Consider a BCI-algebra X = {0, a, b, c} with Hence x ∗ y ∈ F∈ (A; γ), and therefore F∈ (A; γ) is a subalgebra the binary operation ∗ which is given in Table 1 (see [5]). F of X for all γ ∈ [ 1−k 2 , 1). Conversely, suppose that the nonempty neutrosophic ∈-subsets Table 1: Cayley table for the binary operation “∗” T∈ (A; α), I∈ (A; β) and F∈ (A; γ) are subalgebras of X for all 1−kT 1−kI 1−kF α ∈ (0, 2 ], β ∈ (0, 2 ] and γ ∈ [ 2 , 1). If there exist a, b ∈ X such that ∗ 0 a b c ^ 0 0 a b c T AT (a ∗ b) < {AT (a), AT (b), 1−k 2 }, a a 0 c b b b c 0 a then a, b ∈ T∈ (A; αT ) by taking c c b a 0 ^ T αT := {AT (a), AT (b), 1−k 2 }. Let A = (AT , AI , AF ) be a neutrosophic set in X ∈ BI (X) Since T∈ (A; αT ) is a subalgebra of X, it follows that a ∗ b ∈ defined by Table 2 T∈ (A; αT ), that is, AT (a ∗ b) ≥ αT . This is a contradiction, and hence ^ Table 2: Tabular representation of “A = (AT , AI , AF )” T AT (x ∗ y) ≥ {AT (x), AT (y), 1−k 2 } for all x, y ∈ X. Similarly, we can verify that ^ I AI (x ∗ y) ≥ {AI (x), AI (y), 1−k 2 } for all x, y ∈ X. Now, assume that there exist a, b ∈ X such that _ F AF (a ∗ b) > {AF (a), AF (b), 1−k 2 }. W F Then AF (a ∗ b) > γF ≥ {AF (a), AF (b), 1−k 2 } for some 1−kF γF ∈ [ 2 , 1). Hence a, b ∈ F∈ (A; γF ), and so a ∗ b ∈ F∈ (A; γF ) since F∈ (A; γF ) is a subalgebra of X. It follows that

X 0 a b c

AT (x) 0.6 0.7 0.3 0.3

AI (x) 0.5 0.3 0.6 0.3

AF (x) 0.2 0.6 0.6 0.4

If kT = 0.36, then T∈ (A; α) =

X {0, a}

if α ∈ (0, 0.3], if α ∈ (0.3, 0.32].

S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points

Neutrosophic Sets and Systems, Vol. 20, 2018

If kI = 0.32, then

and

X {0, b}

if β ∈ (0, 0.3], if β ∈ (0.3, 0.34].

  {0} {0, c} F∈ (A; γ) =  X

if γ ∈ [0.32, 0.4), if γ ∈ [0.4, 0.6), if γ ∈ [0.6, 1].

I∈ (A; β) =

  {0} {0, c} F∈ (A; γ) =  X

if β ∈ [0.38, 0.4), if β ∈ [0.4, 0.6), if β ∈ [0.6, 1).

If kF = 0.36, then Since X, {0}, {0, a}, {0, b} and {0, c} are subalgebras of X, we know from Theorem 3.12 that A = (AT , AI , AF ) is an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra of X for kT = 0.2, kI = 0.3 and kF = 0.24. Note that

We know that T∈ (A; α), I∈ (A; β) and F∈ (A; γ) are subalgebras a∗b∈ / Tq0.2 (A; 0.25 ∧ 0.4) of X for all α ∈ (0, 0.32], β ∈ (0, 0.34] and γ ∈ [0.32, 1). It follows from Theorem 3.12 that A = (AT , AI , AF ) is an (∈, for a ∈ T∈ (A; 0.4) and b ∈ T∈ (A; 0.25), ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra of X for kT = 0.36, b∗c∈ / Iq0.3 (A; 0.5 ∧ 0.27) kI = 0.32 and kF = 0.36. Since for b ∈ I∈ (A; 0.5) and c ∈ I∈ (A; 0.27), and/or

AT (0) = 0.6 < 0.7 = AT (a) ∧ AT (a)

a∗c∈ / Fq0.24 (A; 0.6 ∨ 0.44)

and/or AI (0) = 0.5 > 0.3 = AI (c) ∨ AI (c),

for a ∈ F∈ (A; 0.6) and c ∈ F∈ (A; 0.44). Hence A = (AT , AI , we know that A = (AT , AI , AF ) is not an (∈, ∈)-neutrosophic AF ) is not an (∈, q(0.2,0.3,0.24) )-neutrosophic subalgebra of X. subalgebra of X by Lemma 3.3. Theorem 3.19. If 0 ≤ kT < jT < 1, 0 ≤ kI < jI < 1 and Definition 3.16. A neutrosophic set A = (AT , AI , AF ) in X ∈ 0 ≤ jF < kF < 1, then every (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic B(X) is called an (∈, q(kT ,kI ,kF ) )-neutrosophic subalgebra of subalgebra is an (∈, ∈ ∨q(jT ,jI ,jF ) )-neutrosophic subalgebra. X if x ∈ T∈ (A; αx ), y ∈ T∈ (A; αy ) ⇒ x ∗ y ∈ TqkT (A; αx ∧ αy ), x ∈ I∈ (A; βx ), y ∈ I∈ (A; βy ) ⇒ x ∗ y ∈ IqkI (A; βx ∧ βy ), x ∈ F∈ (A; γx ), y ∈ F∈ (A; γy ) ⇒ x ∗ y ∈ FqkF (A; γx ∨ γy )

Proof. Straightforward. (3.17)

The following example shows that if 0 ≤ kT < jT < 1, 0 ≤ kI < jI < 1 and 0 ≤ jF < kF < 1, then an (∈, ∈ ∨q(jT ,jI ,jF ) )-neutrosophic subalgebra may not be an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra.

for all x, y ∈ X, αx , αy , βx , βy ∈ (0, 1] and γx , γy ∈ [0, 1).

Example 3.20. Let X be the BCI-algebra given in Example 3.15 and let A = (AT , AI , AF ) be a neutrosophic set in X An (∈, q(kT ,kI ,kF ) )-neutrosophic subalgebra with kT = kI = defined by Table 3 kF = k is called an (∈, qk )-neutrosophic subalgebra. Theorem 3.17. Every (∈, q(kT ,kI ,kF ) )-neutrosophic subalgebra is an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra.

Table 3: Tabular representation of “A = (AT , AI , AF )”

Proof. Straightforward. The converse of Theorem 3.17 is not true as seen in the following example. Example 3.18. Consider the BCI-algebra X = {0, a, b, c} and the neutrosophic set A = (AT , AI , AF ) which are given in Example 3.15. Taking kT = 0.2, kI = 0.3 and kF = 0.24 imply that X if α ∈ (0, 0.3], T∈ (A; α) = {0, a} if α ∈ (0.3, 0.4], I∈ (A; β) =

X {0, b}

if β ∈ (0, 0.3], if β ∈ (0.3, 0.35],

X 0 a b c

AT (x) 0.42 0.40 0.48 0.40

AI (x) 0.40 0.44 0.36 0.36

AF (x) 0.44 0.66 0.66 0.33

If kT = 0.04, then   X {0, b} T∈ (A; α) =  {b}

if α ∈ (0, 0.40], if α ∈ (0.40, 0.42], if α ∈ (0.42, 0.48].

Note that T∈ (A; α) is not a subalgebra of X for α ∈ (0.42, 0.48].

S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points

Neutrosophic Sets and Systems, Vol. 20, 2018

If kI = 0.08, then  X    {0, a} I∈ (A; β) = {a}    ∅

if β if β if β if β

∈ (0, 0.36], ∈ (0.36, 0.40], ∈ (0.40, 0.44], ∈ (0.44, 0.46].

subalgebra of X if and only if the neutrosophic set AS = (AST , ASI , ASF ) is an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra of X.

Proof. Let S be a subalgebra of X. Then neutrosophic ∈subsets T∈ (AST ; α), I∈ (AST ; β) and F∈ (AST ; γ) are obviously 1−kI T subalgebras of X for all α ∈ (0, 1−k 2 ], β ∈ (0, 2 ] and Note that I∈ (A; β) is not a subalgebra of X for β ∈ (0.40, 0.44]. γ ∈ [ 1−kF , 1). Hence A = (A , A , A ) is an (∈, S ST SI SF 2 If kF = 0.42, then ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra of X by Theorem 3.12.  Conversely, assume that AS = (AST , ASI , ASF ) is an (∈, ∅ if γ ∈ [0.29, 0.33),    ∈ ∨q (kT ,kI ,kF ) )-neutrosophic subalgebra of X. Let x, y ∈ S. {c} if γ ∈ [0.33, 0.44), F∈ (A; γ) = Then {0, c} if γ ∈ [0.44, 0.66),    ^ X if γ ∈ [0.66, 1). T AST (x ∗ y) ≥ {AST (x), AST (y), 1−k 2 } Note that F∈ (A; γ) is not a subalgebra of X for γ ∈ [0.33, 0.44). Therefore A = (AT , AI , AF ) is not an (∈, ∈ ∨q(kT ,kI ,kF ) )neutrosophic subalgebra of X for kT = 0.04, kI = 0.08 and kF = 0.42. If jT = 0.16, then X if α ∈ (0, 0.40], T∈ (A; α) = and {0, b} if α ∈ (0.40, 0.42].

ASI (x ∗ y) ≥

I∈ (A; β) =

X {0, a}

if β ∈ (0, 0.36], if β ∈ (0.36, 0.40].

If jF = 0.12, then

1−kT 2

^ I {ASI (x), ASI (y), 1−k 2 }

=1∧

ASF (x ∗ y) ≤

If jI = 0.20, then

1−kI 2

F {ASF (x), ASF (y), 1−k 2 }

=0∨

1−kF 2

which imply that AST (x ∗ y) = 1, ASI (x ∗ y) = 1 and ASF (x ∗ y) = 0.

F∈ (A; γ) =

=1∧

{0, c} X

if γ ∈ [0.44, 0.66), if γ ∈ [0.66, 1).

Hence x ∗ y ∈ S, and so S is a subalgebra of X.

Theorem 3.22. Let S be a subalgebra of X ∈ B(X). For every Therefore A = (AT , AI , AF ) is an (∈, ∈ ∨q(jT ,jI ,jF ) )1−kI 1−kF T α ∈ (0, 1−k 2 ], β ∈ (0, 2 ] and γ ∈ [ 2 , 1), there exneutrosophic subalgebra of X for jT = 0.16, jI = 0.20 and ists an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra A = (AT , jF = 0.12. AI , AF ) of X such that T∈ (A; α) = S, I∈ (A; β) = S and Given a subset S of X, consider a neutrosophic set AS = F∈ (A; γ) = S. (AST , ASI , ASF ) in X defined by Proof. Let A = (AT , AI , AF ) be a neutrosophic set in X de fined by (1, 1, 0) if x ∈ S, AS (x) := (0, 0, 1) otherwise, (α, β, γ) if x ∈ S, A(x) := (0, 0, 1) otherwise, that is, that is, 1 if x ∈ S, AST (x) := 0 otherwise, α if x ∈ S, AT (x) := 0 otherwise, 1 if x ∈ S, ASI (x) := 0 otherwise, β if x ∈ S, AI (x) := 0 otherwise, and and 0 if x ∈ S, ASF (x) := 1 otherwise. γ if x ∈ S, AF (x) := 1 otherwise. Theorem 3.21. A nonempty subset S of X ∈ B(X) is a S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points

Neutrosophic Sets and Systems, Vol. 20, 2018

Obviously, T∈ (A; α) = S, I∈ (A; β) = S and F∈ (A; γ) = S. Suppose that ^ T AT (a ∗ b) < {AT (a), AT (b), 1−k 2 }

Case 1. AI (x) ≥ β and AI (y) ≥ β. If β > AI (x ∗ y) ≥

1−kI 2 ,

I {AI (x), AI (y), 1−k 2 }=

then

1−kI 2 ,

I I + 1−k = 1 − kI . Hence x ∗ y ∈ and so AI (x ∗ y) + β > 1−k 2 2 for some a, b ∈ X. Since #Im(A ) = 2, it follows that 1−kI T I (A; β). If β ≤ , then V q kI 2 T {AT (a), AT (b), 1−k 2 } = α and AT (a ∗ b) = 0. Hence ^ AT (a) = α = AT (b), and so a, b ∈ S. Since S is a subalgebra I AI (x ∗ y) ≥ {AI (x), AI (y), 1−k 2 } ≥ β, of X, we have a ∗ b ∈ S. Thus AT (a ∗ b) = α, a contradiction. Therefore and thus x ∗ y ∈ I∈ (A; β). Hence ^ 1−kT AT (x ∗ y) ≥ {AT (x), AT (y), 2 } x ∗ y ∈ I∈ (A; β) ∪ IqkI (A; β) = I∈∨ qkI (A; β).

for all x, y ∈ X. Similarly, we can verify that ^ I AI (x ∗ y) ≥ {AI (x), AI (y), 1−k 2 } for all x, y ∈ X. Assume that there exist a, b ∈ X such that _ F AF (a ∗ b) > {AF (a), AF (b), 1−k 2 }.

Case 2. AI (x) ≥ β and AI (y) + β + kI > 1. If β > then ^ I AI (x ∗ y) ≥ {AI (x), AI (y), 1−k 2 } I > (1 − β − kI ) ∧ = AI (y) ∧ 1−k 2 = 1 − β − kI ,

1−kI 2 ,

1−kI 2

W F Then AF (a ∗ b) = 1 and {AF (a), AF (b), 1−k 2 } = γ since #Im(AF ) = 2. It follows that AF (a) = γ = AF (b) and so that and so x ∗ y ∈ I (A; β). If β ≤ 1−kI , then qkI 2 a, b ∈ S. Hence a ∗ b ∈ S, and so AF (a ∗ b) = γ, which is a ^ contradiction. Thus I AI (x ∗ y) ≥ {AI (x), AI (y), 1−k 2 } _ ^ 1−kF AF (x ∗ y) ≤ {AF (x), AF (y), 2 } ≥ {β, 1 − β − k , 1−kI } = β, I

for all x, y ∈ X. Therefore A = (AT , AI , AF ) is an (∈, ∈ and thus x ∗ y ∈ I (A; β). Therefore x ∗ y ∈ I ∈ ∈∨ qkI (A; β). ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra of X by Theorem 3.10. Case 3. AI (x) + β + kI > 1 and AI (y) ≥ β. We have Corollary 3.23. Let S be a subalgebra of X ∈ B(X). For every x ∗ y ∈ I∈∨ qkI (A; β) by the similar way to the Case 2. α ∈ (0, 0.5], β ∈ (0, 0.5] and γ ∈ [0.5, 1), there exists an (∈, ∈ ∨q)-neutrosophic subalgebra A = (AT , AI , AF ) of X such Case 4. AI (x) + β + kI > 1 and AI (y) + β + kI > 1. If 1−kI I that T∈ (A; α) = S, I∈ (A; β) = S and F∈ (A; γ) = S. β > 1−k 2 , then 1 − β − kI < 2 , and so Proof. It follows from taking kT = kI = kF = 0 in Theorem 3.22.

AI (x ∗ y) ≥

I {AI (x), AI (y), 1−k 2 } > 1 − β − kI ,

I Theorem 3.24. Given a neutrosophic set A = (AT , AI , AF ) in i.e., x ∗ y ∈ IqkI (A; β). If β ≤ 1−k 2 , then X ∈ B(X), the following are equivalent. ^ I AI (x ∗ y) ≥ {AI (x), AI (y), 1−k 2 } (1) A = (AT , AI , AF ) is an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic I subalgebra of X. ≥ (1 − β − kI ) ∧ 1−k 2 I = 1−k ≥ β, 2 (2) The neutrosophic (∈ ∨ qk )-subsets T∈∨ qkT (A; α), I∈∨ qkI (A; β) and F∈∨ qkF (A; γ) are subalgebras of X for i.e., x ∗ y ∈ I∈ (A; β). Hence x ∗ y ∈ I∈∨ qkI (A; β). Conall α, β ∈ (0, 1] and γ ∈ [0, 1). sequently, I∈∨ qkI (A; β) is a subalgebra of X. Similarly, we Proof. Assume that A = (AT , AI , AF ) is an (∈, ∈ can prove that if x, y ∈ T∈∨ qkT (A; α) for α ∈ (0, 1], then ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra of X. Let x, y ∈ x ∗ y ∈ T∈∨ qkT (A; α), that is, T∈∨ qkT (A; α) is a subalgebra I∈∨ qkI (A; β) for β ∈ (0, 1]. Then AI (x) ≥ β or AI (x) + β + of X. Let x, y ∈ F∈∨ qkF (A; γ) for γ ∈ [0, 1). Then AF (x) ≤ γ kI > 1, and AI (y) ≥ β or AI (y) + β + kI > 1. Using Theorem or AF (x)+γ +kF < 1, and AF (y) ≤ γ or AF (y)+γ +kF < 1. Using Theorem 3.10, we have 3.10, we have ^ _ I F AI (x ∗ y) ≥ {AI (x), AI (y), 1−k AF (x ∗ y) ≤ {AF (x), AF (y), 1−k 2 }. 2 }.

S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points

Neutrosophic Sets and Systems, Vol. 20, 2018

Case 1. AF (x) ≤ γ and AF (y) ≤ γ. If γ < AF (x ∗ y) ≤

F {AF (x), AF (y), 1−k 2 }=

and so AF (x ∗ y) + γ < x ∗ y ∈ FqkF (A; γ). If γ ≥ AF (x ∗ y) ≤

1−kF 2

1−kF 2 1−kF 2 ,

or AT (x ∗ y) + α + kT > 1, a contradiction. Hence ^ T AT (x ∗ y) ≥ {AT (x), AT (y), 1−k 2 }

, then

1−kF 2

F + 1−k = 1 − kF . Hence for all x, y ∈ X. Similarly, we can verify that 2 ^ then I AI (x ∗ y) ≥ {AI (x), AI (y), 1−k 2 }

F {AF (x), AF (y), 1−k 2 } ≤ γ,

for all x, y ∈ X. Now assume that there exist a, b ∈ X and F γ ∈ ( 1−k 2 , 1) such that

and thus x ∗ y ∈ F∈ (A; γ). Hence

AF (a ∗ b) > γ ≥

x ∗ y ∈ F∈ (A; γ) ∪ FqkF (A; γ) = F∈∨ qkF (A; γ). Case 2. AF (x) ≤ γ and AF (y) + γ + kF < 1. If γ < then _ F AF (x ∗ y) ≤ {AF (x), AF (y), 1−k 2 } F = AF (y) ∨ 1−k < (1 − γ − kF ) ∨ 2 = 1 − γ − kF ,

and so x ∗ y ∈ FqkF (A; γ). If γ ≥

1−kF 2

, then

_ F {AF (x), AF (y), 1−k 2 } _ F ≤ {γ, 1 − γ − kF , 1−k 2 } = γ,

AF (x ∗ y) ≤

and thus x ∗ y ∈ F∈ (A; γ). Therefore x ∗ y ∈ F∈∨ qkF (A; γ).

F {AF (a), AF (b), 1−k 2 }.

Then a, b ∈ F∈ (A; γ) ⊆ F∈∨ qkF (A; γ), which implies that a ∗ b ∈ F∈∨ qkF (A; γ). Thus AF (a ∗ b) ≤ γ or AF (a ∗ b) + γ + kF < 1, which is a contradiction. Hence _ F AF (x ∗ y) ≥ {AF (x), AF (y), 1−k 2 } for all x, y ∈ X. Using Theorem 3.10, we conclude that A = (AT , AI , AF ) is an (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra of X.

Conclusions

Similarly, if AI (x) + β + kI < 1 and AI (y) ≤ β, then x ∗ y ∈ Neutrosophic set theory is a nice mathematical tool which can F∈∨ qkF (A; γ). be applied to several fields. The aim of this paper is to consider a general form of neutrosophic points, and to discuss generalFinally, assume that AF (x) + γ + kF < 1 and AF (y) + γ + izations of the papers [4] and [6]. We have introduce the no1−kF 1−kF kF < 1. If γ < 2 , then 1 − γ − kF > 2 , and so tions of (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra, and (∈, _ q(kT ,kI ,kF ) )-neutrosophic subalgebra in BCK/BCI-algebras, 1−kF AF (x ∗ y) ≤ {AF (x), AF (y), 2 } < 1 − γ − kF , and have investigated several properties. We have discussed characterizations of (∈, ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgeF i.e., x ∗ y ∈ FqkF (A; γ). If γ ≥ 1−k , then bra. We have considered relations between (∈, ∈)-neutrosophic 2 subalgebra, (∈, q(kT ,kI ,kF ) )-neutrosophic subalgebra and (∈, _ F AF (x ∗ y) ≤ {AF (x), AF (y), 1−k } ∈ ∨q(kT ,kI ,kF ) )-neutrosophic subalgebra. We hope the idea and 2 result in this paper can be a mathematical tool for dealing with F ≤ (1 − γ − kF ) ∨ 1−k 2 several informations containing uncertainty such as medical diagF = 1−k ≤ γ, nosis, decision making, graph theory, etc. So, based on the results 2 in this article, our future research will be focused to solve real-life i.e., x ∗ y ∈ F∈ (A; γ). Hence x ∗ y ∈ F∈∨ qkF (A; γ). Therefore problems under the opinions of experts in a neutrosophic set enviF∈∨ qkF (A; γ) is a subalgebra of X. ronment such as medical diagnosis, decision making, graph theory etc. In particular, Bucolo et al. [2] suggested a generalization Conversely, suppose that (2) is valid. If it is possible, let of the synchronization principles for the class of array of fuzzy ^ 1−kT logic chaotic based dynamical systems and evaluated as alternaAT (x ∗ y) < α ≤ {AT (x), AT (y), 2 } tive approach to build locally connected fuzzy complex systems 1−kT by manipulating both the rules driving the cells and the architecfor some α ∈ (0, 2 ). Then ture of the system. We will also try to study complex dynamics through neutrosophic environment. The future works also may x, y ∈ T∈ (A; α) ⊆ T∈∨ qkT (A; α), use the study neutrosophic set environment on several related alwhich implies that x ∗ y ∈ T∈∨ qkT (A; α). Thus AT (x ∗ y) ≥ α gebraic structures, for example, M V -algebras, BL-algebras, R0 algebras, EQ-algebras, equality algebras, M T L-algebras etc. S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points

Neutrosophic Sets and Systems, Vol. 20, 2018

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Received : March 23, 2018. Accepted : April 13, 2018.

S.J. Kim, S.Z. Song, Y.B. Jun, Generalizations of neutrosophic subalgebras in BCK/BCI-algebras based on neutrosophic points