MBJ-neutrosophic ideals of BCK/BCI-algebras

Young Bae Jun; Eun Hwan Roh

doi:10.1515/math-2019-0106

Article Open Access

MBJ-neutrosophic ideals of BCK/BCI-algebras

Young Bae Jun and Eun Hwan Roh

Published/Copyright: November 28, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

The notion of MBJ-neutrosophic ideal is introduced, and its properties are investigated. Conditions for an MBJ-neutrosophic set to be an MBJ-neutrosophic ideal are provided. In a BCK/BCI-algebra, a condition for an MBJ-neutrosophic set to be an MBJ-neutrosophic ideal is given. In a BCK-algebra, a condition for an MBJ-neutrosophic subalgebra to be an MBJ-neutrosophic ideal is given. In a BCI-algebra, conditions for an MBJ-neutrosophic ideal to be an MBJ-neutrosophic subalgebra are considered. In an (S)-BCK-algebra, we show that every MBJ-neutrosophic ideal is an MBJ-neutrosophic ∘-subalgebra, and a characterization of an MBJ-neutrosophic ideal is established.

Keywords: MBJ-neutrosophic set; MBJ-neutrosophic subalgebra; MBJ-neutrosophic ideal; MBJ-neutro-sophic ∘-subalgebra

MSC 2010: 06F35; 03G25; 03E72

1 Introduction

Different types of uncertainties are encountered in many complex systems and/or in many practical situations like behavioral, biologial and chemical etc. In order to handle uncertainties in many real applications, the fuzzy set was introduced by L.A. Zadeh [1] in 1965. The intuitionistic fuzzy set on a universe X was introduced by K. Atanassov in 1983 as a generalization of fuzzy set. As a more general platform that extends the notions of classic set, (intuitionistic) fuzzy set and interval valued (intuitionistic) fuzzy set, the notion of neutrosophic set is developed by Smarandache [2, 3, 4]. Neutrosophic algebraic structures in BCK/BCI-algebras are discussed in the papers [5, 6, 7, 8, 9, 10, 11, 12, 13, 14] and [15]. In [16], the notion of MBJ-neutrosophic sets is introduced as another generalization of neutrosophic set, it is applied to BCK/BCI-algebras. Mohseni et al. [16] introduced the concept of MBJ-neutrosophic subalgebras in BCK/BCI-algebras, and investigated related properties. They gave a characterization of MBJ-neutrosophic subalgebra, and established a new MBJ-neutrosophic subalgebra by using an MBJ-neutrosophic subalgebra of a BCI-algebra. They considered the homomorphic inverse image of MBJ-neutrosophic subalgebra, and discussed translation of MBJ-neutrosophic subalgebra.

In this paper, we apply the notion of MBJ-neutrosophic sets to ideals of BCK/BI-algebras. We introduce the concept of MBJ-neutrosophic ideals in BCK/BCI-algebras, and investigate several properties. We provide a condition for an MBJ-neutrosophic subalgebra to be an MBJ-neutrosophic ideal in a BCK-algebra. We provide conditions for an MBJ-neutrosophic set to be an MBJ-neutrosophic ideal in a BCK/BCI-algebra. We discuss relations between MBJ-neutrosophic subalgebras, MBJ-neutrosophic ∘-subalgebras and MBJ-neutrosophic ideals. In a BCI-algebra, we provide conditions for an MBJ-neutrosophic ideal to be an MBJ-neutrosophic subalgebra. In an (S)-BCK-algebra, we consider a characterization of an MBJ-neutrosophic ideal.

2 Preliminaries

By a BCI-algebra, we mean a set X with a binary operation * and a special element 0 that satisfies the following conditions:

((x * y) * (x * z)) * (z * y) = 0,
(x * (x * y)) * y = 0,
x * x = 0,
x * y = 0, y * x = 0 ⇒ x = y

for all x, y, z ∈ X. If a BCI-algebra X satisfies the following identity:

(V) (∀x ∈ X) (0 * x = 0),

then X is called a BCK-algebra.

By a weakly BCK-algebra (see [17]), we mean a BCI-algebra X satisfying 0 * x ≤ x for all x ∈ X.

Every BCK/BCI-algebra X satisfies the following conditions:

(∀x∈X)x∗0=x, (2.1)

(∀x,y,z∈X)x≤y⇒x∗z≤y∗z,z∗y≤z∗x, (2.2)

(∀x,y,z∈X)(x∗y)∗z=(x∗z)∗y, (2.3)

(∀x,y,z∈X)(x∗z)∗(y∗z)≤x∗y, (2.4)

where x ≤ y if and only if x * y = 0. Any BCI-algebra X satisfies the following conditions (see [17]):

(∀x,y∈X)(x∗(x∗(x∗y))=x∗y), (2.5)

(∀x,y∈X)(0∗(x∗y)=(0∗x)∗(0∗y)). (2.6)

A BCI-algebra X is said to be p-semisimple (see [17]) if

(∀x∈X)(0∗(0∗x)=x). (2.7)

In a p-semisimple BCI-algebra X, the following holds:

(∀x,y∈X)(0∗(x∗y)=y∗x, x∗(x∗y)=y). (2.8)

A BCI-algebra X is said to be associative (see [17]) if

(∀x,y,z∈X)((x∗y)∗z=x∗(y∗z)). (2.9)

By an (S)-BCK-algebra, we mean a BCK-algebra X such that, for any x, y ∈ X, the set

{z∈X∣z∗x≤y}

has the greatest element, written by x ∘ y (see [18]).

A nonempty subset S of a BCK/BCI-algebra X is called a subalgebra of X if x * y ∈ S for all x, y ∈ S. A subset I of a BCK/BCI-algebra X is called an ideal of X if it satisfies:

0∈I, (2.10)

(∀x∈X)∀y∈Ix∗y∈I ⇒ x∈I. (2.11)

A subset I of a BCI-algebra X is called a closed ideal of X (see [17]) if it is an ideal of X which satisfies:

(∀x∈X)(x∈I ⇒ 0∗x∈I). (2.12)

By an interval number we mean a closed subinterval ã = [a^–, a⁺] of I, where 0 ≤ a^– ≤ a⁺ ≤ 1. Denote by [I] the set of all interval numbers. Let us define what is known as refined minimum (briefly, rmin) and refined maximum (briefly, rmax) of two elements in [I]. We also define the symbols “⪰”, “⪯”, “ = ” in case of two elements in [I]. Consider two interval numbers a~1:=a1−,a1+ and a~2:=a2−,a2+. Then

rmina~1,a~2=mina1−,a2−,mina1+,a2+,rmaxa~1,a~2=maxa1−,a2−,maxa1+,a2+,a~1⪰a~2 ⇔ a1−≥a2−, a1+≥a2+,

and similarly we may have ã₁ ⪯ ã₂ and ã₁ = ã₂. To say ã₁ ≻ ã₂ (resp. ã₁ ≺ ã₂) we mean ã₁ ⪰ ã₂ and ã₁ ≠ ã₂ (resp. ã₁ ⪯ ã₂ and ã₁ ≠ ã₂). Let ã_i ∈ [I] where i ∈ Λ. We define

rinfi∈Λa~i=infi∈Λai−,infi∈Λai+ and rsupi∈Λa~i=supi∈Λai−,supi∈Λai+.

Let X be a nonempty set. A function A : X → [I] is called an interval-valued fuzzy set (briefly, an IVF set) in X. Let [I]^X stand for the set of all IVF sets in X. For every A ∈ [I]^X and x ∈ X, A(x) = [A^–(x), A⁺(x)] is called the degree of membership of an element x to A, where A^– : X → I and A⁺ : X → I are fuzzy sets in X which are called a lower fuzzy set and an upper fuzzy set in X, respectively. For simplicity, we denote A = [A^–, A⁺].

Let X be a non-empty set. A neutrosophic set (NS) in X (see [3]) is a structure of the form:

A:={〈x;AT(x),AI(x),AF(x)〉∣x∈X},

where A_T : X → [0, 1] is a truth membership function, A_I : X → [0, 1] is an indeterminate membership function, and A_F : X → [0, 1] is a false membership function. For the sake of simplicity, we shall use the symbol A = (A_T, A_I, A_F) for the neutrosophic set

A:={〈x;AT(x),AI(x),AF(x)〉∣x∈X}.

We refer the reader to the books [17, 18] for further information regarding BCK/BCI-algebras, and to the site “http://fs.gallup.unm.edu/neutrosophy.htm” for further information regarding neutrosophic set theory.

Let X be a non-empty set. By an MBJ-neutrosophic set in X (see [16]), we mean a structure of the form:

A:={〈x;MA(x),B~A(x),JA(x)〉∣x∈X},

where M_A and J_A are fuzzy sets in X, which are called a truth membership function and a false membership function, respectively, and B̃_A is an IVF set in X which is called an indeterminate interval-valued membership function.

For the sake of simplicity, we shall use the symbol 𝓐 = (M_A, B̃_A, J_A) for the MBJ-neutrosophic set

A:={〈x;MA(x),B~A(x),JA(x)〉∣x∈X}.

Let X be a BCK/BCI-algebra. An MBJ-neutrosophic set 𝓐 = (M_A, B̃_A, J_A) in X is called an MBJ-neutrosophic subalgebra of X (see [16]) if it satisfies:

(∀x,y∈X)MA(x∗y)≥min{MA(x),MA(y)},B~A(x∗y)⪰rmin{B~A(x),B~A(y)},JA(x∗y)≤max{JA(x),JA(y)}. (2.13)

3 MBJ-neutrosophic ideals of BCK/BCI-algebras

Definition 3.1

Let X be a BCK/BCI-algebra. An MBJ-neutrosophic set 𝓐 = (M_A, B̃_A, J_A) in X is called an MBJ-neutrosophic ideal of X if it satisfies:

(∀x∈X)MA(0)≥MA(x)B~A(0)⪰B~A(x)JA(0)≤JA(x) (3.1)

and

(∀x,y∈X)MA(x)≥min{MA(x∗y),MA(y)}B~A(x)⪰rmin{B~A(x∗y),B~A(y)}JA(x)≤max{JA(x∗y),JA(y)}. (3.2)

An MBJ-neutrosophic ideal 𝓐 = (M_A, B̃_A, J_A) of a BCI-algebra X is said to be closed if

(∀x∈X)MA(0∗x)≥MA(x)B~A(0∗x)⪰B~A(x)JA(0∗x)≤JA(x). (3.3)

Example 3.2

Consider a set X = {0, 1, 2, a} with the binary operation * which is given in Table 1. Then (X; *, 0) is a BCI-algebra (see [17]). Let 𝓐 = (M_A, B̃_A, J_A) be an MBJ-neutrosophic set in X defined by Table 2.

Table 1

Cayley table for the binary operation “*”.

*	0	1	2	a
0	0	0	0	a
1	1	0	0	a
2	2	2	0	a
a	a	a	a	0

Table 2

MBJ-neutrosophic set 𝓐 = (M_A, B̃_A, J_A).

X	M_A(x)	B̃_A(x)	J_A(x)
0	0.7	[1.0, 1.0]	0.2
1	0.5	[0.2, 0.6]	0.2
2	0.4	[0.2, 0.6]	0.7
a	0.3	[0.2, 0.6]	0.7

It is routine to verify that 𝓐 = (M_A, B̃_A, J_A) is a closed MBJ-neutrosophic ideal of X.

Proposition 3.3

Let X be a BCK/BCI-algebra. Then every MBJ-neutrosophic ideal 𝓐 = (M_A, B̃_A, J_A) of X satisfies the following assertion.

x∗y≤z ⇒MA(x)≥min{MA(y),MA(z)},B~A(x)⪰rmin{B~A(y),B~A(z)},JA(x)≤max{JA(y),JA(z)} (3.4)

for all x, y, z ∈ X.

Proof

Let x, y, z ∈ X be such that x * y ≤ z. Then

MA(x∗y)≥min{MA((x∗y)∗z),MA(z)}=min{MA(0),MA(z)}=MA(z),B~A(x∗y)⪰rmin{B~A((x∗y)∗z),B~A(z)}=rmin{B~A(0),B~A(z)}=B~A(z),

and

JA(x∗y)≤max{JA((x∗y)∗z),JA(z)}=max{JA(0),JA(z)}=JA(z).

It follows that

MA(x)≥min{MA(x∗y),MA(y)}=min{MA(y),MA(z)},B~A(x)⪰rmin{B~A(x∗y),B~A(y)}=rmin{B~A(y),B~A(z)},

and

JA(x)≤max{JA(x∗y),JA(y)}=max{JA(y),JA(z)}.

This completes the proof.□

Theorem 3.4

Every MBJ-neutrosophic set in a BCK/BCI-algebra X satisfying (3.1) and (3.4) is an MBJ-neutrosophic ideal of X.

Proof

Let 𝓐 = (M_A, B̃_A, J_A) be an MBJ-neutrosophic set in X satisfying (3.1) and (3.4). Note that x * (x * y) ≤ y for all x, y ∈ X. It follows from (3.4) that

MA(x)≥min{MA(x∗y),MA(y)},B~A(x)⪰rmin{B~A(x∗y),B~A(y)},

and

JA(x)≤max{JA(x∗y),JA(y)}.

Therefore 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic ideal of X.□

Theorem 3.5

Given an MBJ-neutrosophic set 𝓐 = (M_A, B̃_A, J_A) in a BCK/BCI-algebra X, if (M_A, J_A) is an intuitionistic fuzzy ideal of X, and BA− and BA+ are fuzzy ideals of X, then 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic ideal of X.

Proof

It is sufficient to show that B̃_A satisfies the condition

(∀x∈X)(B~A(0)⪰B~A(x)) (3.5)

and

(∀x,y∈X)(B~A(x)⪰rmin{B~A(x∗y),B~A(y)}). (3.6)

For any x, y ∈ X, we get

B~A(0)=[BA−(0),BA+(0)]⪰[BA−(x),BA+(x)]=B~A(x)

and

B~A(x)=[BA−(x),BA+(x)]⪰[min{BA−(x∗y),BA−(y)},min{BA+(x∗y),BA+(y)}]=rmin{[BA−(x∗y),BA+(x∗y)],[BA−(y),BA+(y)]=rmin{B~A(x∗y),B~A(y)}.

Therefore 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic ideal of X.□

If 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic ideal of a BCK/BCI-algebra X, then

BA−(x),BA+(x)]=B~A(x)⪰rmin{B~A(x∗y),B~A(y)}=rmin{[BA−(x∗y),BA+(x∗y),[BA−(y),BA+(y)]}=[min{BA−(x∗y),BA−(y)},min{BA+(x∗y),BA+(y)}

for all x, y ∈ X. It follows that BA− (x) ≥ min{ BA− (x * y), BA− (y)} and BA+ (x) ≥ min{ BA+ (x * y), BA+ (y)}. Thus BA− and BA+ are fuzzy ideals of X. But (M_A, J_A) is not an intuitionistic fuzzy ideal of X as seen in Example 3.2. This shows that the converse of Theorem 3.5 is not true.

Given an MBJ-neutrosophic set 𝓐 = (M_A, B̃_A, J_A) in a BCK/BCI-algebra X, we consider the following sets.

U(MA;t):={x∈X∣MA(x)≥t},U(B~A;[δ1,δ2]):={x∈X∣B~A(x)⪰[δ1,δ2]},L(JA;s):={x∈X∣JA(x)≤s},

where t, s ∈ [0, 1] and [δ₁, δ₂] ∈ [I].

Theorem 3.6

An MBJ-neutrosophic set 𝓐 = (M_A, B̃_A, J_A) in a BCK/BCI-algebra X is an MBJ-neutrosophic ideal of X if and only if the non-empty sets U(M_A; t), U(B̃_A;[δ₁, δ₂]) and L(J_A; s) are ideals of X for all t, s ∈ [0, 1] and [δ₁, δ₂] ∈ [I].

Proof

Suppose that 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic ideal of X. Let t, s ∈ [0, 1] and [δ₁, δ₂] ∈ [I] be such that U(M_A; t), U(B̃_A;[δ₁, δ₂]) and L(J_A; s) are non-empty. Obviously, 0 ∈ U(M_A; t) ∩ U(B̃_A;[δ₁, δ₂]) ∩ L(J_A; s). For any x, y, a, b, u, v ∈ X, if x * y ∈ U(M_A; t), y ∈ U(M_A; t), a * b ∈ U(B̃_A;[δ₁, δ₂]), b ∈ U(B̃_A;[δ₁, δ₂]), u * v ∈ L(J_A; s) and v ∈ L(J_A; s), then

MA(x)≥min{MA(x∗y),MA(y)}≥min{t,t}=t,B~A(a)⪰rmin{B~A(a∗b),B~A(b)}⪰rmin{[δ1,δ2],[δ1,δ2]}=[δ1,δ2],JA(u)≤max{JA(u∗v),JA(v)}≤min{s,s}=s,

and so x ∈ U(M_A; t), a ∈ U(B̃_A;[δ₁, δ₂]) and u ∈ L(J_A; s). Therefore U(M_A; t), U(B̃_A;[δ₁, δ₂]) and L(J_A; s) are ideals of X.

Conversely, assume that the non-empty sets U(M_A; t), U(B̃_A;[δ₁, δ₂]) and L(J_A; s) are ideals of X for all t, s ∈ [0, 1] and [δ₁, δ₂] ∈ [I]. Assume that M_A(0) < M_A(a), B̃_A(0) ≺ B̃_A(a) and J_A(0) > J_A(a) for some a ∈ X. Then 0 ∉ U(M_A; M_A(a)) ∩ U(B̃_A; B̃_A(a)) ∩ L(J_A; J_A(a), which is a contradiction. Hence M_A(0) ≥ M_A(x), B̃_A(0) ⪰ B̃_A(x) and J_A(0) ≤ J_A(x) for all x ∈ X. If

MA(a0)<min{MA(a0∗b0),MA(b0)}

for some a₀, b₀ ∈ X, then a₀ * b₀ ∈ U(M_A; t₀) and b₀ ∈ U(M_A; t₀) but a₀ ∉ U(M_A; t₀) for t₀ := min{M_A(a₀ * b₀), M_A(b₀)}. This is a contradiction, and thus M_A(a) ≥ min{M_A(a * b), M_A(b)} for all a, b ∈ X. Similarly, we can show that J_A(a) ≤ max{J_A(a * b), J_A(b)} for all a, b ∈ X. Suppose that B̃_A(a₀) ≺ rmin {B̃_A(a₀ * b₀), B̃_A(b₀)} for some a₀, b₀ ∈ X. Let B̃_A(a₀ * b₀) = [λ₁, λ₂], B̃_A(b₀) = [λ₃, λ₄] and B̃_A(a₀) = [δ₁, δ₂]. Then

δ1,δ2]≺rmin{[λ1,λ2],[λ3,λ4]}=[min{λ1,λ3},min{λ2,λ4}

and so δ₁ < min{λ₁, λ₃} and δ₂ < min{λ₂, λ₄}. Taking

[γ1,γ2]:=12B~A(a0)+rmin{B~A(a0∗b0),B~A(b0)}

implies that

[γ1,γ2]=12[δ1,δ2]+[min{λ1,λ3},min{λ2,λ4}]=12(δ1+min{λ1,λ3}),12(δ2+min{λ2,λ4}).

It follows that

min{λ1,λ3}>γ1=12(δ1+min{λ1,λ3})>δ1

and

min{λ2,λ4}>γ2=12(δ2+min{λ2,λ4})>δ2.

Hence [min{λ₁, λ₃}, min{λ₂, λ₄}] ≻ [γ₁, γ₂] ≻ [δ₁, δ₂] = B̃_A(a₀), and therefore a₀ ∉ U(B̃_A; [γ₁, γ₂]). On the other hand,

B~A(a0∗b0)=[λ1,λ2]⪰[min{λ1,λ3},min{λ2,λ4}]≻[γ1,γ2]

and

B~A(b0)=[λ3,λ4]⪰[min{λ1,λ3},min{λ2,λ4}]≻[γ1,γ2],

that is, a₀ * b₀, b₀ ∈ U(B̃_A; [γ₁, γ₂]). This is a contradiction, and therefore B̃_A(x) ⪰ rmin {B̃_A(x * y), B̃_A(y)} for all x, y ∈ X. Consequently 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic ideal of X.□

Theorem 3.7

Given an ideal I of a BCK/BCI-algebra X, let 𝓐 = (M_A, B̃_A, J_A) be an MBJ-neutrosophic set in X defined by

MA(x)=t if x∈I,0 otherwise,B~A(x)=[γ1,γ2] if x∈I,[0,0] otherwise,JA(x)=s if x∈I,1 otherwise, (3.7)

where t ∈ (0, 1], s ∈ [0, 1) and γ₁, γ₂ ∈ (0, 1] with γ₁ < γ₂. Then 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic ideal of X such that U(M_A; t) = U(B̃_A; [γ₁, γ₂]) = L(J_A; s) = I.

Proof

Let x, y ∈ X. If x * y ∈ I and y ∈ I, then x ∈ I and so

MA(x)=t=min{MA(x∗y),MA(y)},B~A(x)=[γ1,γ2]=rmin{[γ1,γ2],[γ1,γ2]}=rmin{B~A(x∗y),B~A(y)},JA(x)=s=max{JA(x∗y),JA(y)}.

If any one of x * y and y is contained in I, say x * y ∈ K, then M_A(x * y) = t, B̃_A(x * y) = [γ₁, γ₂], J_A(x * y) = s, M_A(y) = 0, B̃_A(y) = [0, 0] and J_A(y) = 1. Hence

MA(x)≥0=min{t,0}=min{MA(x∗y),MA(y)},B~A(x)⪰[0,0]=rmin{[γ1,γ2],[0,0]}=rmin{B~A(x∗y),B~A(y)},JA(x)≤1=max{s,1}=max{JA(x∗y),JA(y)}.

If x * y, y ∉ K, then M_A(x * y) = 0 = M_A(y), B̃_A(x * y) = [0, 0] = B̃_A(y) and J_A(x * y) = 1 = J_A(y). It follows that

MA(x)≥0=min{0,0}=min{MA(x∗y),MA(y)},B~A(x)⪰[0,0]=rmin{[0,0],[0,0]}=rmin{B~A(x∗y),B~A(y)},JA(x)≤1=max{1,1}=max{JA(x∗y),JA(y)}.

It is obvious that M_A(0) ≥ M_A(x), B̃_A(0)⪰ B̃_A(x) and J_A(0) ≤ J_A(x) for all x ∈ X. Therefore 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic ideal of X. Obviously, we have U(M_A; t) = U(B̃_A; [γ₁, γ₂]) = L(J_A; s) = I.□

Theorem 3.8

For any non-empty subset I of X, let 𝓐 = (M_A, B̃_A, J_A) be an MBJ-neutrosophic set in X which is given in (3.7). If 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic ideal of X, then I is an ideal of X.

Proof

Obviously, 0 ∈ I. Let x, y ∈ X be such that x * y ∈ I and y ∈ I. Then M_A(x * y) = t = M_A(y), B̃_A(x * y) = [γ₁, γ₂] = B̃_A(y) and J_A(x * y) = s = J_A(y). Thus

MA(x)≥min{MA(x∗y),MA(y)}=t,B~A(x)⪰rmin{B~A(x∗y),B~A(y)}=[γ1,γ2],JA(x)≤max{JA(x∗y),JA(y)}=s,

and hence x ∈ I. Therefore I is an ideal of X.□

Theorem 3.9

In a BCK-algebra, every MBJ-neutrosophic ideal is an MBJ-neutrosophic subalgebra.

Proof

Let 𝓐 = (M_A, B̃_A, J_A) be an MBJ-neutrosophic ideal of a BCK-algebra X. Since (x * y) * x ≤ y for all x, y ∈ X, it follows from Proposition 3.3 that

MA(x∗y)≥min{MA(x),MA(y)},B~A(x∗y)⪰rmin{B~A(x),B~A(y)},JA(x∗y)≤max{JA(x),JA(y)}

for all x, y ∈ X. Hence 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic subalgebra of a BCK-algebra X.□

The converse of Theorem 3.9 may not be true as seen in the following example.

Example 3.10

Consider a BCK-algebra X = {0, 1, 2, 3} with the binary operation * which is given in Table 3. Let 𝓐 = (M_A, B̃_A, J_A) be an MBJ-neutrosophic set in X defined by Table 4. Then 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic subalgebra of X, but it is not an MBJ-neutrosophic ideal of X since

Table 3

Cayley table for the binary operation “*”.

*	0	1	2	3
0	0	0	0	0
1	1	0	0	1
2	2	1	0	2
3	3	3	3	0

Table 4

MBJ-neutrosophic set 𝓐 = (M_A, B̃_A, J_A).

X	M_A(x)	B̃_A(x)	J_A(x)
0	0.7	[0.3, 0.8]	0.2
1	0.4	[0.2, 0.6]	0.3
2	0.4	[0.3, 0.8]	0.4
3	0.6	[0.2, 0.6]	0.5

B~A(1)⋡rmin{B~A(1∗2),B~A(2)}.

We provide a condition for an MBJ-neutrosophic subalgebra to be an MBJ-neutrosophic ideal in a BCK-algebra.

Theorem 3.11

Let 𝓐 = (M_A, B̃_A, J_A) be an MBJ-neutrosophic subalgebra of a BCK-algebra X satisfying the condition (3.4). Then 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic ideal of X.

Proof

For any x ∈ X, we get

MA(0)=MA(x∗x)≥min{MA(x),MA(x)}=MA(x),

B~A(0)=B~A(x∗x)⪰rmin{B~A(x),B~A(x)}=rmin{[BA−(x),BA+(x)],[BA−(x),BA+(x)]}=[BA−(x),BA+(x)]=B~A(x),

and

JA(0)=JA(x∗x)≤max{JA(x),JA(x)}=JA(x).

Since x * (x * y) ≤ y for all x, y ∈ X, it follows from (3.4) that

MA(x)≥min{MA(x∗y),MA(y)},B~A(x)⪰rmin{B~A(x∗y),B~A(y)},JA(x)≤max{JA(x∗y),JA(y)}

for all x, y ∈ X. Therefore 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic ideal of X.□

Theorem 3.9 is not true in a BCI-algebra as seen in the following example.

Example 3.12

Let (Y, *, 0) be a BCI-algebra and let (ℤ, –, 0) be an adjoint BCI-algebra of the additive group (ℤ, +, 0) of integers. Then X = Y × ℤ is a BCI-algebra and I = Y × ℕ is an ideal of X where ℕ is the set of all non-negative integers (see [17]). Let 𝓐 = (M_A, B̃_A, J_A) be an MBJ-neutrosophic set in X defined by

MA(x)=t if x∈I,0 otherwise,B~A(x)=[γ1,γ2] if x∈I,[0,0] otherwise,JA(x)=s if x∈I,1 otherwise, (3.8)

where t ∈ (0, 1], s ∈ [0, 1) and γ₁, γ₂ ∈ (0, 1] with γ₁ < γ₂. Then 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic ideal of X by Theorem 3.7. But it is not an MBJ-neutrosophic subalgebra of X since

MA((0,0)∗(0,1))=MA((0,−1))=0<t=min{MA((0,0)),MA(0,1))},B~A((0,0)∗(0,2))=B~A((0,−2))=[0,0]≺[γ1,γ2]=rmin{B~A((0,0)),B~A(0,2))},

and/or

JA((0,0)∗(0,3))=JA((0,−3))=1>s=max{JA((0,0)),JA(0,3))}.

Definition 3.13

An MBJ-neutrosophic ideal 𝓐 = (M_A, B̃_A, J_A) of a BCI-algebra X is said to be closed if

(∀x∈X)(MA(0∗x)≥MA(x),B~A(0∗x)⪰B~A(x),JA(0∗x)≤JA(x)). (3.9)

Theorem 3.14

In a BCI-algebra, every closed MBJ-neutrosophic ideal is an MBJ-neutrosophic subalgebra.

Proof

Let 𝓐 = (M_A, B̃_A, J_A) be a closed MBJ-neutrosophic ideal of a BCI-algebra X. Using (3.2), (2.3), (III) and (3.3), we have

MA(x∗y)≥min{MA((x∗y)∗x),MA(x)}=min{MA(0∗y),MA(x)}≥min{MA(y),MA(x)},

B~A(x∗y)⪰rmin{B~A((x∗y)∗x),B~A(x)}=rmin{B~A(0∗y),B~A(x)}⪰rmin{B~A(y),B~A(x)},

and

JA(x∗y)≤max{JA((x∗y)∗x),JA(x)}=max{JA(0∗y),JA(x)}≤max{JA(y),JA(x)}

for all x, y ∈ X. Hence 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic subalgebra of X.□

Theorem 3.15

In a weakly BCK-algebra, every MBJ-neutrosophic ideal is closed.

Proof

Let 𝓐 = (M_A, B̃_A, J_A) be an MBJ-neutrosophic ideal of a weakly BCK-algebra X. For any x ∈ X, we obtain

MA(0∗x)≥min{MA((0∗x)∗x),MA(x)}=min{MA(0),MA(x)}=MA(x),B~A(0∗x)⪰rmin{B~A((0∗x)∗x),B~A(x)}=rmin{B~A(0),B~A(x)}=B~A(x),

and

JA(0∗x)≤max{JA((0∗x)∗x),JA(x)}=max{JA(0),JA(x)}=JA(x).

Therefore 𝓐 = (M_A, B̃_A, J_A) is a closed MBJ-neutrosophic ideal of X.□

Corollary 3.16

In a weakly BCK-algebra, every MBJ-neutrosophic ideal is an MBJ-neutrosophic subalgebra.

The following example shows that any MBJ-neutrosophic subalgebra is not an MBJ-neutrosophic ideal in a BCI-algebra.

Example 3.17

Consider a BCI-algebra X = {0, a, b, c, d, e} with the *-operation in Table 5. Let 𝓐 = (M_A, B̃_A, J_A) be an MBJ-neutrosophic set in X defined by Table 6. It is routine to verify that 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic subalgebra of X. But it is not an MBJ-neutrosophic ideal of X since

Table 5

Cayley table for the binary operation “*”.

*	0	a	b	c	d	e
0	0	0	c	b	c	c
a	a	0	c	b	c	c
b	b	b	0	c	0	0
c	c	c	b	0	b	b
d	d	b	a	c	0	a
e	e	b	a	c	a	0

Table 6

MBJ-neutrosophic set 𝓐 = (M_A, B̃_A, J_A).

X	M_A(x)	B̃_A(x)	J_A(x)
0	0.7	[0.4, 0.9]	0.3
a	0.4	[0.04, 0.45]	0.6
b	0.7	[0.4, 0.9]	0.3
c	0.7	[0.4, 0.9]	0.3
d	0.4	[0.04, 0.45]	0.6
e	0.4	[0.04, 0.45]	0.6

MA(d)<min{MA(d∗c),MA(c)},B~A(d)≺rmin{B~A(d∗c),B~A(c)},

and/or

JA(d)>max{JA(d∗c),JA(c)}.

Theorem 3.18

In a p-semisimple BCI-algebra X, the following are equivalent.

𝓐 = (M_A, B̃_A, J_A) is a closed MBJ-neutrosophic ideal of X.
𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic subalgebra of X.

Proof

(1) ⇒ (2). See Theorem 3.14.

(2) ⇒ (1). Suppose that 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic subalgebra of X. For any x ∈ X, we get

MA(0)=MA(x∗x)≥min{MA(x),MA(x)}=MA(x),B~A(0)=B~A(x∗x)⪰rmin{B~A(x),B~A(x)}=B~A(x),

and

JA(0)=JA(x∗x)≤max{JA(x),JA(x)}=JA(x).

Hence M_A(0 * x) ≥ min{M_A(0), M_A(x)} = M_A(x), B̃_A(0 * x) ⪰ rmin{B̃_A(0), B̃_A(x)} = B̃_A(x) and J_A(0 * x) ≤ max{J_A(0), J_A(x)} = J_A(x) for all x ∈ X. Let x, y ∈ X. Then

MA(x)=MA(y∗(y∗x))≥min{MA(y),MA(y∗x)}=min{MA(y),MA(0∗(x∗y))}≥min{MA(x∗y),MA(y)},

B~A(x)=B~A(y∗(y∗x))⪰rmin{B~A(y),B~A(y∗x)}=rmin{B~A(y),B~A(0∗(x∗y))}⪰rmin{B~A(x∗y),B~A(y)}

and

JA(x)=JA(y∗(y∗x))≤max{JA(y),JA(y∗x)}=max{JA(y),JA(0∗(x∗y))}≤max{JA(x∗y),JA(y)}.

Therefore 𝓐 = (M_A, B̃_A, J_A) is a closed MBJ-neutrosophic ideal of X.□

Since every associative BCI-algebra is p-semisimple, we have the following corollary.

Corollary 3.19

In an associative BCI-algebra X, the following are equivalent.

𝓐 = (M_A, B̃_A, J_A) is a closed MBJ-neutrosophic ideal of X.
𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic subalgebra of X.

Corollary 3.20

In a BCI-algebra X, consider the following conditions

every element x in X is minimal.
X = {0 * x | x ∈ X}.
(∀x, y ∈ X) (x * (0 * y) = y * (0 * x)).
(∀x ∈ X) (0 * x = 0 ⇒ x = 0).
(∀a, x ∈ X) (a * (a * x) = x).
(∀a ∈ X) X = {a * x | x ∈ X}.
(∀x, y, a, b ∈ X) ((x * y) * (a * b) = (x * a) * (y * b)).
(∀x, y ∈ X) (0 * (y * x) = x * y).

If one of the conditions above is valid, then the following are equivalent.

𝓐 = (M_A, B̃_A, J_A) is a closed MBJ-neutrosophic ideal of X.
𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic subalgebra of X.

Definition 3.21

Let X be an (S)-BCK-algebra. An MBJ-neutrosophic set 𝓐 = (M_A, B̃_A, J_A) in X is called an MBJ-neutrosophic ∘-subalgebra of X if the following assertions are valid.

MA(x∘y)≥min{MA(x),MA(y)},B~A(x∘y)⪰rmin{B~A(x),B~A(y)},JA(x∘y)≤max{JA(x),JA(y)} (3.10)

for all x, y ∈ X.

Lemma 3.22

Every MBJ-neutrosophic ideal of a BCK/BCI-algebra X satisfies the following assertion.

(∀x,y∈X)x≤y ⇒ MA(x)≥MA(y),B~A(x)⪰B~A(y),JA(x)≤JA(y). (3.11)

Proof

Assume that x ≤ y for all x, y ∈ X. Then x * y = 0, and so

MA(x)≥min{MA(x∗y),MA(y)}=min{MA(0),MA(y)}=MA(y),B~A(x)⪰rmin{B~A(x∗y),B~A(y)}=rmin{B~A(0),B~A(y)}=B~A(y),

and

JA(x)≤max{JA(x∗y),JA(y)}=max{JA(0),JA(y)}=JA(y).

This completes the proof.□

Theorem 3.23

In an (S)-BCK-algebra, every MBJ-neutrosophic ideal is an MBJ-neutrosophic ∘-subalgebra.

Proof

Let 𝓐 = (M_A, B̃_A, J_A) be an MBJ-neutrosophic ideal of an (S)-BCK-algebra X. Note that (x ∘ y) * x ≤ y for all x, y ∈ X. Using Lemma 3.22 and (3.2) implies that

MA(x∘y)≥min{MA((x∘y)∗x),MA(x)}≥min{MA(y),MA(x)},B~A(x∘y)⪰rmin{B~A((x∘y)∗x),B~A(x)}⪰rmin{B~A(y),B~A(x)},

and

JA(x∘y)≤max{JA((x∘y)∗x),JA(x)}≤max{JA(y),JA(x)}.

Therefore 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic ∘$-subalgebra of X.□

We provide a characterization of an MBJ-neutrosophic ideal in an (S)-BCK-algebra.

Theorem 3.24

Let 𝓐 = (M_A, B̃_A, J_A) be an MBJ-neutrosophic set in an (S)-BCK-algebra X. Then 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic ideal of X if and only if the following assertions are valid.

MA(x)≥min{MA(y),MA(z)},B~A(x)⪰rmin{B~A(y),B~A(z)},JA(x)≤max{JA(y),JA(z)} (3.12)

for all x, y, z ∈ X with x ≤ y ∘ z.

Proof

Assume that 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic ideal of X and let x, y, z ∈ X be such that x ≤ y ∘ z. Using (3.1), (3.2) and Theorem 3.23, we have

MA(x)≥min{MA(x∗(y∘z)),MA(y∘z)}=min{MA(0),MA(y∘z)}=MA(y∘z)≥min{MA(y),MA(z)},

B~A(x)⪰rmin{B~A(x∗(y∘z)),B~A(y∘z)}=rmin{B~A(0),B~A(y∘z)}=B~A(y∘z)⪰rmin{B~A(y),B~A(z)},

and

JA(x)≤max{JA(x∗(y∘z)),JA(y∘z)}=max{JA(0),JA(y∘z)}=JA(y∘z)≤max{JA(y),JA(z)}.

Conversely, let 𝓐 = (M_A, B̃_A, J_A) be an MBJ-neutrosophic set in an (S)-BCK-algebra X satisfying the condition (3.12) for all x, y, z ∈ X with x ≤ y ∘ z. Sine 0 ≤ x ∘ x for all x ∈ X, it follows from (3.12) that

MA(0)≥min{MA(x),MA(x)}=MA(x),B~A(0)⪰rmin{B~A(x),B~A(x)}=B~A(x),

and

JA(0)≤max{JA(x),JA(x)}=JA(x).

Note that x ≤ (x * y) ∘ y for all x, y ∈ X. Hence we have

MA(x)≥min{MA(x∗y),MA(y)},B~A(x)⪰min{B~A(x∗y),B~A(y)} and JA(x)≤max{JA(x∗y),JA(y)}.

Therefore 𝓐 = (M_A, B̃_A, J_A) is an MBJ-neutrosophic ideal of X.□

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Received: 2018-09-06

Accepted: 2019-02-19

Published Online: 2019-11-28

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0106

Keywords for this article

MBJ-neutrosophic set; MBJ-neutrosophic subalgebra; MBJ-neutrosophic ideal; MBJ-neutro-sophic ∘-subalgebra

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