# Left (Right)-Quasi Neutrosophic Triplet Loops (Groups) and Generalized BE-Algebras

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Concepts

**Definition**

**1.**

**Remark**

**1.**

**Definition**

**2.**

- (1)
- If * is well-defined, that is, for any a, b ∈ N, one has a * b ∈ N. Then, N is called a neutrosophic triplet loop.
- (2)
- If N is a neutrosophic triplet loop, and * is associative, that is, (a * b) * c= a * (b * c) for all a, b, c ∈ N. Then, N is called a neutrosophic triplet group.
- (3)
- If N is a neutrosophic triplet group, and * is commutative, that is, a * b = b * a for all a, b ∈ N. Then, N is called a commutative neutrosophic triplet group.

**Definition**

**3.**

**Definition**

**4.**

_{→}y ⇔ x → y = 1. x ≤

_{⇝}y ⇔ x ⇝ y = 1.

_{→}= ≤

_{⇝}.

## 3. Various Quasi Neutrosophic Triplet Loops (Groups)

**Definition**

**5.**

- (1)
- If exist b, c ∈ N, such that a * b = a and a * c = b, then a is called an NT-element with (r-r)- property;
- (2)
- If exist b, c ∈ N, such that a * b = a and c * a = b, then a is called an NT-element with (r-l)- property;
- (3)
- If exist b, c ∈ N, such that b * a = a and c * a = b, then a is called an NT-element with (l-l)- property;
- (4)
- If exist b, c ∈ N, such that b * a = a and a * c = b, then a is called an NT-element with (l-r)- property;
- (5)
- If exist b, c ∈ N, such that a * b = b * a = a and c * a = b, then a is called an NT-element with (lr-l)-property;
- (6)
- If exist b, c ∈ N, such that a * b = b * a = a and a * c = b, then a is called an NT-element with (lr-r)-property;
- (7)
- If exist b, c ∈ N, such that b * a = a and a * c = c * a = b, then a is called an NT-element with (l-lr)-property;
- (8)
- If exist b, c ∈ N, such that a * b = a and a * c = c * a = b, then a is called an NT-element with (r-lr)-property;
- (9)
- If exist b, c ∈ N, such that a * b = b * a = a and a * c = c * a = b, then a is called an NT-element with (lr-lr)-property.

**Example**

**1.**

**Definition**

**6.**

**Remark**

**2.**

_{(r-r)}(a), denote a (r-r)-opposite of ‘a’ by anti

_{(r-r)}(a), where ‘a’ is an NT-element with (r-r)-property. If neut

_{(r-r)}(a) and anti

_{(r-r)}(a) are not unique, then denote the set of all (r-r)-neutral of ‘a’ by {neut

_{(r-r)}(a)}, denote the set of all (r-r)-opposite of ‘a’ by {anti

_{(r-r)}(a)}.

_{(r-r)}(a) = a, anti

_{(r-r)}(a) = a; neut

_{(r-r)}(b) = c, {anti

_{(r-r)}(b)} = {a, d};

_{(r-r)}(c) = a, anti

_{(r-r)}(c) = d; neut

_{(r-r)}(d) = b, anti

_{(r-r)}(d) = c.

**Theorem**

**1.**

**Proof.**

_{(l-lr)}(a) * a = a, anti

_{(l-lr)}(a) * a = a * anti

_{(l-lr)}(a) = neut

_{(l-lr)}(a).

_{(l-lr)}(a) ∈ {neut

_{(l-lr)}(a)}, anti

_{(l-lr)}(a) ∈ {anti

_{(l-lr)}(a)}. Applying associative law we get the following:

_{(l-lr)}(a) = a * (anti

_{(l-lr)}(a) * a) = (a * anti

_{(l-lr)}(a)) * a = neut

_{(l-lr)}(a) * a = a.

_{(l-lr)}(a) is a right neutral of ‘a’. From the arbitrariness of a, it is known that (N, *) is a neutrosophic triplet group.

**Theorem**

**2.**

_{(r-lr)}(a)}, ∀ p ∈ {anti

_{(r-lr)}(a)}.

- (1)
- for any a ∈ N, s ∈ {neut
_{(r-lr)}(a)} ⇒s * s = s. - (2)
- for any a ∈ N, s, t ∈{neut
_{(r-lr)}(a)} ⇒s * t = t. - (3)
- when * is commutative, for any a ∈ N, neut
_{(r-lr)}(a) is unique.

**Proof.**

_{(r-lr)}(a)}, then a * s = a, and exist p ∈ N, such that p * a = a *p = s. Thus,

_{(r-lr)}(a)}, then a * s = a, a * t = a, and exist p, q ∈ N, such that p * a = a * p = s, q * a = a * q = t. Thus,

_{(r-lr)}(a)}. Applying Theorem (2) to s and t we have s * t = t. Moreover, applying Therorem (2) to t and s we have t * s = s. Hence, when * is commutative, s * t = t * s. Therefore, s = t, that is, neut

_{(r-lr)}(a) is unique. □

**Corollary**

**1.**

**Proof.**

**Example**

**2.**

**Example**

**3.**

## 4. Quasi Neutrosophic Triplet Structures in BE-Algebras and CI-Algebras

#### 4.1. BE-Algebras (CI-Algebras) and (l-l)-Quasi Neutrosophic Triplet Loops

**Theorem**

**3.**

**Proof.**

_{(l-l)}(x)}, x ∈ {anti

_{(l-l)}(x)}, for any x ∈ X.

**Example**

**4.**

_{(l-l)}(a)} = {1}, {anti

_{(l-l)}(a)} = {a, c}; {neut

_{(l-l)}(b)} = {1}, {anti

_{(l-l)}(b)} = {b, c};

_{(l-l)}(c)} = {1}, {anti

_{(l-l)}(c)} = {c}; {neut

_{(l-l)}(1)} = {1}, {anti

_{(l-l)}(1)} = {1}.

**Example**

**5.**

_{(l-l)}(a)} = {1}, {anti

_{(l-l)}(a)} = {a}; {neut

_{(l-l)}(b)} = {1}, {anti

_{(l-l)}(b)} = {b};

_{(l-l)}(c)} = {1}, {anti

_{(l-l)}(c)} = {c}; {neut

_{(l-l)}(1)} = {1}, {anti

_{(l-l)}(1)} = {1}.

**Definition**

**7.**

**Lemma**

**1.**

- (1)
- x → ((x → y) → y) = 1,
- (2)
- 1 → x = 1 (or equivalently, 1 ≤ x) implies x = 1,
- (3)
- (x → y) → 1 = (x → 1) → (y → 1).

**Lemma**

**2.**

- (1)
- a = (a → 1) → 1,
- (2)
- (a → b) → 1 = b → a,
- (3)
- ((a → b) → 1) → 1 = a → b,
- (4)
- for any x ∈ X, (a → x) → (b → x) = b → a,
- (5)
- for any x ∈ X, (a → x) → b = (b → x) → a,
- (6)
- for any x ∈ X, (a → x) → (y → b) = (b → x) → (y → a).

**Definition**

**8.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

- (1)
- For any x, y ∈ X, if x → y = y → x = 1, then, by Definitions 8 and 3, we have the following:x = x → 1 = x → (y → x) = y → (x → x) = y → 1 = y.
- (2)
- For any x, y, z ∈ X, by Proposition 1 and Lemma 2 (4), we can get the following:(y → z) → ((z → x) → (y → x)) = (y → z) → (y → z) = 1.

**Theorem**

**4.**

**Proof.**

**Example**

**6.**

_{(l-l)}(a)} = {1}, {anti

_{(l-l)}(a)} = {a,b}; {neut

_{(l-l)}(b)} = {1}, {anti

_{(l-l)}(b)} = {a,b,c};

_{(l-l)}(c)} = {1}, {anti

_{(l-l)}(c)} = {c,d,e}; {neut

_{(l-l)}(d)} = {1}, {anti

_{(l-l)}(d)} = {d,e};

_{(l-l)}(e)}={1}, {anti

_{(l-l)}(e)}={d,e}; {neut

_{(l-l)}(1)}={1}, {anti

_{(l-l)}(1)}={1}.

#### 4.2. BE-Algebras (CI-Algebras) and Their Adjoint Semi-Groups

_{a}to denote the self-map of X defined by the following:

_{a}: X → X; ↦ a → x, for all x ∈ X.

**Theorem**

**5.**

_{a}* …* p

_{b}of self-map of X with a, …, b ∈X, where * represents the composition operation of mappings. Then, (M(X), *) is a commutative semigroup with identity p

_{1}.

**Proof.**

_{1}: X→X ↦ 1 → x, for all x ∈ X.

_{1}(x)=x for any x∈X. Hence, p

_{1}*m= p

_{1}*m=m for any m∈M(X).

_{a}* p

_{b})(x) = p

_{a}(b → x) = a → (b → x) = b → (a → x) = p

_{b}(a → x) = (p

_{b}* p

_{a})(x).

_{1}. □

**Example**

**7.**

_{a}: X → X; a $\mapsto $ 1, b $\mapsto $ 1, c $\mapsto $ 1, 1 $\mapsto $ 1. It is abbreviated to p

_{a}= (1, 1, 1, 1).

_{b}: X → X; a $\mapsto $ c, b $\mapsto $ 1, c $\mapsto $ a, 1 $\mapsto $ 1. It is abbreviated to p

_{b}= (c, 1, a, 1).

_{c}: X → X; a $\mapsto $ 1, b $\mapsto $ 1, c $\mapsto $ 1, 1 $\mapsto $ 1. It is abbreviated to p

_{c}= (1, 1, 1, 1).

_{1}: X → X; a $\mapsto $ a, b $\mapsto $ b, c $\mapsto $ c, 1 $\mapsto $ 1. It is abbreviated to p

_{1}= (a, b, c, 1).

_{a}* p

_{a}= p

_{a}, p

_{a}* p

_{b}= p

_{a}, p

_{a}* p

_{c}= p

_{a}; p

_{b}* p

_{b}= (a, 1, c, 1), p

_{b}* p

_{c}= p

_{c}= p

_{a}; p

_{a}* (p

_{b}* p

_{b}) = p

_{a}, p

_{b}* (p

_{b}* p

_{b}) = p

_{b}, p

_{c}* (p

_{b}* p

_{b}) = p

_{c}= p

_{a}. Denote p

_{bb}= p

_{b}* p

_{b}= (a, 1, c, 1), then M(X) = {p

_{a}, p

_{b}, p

_{bb}, p

_{1}}, and its Cayley table is Table 8. Obviously, (M(X), *) is a commutative neutrosophic triplet group and

_{a}) = p

_{a}, anti(p

_{a}) = p

_{a}; neut(p

_{b}) = p

_{bb}, anti(p

_{b}) = p

_{b}; neut(p

_{bb}) = p

_{bb}, anti(p

_{bb}) = p

_{bb}; neut(p

_{1}) = p

_{1}, anti(p

_{1}) = p

_{1}.

**Example**

**8.**

_{a}: X → X; a $\mapsto $ 1, b $\mapsto $ a, 1 $\mapsto $ b. It is abbreviated to p

_{a}= (1, a, b).

_{b}: X → X; a $\mapsto $ b, b $\mapsto $ 1, 1 $\mapsto $ a. It is abbreviated to p

_{b}= (b, 1, a).

_{1}: X → X; a $\mapsto $ a, b $\mapsto $ b, 1 $\mapsto $ 1. It is abbreviated to p

_{1}= (a, b, 1).

_{a}* p

_{a}= p

_{b}, p

_{a}* p

_{b}= p

_{1}; p

_{b}* p

_{b}= p

_{a}. Then M(X) = {p

_{a}, p

_{b}, p

_{1}} and its Cayley table is Table 10. Obviously, (M(X), *) is a commutative group with identity p

_{1}and (p

_{a})

^{−1}= p

_{b}, (p

_{b})

^{−1}= p

_{a}.

**Theorem**

**6.**

_{1}, where M(X) = {p

_{a}| a ∈X} and |M(X)| = |X|.

**Proof.**

= (x → 1) → (((a → 1) → b) → 1)

= (x → 1) → (((a → 1) → 1) → (b → 1))

= (x →1) → (a → (b → 1))

= a → ((x → 1) → (b → 1))

= a → (b → x).

_{a}≠ p

_{b}, ∀ a, b ∈ X.

_{a}= p

_{b}, a, b ∈ X. Then, for all x in X, p

_{a}(x) = p

_{b}(x). Hence,

_{a}(b) = p

_{b}(b) = b → b = 1.

_{a}* p

_{b}= p

_{c}, where c = (a → 1) → b. This means that M(X) $\subseteq $ {p

_{a}|a ∈ X}. By the definition of M(X), {p

_{a}a ∈ X} $\subseteq $ M(X). Hence, M(X) = {p

_{a}|a ∈ X}.

## 5. Quasi Neutrosophic Triplet Structures in Pseudo BE-Algebras and Pseudo CI-Algebras

#### 5.1. Pseudo BE-Algebras (Pseudo CI-Algebras) and (l-l)-Quasi Neutrosophic Triplet Loops

**Theorem**

**7.**

**Example**

**9.**

**Definition**

**9.**

**Proposition**

**3.**

- (1)
- x ≤ (x → y) ⇝ y, x ≤ (x ⇝ y) → y,
- (2)
- x ≤ y → z ⇔y ≤ x ⇝ z,
- (3)
- (x → y) → 1 = (x → 1) ⇝ (y ⇝ 1), (x ⇝ y) ⇝ 1 = (x ⇝ 1) → (y → 1),
- (4)
- x → 1 = x ⇝ 1,
- (5)
- x ≤ y implies x → 1 = y → 1.

**Proposition**

**4.**

- (1)
- a = (a → 1) → 1,
- (2)
- for any x ∈ X, (a → x) ⇝ x = a, (a ⇝ x) → x = a,
- (3)
- for any x ∈ X, (a → x) ⇝ 1 = x → a, (a ⇝ x) → 1 = x⇝ a,
- (4)
- for any x ∈ X, x → a = (a → 1) ⇝ (x → 1), x ⇝ a = (a ⇝ 1) → (x ⇝ 1).

**Definition**

**10.**

**Proposition**

**5.**

**Proof.**

**Lemma**

**3.**

- (1)
- x → (y → z) = (x → y) → z, for all x, y, z in X;
- (2)
- x → 1 = x= x⇝ 1, for every x in X;
- (3)
- x → y= x ⇝ y = y → x, for all x, y in X;
- (4)
- x ⇝ (y ⇝ z) = (x ⇝ y) ⇝ z, for all x, y, z in X.

**Proposition**

**6.**

**Proof.**

**Theorem**

**8.**

**Proof.**

#### 5.2. Pseudo BE-Algebras (Pseudo CI-Algebras) and Their Adjoint Semi-Groups

_{a}

^{→}and p

_{a}

^{⇝}to denote the self-map of X, which is defined by the following:

_{a}

^{→}: X → X; ↦ a → x, for all x ∈ X.

_{a}

^{⇝}: X → X; ↦ a ⇝ x, for all x ∈ X.

**Theorem**

**9.**

^{→}(X) = {finite products p

_{a}

^{→}* …* p

_{b}

^{→}of self-map of X | a, …, b ∈ X},

^{⇝}(X) = {finite products p

_{a}

^{⇝}* …* p

_{b}

^{⇝}of self-map of X | a, …, b ∈ X},

_{a}

^{→}(or p

_{a}

^{⇝}) * …* p

_{b}

^{→}(or p

_{b}

^{⇝}) of self-map of X | a, …, b ∈ X},

^{→}(X), *), (M

^{⇝}(X), *), and (M(X), *) are all semigroups with the identity p

_{1}= p

_{1}

^{→}= p

_{1}

^{⇝}.

**Proof.**

^{→}(X), *), (M

^{⇝}(X), *), and (M(X), *) the adjoint semigroups of X.

**Example**

**10.**

_{a}

^{→}* p

_{a}

^{→}= p

_{a}

^{→}, p

_{a}

^{→}* p

_{b}

^{→}= (1, 1, b, 1), p

_{a}

^{→}* p

_{c}

^{→}= p

_{c}

^{→}, p

_{a}

^{→}* p

_{1}

^{→}= p

_{a}

^{→};

_{b}

^{→}* p

_{a}

^{→}= p

_{c}

^{→}, p

_{b}

^{→}* p

_{b}

^{→}= p

_{b}

^{→}, p

_{b}

^{→}* p

_{c}

^{→}= p

_{c}

^{→}, p

_{b}

^{→}* p

_{1}

^{→}= p

_{b}

^{→};

_{c}

^{→}* p

_{a}

^{→}= p

_{c}

^{→}, p

_{c}

^{→}* p

_{b}

^{→}=p

_{c}

^{→}, p

_{c}

^{→}* p

_{c}

^{→}=p

_{c}

^{→}, p

_{c}

^{→}* p

_{1}

^{→}= p

_{c}

^{→};

_{1}

^{→}* p

_{a}

^{→}= p

_{a}

^{→}, p

_{1}

^{→}* p

_{b}

^{→}= p

_{b}

^{→}, p

_{1}

^{→}* p

_{c}

^{→}= p

_{c}

^{→}, p

_{1}

^{→}* p

_{1}

^{→}= p

_{1}

^{→}.

_{ab}

^{→}= p

_{a}

^{→}* p

_{b}

^{→}= (1, 1, b, 1), then p

_{ab}

^{→}* p

_{a}

^{→}= p

_{c}

^{→}, p

_{ab}

^{→}* p

_{b}

^{→}= p

_{ab}

^{→}, p

_{ab}

^{→}* p

_{ab}

^{→}= p

^{→}, p

_{ab}

^{→}* p

_{c}

^{→}= p

_{c}

^{→}. Hence, M

^{→}(X) = {p

_{a}

^{→}, p

_{b}

^{→}, p

_{ab}

^{→}, p

_{c}

^{→}, p

_{1}

^{→}} and its Cayley table is Table 15. Obviously, (M

^{→}(X), *) is a non-commutative semigroup, but it is not a neutrosophic triplet group.

_{a}

^{⇝}= (1, b, c, 1), p

_{b}

^{⇝}= (a, 1, a, 1), p

_{c}

^{⇝}= (1, 1, 1, 1), p

_{1}

^{⇝}= (a, b, c, 1).

_{a}

^{⇝}* p

_{a}

^{⇝}= p

_{a}

^{⇝}, p

_{a}

^{⇝}* p

_{b}

^{⇝}= p

_{a}

^{⇝}* p

_{c}

^{⇝}= (1, 1, 1, 1), p

_{a}

^{⇝}* p

_{1}

^{⇝}= p

_{a}

^{⇝};

_{b}

^{⇝}* p

_{a}

^{⇝}= (1, 1, a, 1), p

_{b}

^{⇝}* p

_{b}

^{⇝}= p

_{b}

^{⇝}, p

_{b}

^{⇝}* p

_{c}

^{⇝}= p

_{c}

^{⇝}, p

_{b}

^{⇝}* p

_{1}

^{⇝}= p

_{b}

^{⇝};

_{c}

^{⇝}* p

_{a}

^{⇝}= p

_{c}

^{⇝}, p

_{c}

^{⇝}* p

_{b}

^{⇝}= p

_{c}

^{⇝}, p

_{c}

^{⇝}* p

_{c}

^{⇝}= p

_{c}

^{⇝}, p

_{c}

^{⇝}* p

_{1}

^{⇝}= p

_{c}

^{⇝}.

_{ba}

^{⇝}= p

_{b}

^{⇝}* p

_{a}

^{⇝}= (1, 1, a, 1), then p

_{ba}

^{⇝}* p

_{a}

^{⇝}= p

_{ba}

^{⇝}, p

_{a}

^{⇝}* p

_{ba}

^{⇝}= p

_{c}

^{⇝}; p

_{ba}

^{⇝}* p

_{b}

^{⇝}= p

_{c}

^{⇝}, p

_{b}

^{⇝}* p

_{ba}

^{⇝}= p

_{ba}

^{⇝}; p

_{ba}

^{⇝}* p

_{ba}

^{⇝}= p

_{c}

^{⇝}; p

_{ba}

^{⇝}* p

_{c}

^{⇝}= p

_{c}

^{⇝}, p

_{c}

^{⇝}* p

_{ba}

^{⇝}= p

_{c}

^{⇝}. Hence, M

^{⇝}(X) = {p

_{a}

^{⇝}, p

_{b}

^{⇝}, p

_{ba}

^{⇝}, p

_{c}

^{⇝}, p

_{1}

^{⇝}} and its Cayley table is Table 16. Obviously, (M

^{⇝}(X), *) is a non-commutative semigroup, but it is not a neutrosophic triplet group.

_{c}

^{→}= (1, 1, 1, 1) = p

_{c}

^{⇝}, p

_{1}

^{→}= (a, b, c, 1) = p

_{1}

^{⇝};

_{a}

^{→}* p

_{a}

^{⇝}= p

_{a}

^{→}, p

_{a}

^{⇝}* p

_{a}

^{→}= p

_{a}

^{→};

_{a}

^{→}* p

_{b}

^{⇝}= (1, 1, 1, 1) = p

_{c}

^{→}, p

_{b}

^{⇝}* p

_{a}

^{→}= (1, 1, 1, 1) = p

_{c}

^{→};

_{a}

^{⇝}* p

_{b}

^{→}= p

_{b}

^{→}* p

_{a}

^{⇝}= (1, 1, c, 1);

_{a}

^{⇝}* p

_{ab}

^{→}= p

_{ab}

^{→}, p

_{ab}

^{→}* p

_{a}

^{⇝}= p

_{ab}

^{→}; p

_{b}

^{→}* p

_{b}

^{⇝}= p

_{b}

^{⇝}, p

_{b}

^{⇝}* p

_{b}

^{→}= p

_{b}

^{⇝};

_{ab}

^{→}* p

_{b}

^{⇝}= (1, 1, 1, 1) = p

_{c}

^{→}, p

_{b}

^{⇝}* p

_{ab}

^{→}= (1, 1, 1, 1) = p

_{c}

^{→};

_{a}

^{→}* p

_{ba}

^{⇝}= (1, 1, 1, 1) = p

_{c}

^{→}, p

_{ba}

^{⇝}* p

_{a}

^{→}= (1, 1, 1, 1) = p

_{c}

^{→};

_{b}

^{→}* p

_{ba}

^{⇝}= p

_{ba}

^{⇝}, p

_{ba}

^{⇝}* p

_{b}

^{→}= p

_{ba}

^{⇝};

_{ab}

^{→}* p

_{ba}

^{⇝}= (1, 1, 1, 1) = p

_{c}

^{→}, p

_{ba}

^{⇝}* p

_{ab}

^{→}= (1, 1, 1, 1) = p

_{c}

^{→}.

_{a}

^{→}, p

_{a}

^{⇝}, p

_{b}

^{→}, p

_{b}

^{⇝}, p

_{ab}

^{→}, p

_{ba}

^{⇝}, p, p

_{c}

^{→}, p

_{1}

^{→}}, and Table 17 is its Cayley table (it is a non-commutative semigroup, but it is not a neutrosophic triplet group).

**Example**

**11.**

_{a}

^{→}* p

_{a}

^{→}= p

_{a}

^{→}, p

_{a}

^{→}* p

_{b}

^{→}= p

_{a}

^{→}* p

_{c}

^{→}= (1, 1, 1, 1, 1), p

_{a}

^{→}* p

_{d}

^{→}= p

_{a}

^{→}, p

_{a}

^{→}* p

_{1}

^{→}= p

_{a}

^{→};

_{b}

^{→}* p

_{a}

^{→}= (1, 1, 1, 1, 1), p

_{b}

^{→}* p

_{b}

^{→}= p

_{b}

^{→}* p

_{c}

^{→}=p

_{b}

^{→}, p

_{b}

^{→}* p

_{d}

^{→}= (1, 1, 1, 1, 1), p

_{b}

^{→}* p

_{1}

^{→}= p

_{b}

^{→};

_{c}

^{→}* p

_{a}

^{→}= (1, 1, 1, 1, 1), p

_{c}

^{→}* p

_{b}

^{→}= p

_{c}

^{→}* p

_{c}

^{→}=p

_{c}

^{→}, p

_{c}

^{→}* p

_{d}

^{→}= (1, 1, 1, 1, 1), p

_{c}

^{→}* p

_{1}

^{→}= p

_{b}

^{→};

_{d}

^{→}* p

_{a}

^{→}= p

_{d}

^{→}, p

_{d}

^{→}* p

_{b}

^{→}= p

_{d}

^{→}* p

_{c}

^{→}= (1, 1, 1, 1, 1), p

_{d}

^{→}* p

_{d}

^{→}= p

_{d}

^{→}, p

_{d}

^{→}* p

_{1}

^{→}= p

_{d}

^{→}.

_{ab}

^{→}= p

_{a}

^{→}* p

_{b}

^{→}= (1, 1, 1, 1, 1), then p

_{ab}

^{→}* p

_{a}

^{→}= p

_{ab}

^{→}* p

_{b}

^{→}= p

_{ab}

^{→}* p

_{c}

^{→}= p

_{ab}

^{→}* p

_{d}

^{→}= p

_{ab}

^{→}* p

_{ab}

^{→}= p

_{ab}

^{→}* p

_{1}

^{→}= p

_{ab}

^{→}. Hence, M

^{→}(X) = {p

_{a}

^{→}, p

_{b}

^{→}, p

_{ab}

^{→}, p

_{1}

^{→}} and its Cayley table is Table 20. Obviously, (M

^{→}(X), *) is a commutative neutrosophic triplet group.

_{a}

^{⇝}= (1, b, c, 1, 1), p

_{b}

^{⇝}= (d, 1, 1, d, 1), p

_{c}

^{⇝}= (d, 1, 1, d, 1), p

_{d}

^{⇝}= (1, b, c, 1, 1), p

_{1}

^{⇝}= (a, b, c, d, 1).

_{a}

^{⇝}* p

_{a}

^{⇝}= p

_{a}

^{⇝}, p

_{a}

^{⇝}* p

_{b}

^{⇝}= p

_{a}

^{⇝}* p

_{c}

^{⇝}= (1, 1, 1, 1, 1), p

_{a}

^{⇝}* p

_{d}

^{⇝}= p

_{a}

^{⇝};

_{b}

^{⇝}* p

_{a}

^{⇝}= (1, 1, 1, 1, 1), p

_{b}

^{⇝}* p

_{b}

^{⇝}= p

_{b}

^{⇝}* p

_{c}

^{⇝}= p

_{b}

^{⇝}, p

_{b}

^{⇝}* p

_{d}

^{⇝}= (1, 1, 1, 1, 1).

_{ab}

^{⇝}= p

_{a}

^{⇝}* p

_{b}

^{⇝}= (1, 1, 1, 1, 1), then M

^{⇝}(X) = {p

_{a}

^{⇝}, p

_{b}

^{⇝}, p

_{ab}

^{⇝}, p

_{1}

^{⇝}} and its Cayley table is Table 21. Obviously, (M

^{⇝}(X), *) is a commutative neutrosophic triplet group.

_{b}

^{→}= p

_{c}

^{→}= (d, 1, 1, d, 1) = p

_{b}

^{⇝}= p

_{c}

^{⇝}, p

_{a}

^{→}= p

_{d}

^{→}= (1, c, c, 1, 1), p

_{a}

^{⇝}= p

_{d}

^{⇝}= (1, b, c, 1, 1);

_{a}

^{→}* p

_{a}

^{⇝}= p

_{a}

^{→}, p

_{a}

^{⇝}* p

_{a}

^{→}= p

_{a}

^{→}; p

_{a}

^{→}* p

_{b}

^{⇝}= (1, 1, 1, 1, 1) = p

_{ab}

^{→}= p

_{ab}

^{⇝}, p

_{b}

^{⇝}* p

_{a}

^{→}= (1, 1, 1, 1, 1).

_{a}

^{→}, p

_{a}

^{⇝}, p

_{b}

^{→}, p

_{ab}

^{→}, p

_{1}

^{→}}, and Table 22 is its Cayley table (it is a commutative neutrosophic triplet group).

**Remark**

**3.**

^{→}(X), *), (M

^{⇝}(X), *), and (M(X), *) are usually three different semi-groups; (2) (M

^{→}(X), *) and (M

^{⇝}(X), *) are all sub-semi-groups of (M(X), *), which can also be proved from their definitions; (3) (M

^{→}(X), *), (M

^{⇝}(X), *), and (M(X), *) may be neutrosophic triplet groups. Under what circumstances they will become neutrosophic triplet groups, will be examined in the next study.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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* | a | b | c | d |

a | a | a | a | d |

b | c | a | b | c |

c | c | b | d | a |

d | a | d | b | a |

* | 1 | 2 | 3 |

1 | 1 | 1 | 2 |

2 | 1 | 2 | 3 |

3 | 2 | 3 | 3 |

* | 1 | 2 | 3 | 4 |

1 | 3 | 1 | 1 | 3 |

2 | 4 | 2 | 2 | 4 |

3 | 1 | 3 | 3 | 4 |

4 | 3 | 4 | 4 | 2 |

→ | a | b | c | 1 |

a | 1 | b | b | 1 |

b | a | 1 | a | 1 |

c | 1 | 1 | 1 | 1 |

1 | a | b | c | 1 |

→ | a | b | c | 1 |

a | 1 | b | c | 1 |

b | a | 1 | c | 1 |

c | a | b | 1 | 1 |

1 | a | b | c | 1 |

→ | a | b | c | d | e | 1 |

a | 1 | 1 | c | c | c | 1 |

b | 1 | 1 | c | c | c | 1 |

c | d | 1 | 1 | a | b | c |

d | c | c | 1 | 1 | 1 | c |

e | c | c | 1 | 1 | 1 | c |

1 | a | b | c | d | e | 1 |

→ | a | b | c | 1 |

a | 1 | 1 | 1 | 1 |

b | c | 1 | a | 1 |

c | 1 | 1 | 1 | 1 |

1 | a | b | c | 1 |

* | p_{a} | p_{b} | p_{bb} | p_{1} |

p_{a} | p_{a} | p_{a} | p_{a} | p_{a} |

p_{b} | p_{a} | p_{bb} | p_{b} | p_{b} |

p_{bb} | p_{a} | p_{b} | p_{bb} | p_{bb} |

p_{1} | p_{a} | p_{b} | p_{bb} | p_{1} |

→ | a | b | 1 |

a | 1 | a | b |

b | b | 1 | a |

1 | a | b | 1 |

* | p_{a} | p_{b} | p_{1} |

p_{a} | p_{b} | p_{1} | p_{a} |

p_{b} | p_{1} | p_{a} | p_{b} |

p_{1} | p_{a} | p_{b} | p_{1} |

→ | a | b | c | 1 |

a | 1 | 1 | b | 1 |

b | a | 1 | c | 1 |

c | 1 | 1 | 1 | 1 |

1 | a | b | c | 1 |

⇝ | a | b | c | 1 |

a | 1 | 1 | a | 1 |

b | a | 1 | a | 1 |

c | 1 | 1 | 1 | 1 |

1 | a | b | c | 1 |

→ | a | b | c | 1 |

a | 1 | b | b | 1 |

b | a | 1 | c | 1 |

c | 1 | 1 | 1 | 1 |

1 | a | b | c | 1 |

⇝ | a | b | c | 1 |

a | 1 | b | c | 1 |

b | a | 1 | a | 1 |

c | 1 | 1 | 1 | 1 |

1 | a | b | c | 1 |

* | p_{a}^{→} | p_{b}^{→} | p_{ab}^{→} | p_{c}^{→} | p_{1}^{→} |

p_{a}^{→} | p_{a}^{→} | p_{ab}^{→} | p_{ab}^{→} | p_{c}^{→} | p_{a}^{→} |

p_{b}^{→} | p_{c}^{→} | p_{b}^{→} | p_{c}^{→} | p_{c}^{→} | p_{b}^{→} |

p_{ab}^{→} | p_{c}^{→} | p_{ab}^{→} | p_{c}^{→} | p_{c}^{→} | p_{ab}^{→} |

p_{c}^{→} | p_{c}^{→} | p_{c}^{→} | p_{c}^{→} | p_{c}^{→} | p_{c}^{→} |

p_{1}^{→} | p_{a}^{→} | p_{b}^{→} | p_{ab}^{→} | p_{c}^{→} | p_{1}^{→} |

* | p_{a}^{⇝} | p_{b}^{⇝} | p_{ba}^{⇝} | p_{c}^{⇝} | p_{1}^{⇝} |

p_{a}^{⇝} | p_{a}^{⇝} | p_{c}^{⇝} | p_{c}^{⇝} | p_{c}^{⇝} | p_{a}^{⇝} |

p_{b}^{⇝} | p_{ba}^{⇝} | p_{b}^{⇝} | p_{ba}^{⇝} | p_{c}^{⇝} | p_{b}^{⇝} |

p_{ba}^{⇝} | p_{ba}^{⇝} | p_{c}^{⇝} | p_{c}^{⇝} | p_{c}^{⇝} | p_{ba}^{⇝} |

p_{c}^{⇝} | p_{c}^{⇝} | p_{c}^{⇝} | p_{c}^{⇝} | p_{c}^{⇝} | p_{c}^{⇝} |

p_{1}^{⇝} | p_{a}^{⇝} | p_{b}^{⇝} | p_{ba}^{⇝} | p_{c}^{⇝} | p_{1}^{⇝} |

* | p_{a}^{→} | p_{a}^{⇝} | p_{b}^{→} | p_{b}^{⇝} | p_{ab}^{→} | p_{ba}^{⇝} | p | p_{c}^{→} | p_{1}^{→} |

p_{a}^{→} | p_{a}^{→} | p_{a}^{→} | p_{ab}^{→} | p_{c}^{→} | p_{ab}^{→} | p_{c}^{→} | p_{ab}^{→} | p_{c}^{→} | p_{a}^{→} |

p_{a}^{⇝} | p_{a}^{→} | p_{a}^{⇝} | p | p_{c}^{→} | p_{ab}^{→} | p_{ba}^{⇝} | p | p_{c}^{→} | p_{a}^{⇝} |

p_{b}^{→} | p_{c}^{→} | p | p_{b}^{→} | p_{b}^{⇝} | p_{c}^{→} | p_{ba}^{⇝} | p | p_{c}^{→} | p_{b}^{→} |

p_{b}^{⇝} | p_{c}^{→} | p_{ba}^{⇝} | p_{b}^{⇝} | p_{b}^{⇝} | p_{c}^{→} | p_{ba}^{⇝} | p_{ba}^{⇝} | p_{c}^{→} | p_{b}^{⇝} |

p_{ab}^{→} | p_{c}^{→} | p_{ab}^{→} | p_{ab}^{→} | p_{c}^{→} | p_{c}^{→} | p_{c}^{→} | p_{ab}^{→} | p_{c}^{→} | p_{ab}^{→} |

p_{ba}^{⇝} | p_{c}^{→} | p_{ba}^{⇝} | p_{ba}^{⇝} | p_{c}^{→} | p_{c}^{→} | p_{c}^{→} | p_{ba}^{⇝} | p_{c}^{→} | p_{ba}^{⇝} |

p | p_{c}^{→} | p | p | p_{c}^{→} | p_{c}^{→} | p_{c}^{→} | p | p_{c}^{→} | p |

p_{c}^{→} | p_{c}^{→} | p_{c}^{→} | p_{c}^{→} | p_{c}^{→} | p_{c}^{→} | p_{c}^{→} | p_{c}^{→} | p_{c}^{→} | p_{c}^{→} |

p_{1}^{→} | p_{a}^{→} | p_{a}^{⇝} | p_{b}^{→} | p_{b}^{⇝} | p_{ab}^{→} | p_{ba}^{⇝} | p | p_{c}^{→} | p_{1}^{→} |

→ | a | b | c | d | 1 |

a | 1 | c | c | 1 | 1 |

b | d | 1 | 1 | d | 1 |

c | d | 1 | 1 | d | 1 |

d | 1 | c | c | 1 | 1 |

1 | a | b | c | d | 1 |

⇝ | a | b | c | d | 1 |

a | 1 | b | c | 1 | 1 |

b | d | 1 | 1 | d | 1 |

c | d | 1 | 1 | d | 1 |

d | 1 | b | c | 1 | 1 |

1 | a | b | c | d | 1 |

* | p_{a}^{→} | p_{b}^{→} | p_{ab}^{→} | p_{1}^{→} |

p_{a}^{→} | p_{a}^{→} | p_{ab}^{→} | p_{ab}^{→} | p_{a}^{→} |

p_{b}^{→} | p_{ab}^{→} | p_{b}^{→} | p_{ab}^{→} | p_{b}^{→} |

p_{ab}^{→} | p_{ab}^{→} | p_{ab}^{→} | p_{ab}^{→} | p_{ab}^{→} |

p_{1}^{→} | p_{a}^{→} | p_{b}^{→} | p_{ab}^{→} | p_{1}^{→} |

* | p_{a}^{⇝} | p_{b}^{⇝} | p_{ab}^{⇝} | p_{1}^{⇝} |

p_{a}^{⇝} | p_{a}^{⇝} | p_{ab}^{⇝} | p_{ab}^{⇝} | p_{a}^{⇝} |

p_{b}^{⇝} | p_{ab}^{⇝} | p_{b}^{⇝} | p_{ab}^{⇝} | p_{b}^{⇝} |

p_{ab}^{⇝} | p_{ab}^{⇝} | p_{ab}^{⇝} | p_{ab}^{⇝} | p_{ab}^{⇝} |

p_{1}^{⇝} | p_{a}^{⇝} | p_{b}^{⇝} | p_{ab}^{⇝} | p_{1}^{⇝} |

* | p_{a}^{→} | p_{a}^{⇝} | p_{b}^{→} | p_{ab}^{→} | p_{1}^{→} |

p_{a}^{→} | p_{a}^{→} | p_{a}^{→} | p_{ab}^{→} | p_{ab}^{→} | p_{a}^{→} |

p_{a}^{⇝} | p_{a}^{→} | p_{a}^{⇝} | p_{ab}^{→} | p_{ab}^{→} | p_{a}^{⇝} |

p_{b}^{→} | p_{ab}^{→} | p_{ab}^{→} | p_{b}^{→} | p_{ab}^{→} | p_{b}^{→} |

p_{ab}^{→} | p_{ab}^{→} | p_{ab}^{→} | p_{ab}^{→} | p_{ab}^{→} | p_{ab}^{→} |

p_{1}^{→} | p_{a}^{→} | p_{a}^{⇝} | p_{b}^{→} | p_{ab}^{→} | p_{1}^{→} |

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**MDPI and ACS Style**

Zhang, X.; Wu, X.; Smarandache, F.; Hu, M.
Left (Right)-Quasi Neutrosophic Triplet Loops (Groups) and Generalized BE-Algebras. *Symmetry* **2018**, *10*, 241.
https://doi.org/10.3390/sym10070241

**AMA Style**

Zhang X, Wu X, Smarandache F, Hu M.
Left (Right)-Quasi Neutrosophic Triplet Loops (Groups) and Generalized BE-Algebras. *Symmetry*. 2018; 10(7):241.
https://doi.org/10.3390/sym10070241

**Chicago/Turabian Style**

Zhang, Xiaohong, Xiaoying Wu, Florentin Smarandache, and Minghao Hu.
2018. "Left (Right)-Quasi Neutrosophic Triplet Loops (Groups) and Generalized BE-Algebras" *Symmetry* 10, no. 7: 241.
https://doi.org/10.3390/sym10070241