Single–valued neutrosophic filters in EQ

Abstract

This paper introduces the concept of single–valued neutrosophic EQ–subalgebras, single–valued neutrosophic EQ–prefilters and single–valued neutrosophic EQ–filters. We study some properties of single–valued neutrosophic EQ–prefilters and show how to construct single–valued neutrosophic EQ–filters. Finally, the relationship between single–valued neutrosophic EQ–filters and EQ–filters are studied.

1 Introduction

EQ-algebra as an alternative to residuated lattices is a special algebra that was presented for the first time by V. Novák [10, 11]. Its original motivation comes from fuzzy type theory, in which the main connective is fuzzy equality and stems from the equational style of proof in logic [15]. EQ-algebras are intended to become algebras of truth values for fuzzy type theory (FTT) where the main connective is a fuzzy equality. Every EQ–algebra has three operations meet “∧”, multiplication “⊗”, and fuzzy equality “∼” and a unit element, while the implication “→” is derived from fuzzy equality “∼”. This basic structure in fuzzy logic is ordering, represented by ∧–semilattice, with maximal element “1”. Further materials regarding EQ–algebras are available in the literature too [6 , 12]. Algebras including EQ-algebras have played an important role in recent years and have had its comprehensive applications in many aspects including dynamical systems and genetic code of biology [2]. From the point of view of logic, the main difference between residuated lattices and EQ–algebras lies in the way the implication operation is obtained. While in residuated lattices it is obtained from (strong) conjunction, in EQ–algebras it is obtained from equivalence. Consequently, the two kinds of algebras differ in several essential points despite their many similar or identical properties.

Filter theory plays an important role in studying various logical algebras. From logical point of view, filters correspond to sets of provable formulae. Filters are very important in the proof of the completeness of various logic algebras. Many researchers have studied the filter theory of various logical algebras[3 –5].

Neutrosophy, as a newly born science, is a branch of philosophy that studies the origin, nature and scope of neutralities, as well as their interactions with different ideational spectra. It can be defined as the incidence of the application of a law, an axiom, an idea, a conceptual accredited construction on an unclear, indeterminate phenomenon, contradictory to the purpose of making it intelligible. Neutrosophic set and neutrosophic logic are generalizations of the fuzzy set and respectively fuzzy logic (especially of intuitionistic fuzzy set and respectively intuitionistic fuzzy logic) are tools for publications on advanced studies in neutrosophy. In neutrosophic logic, a proposition has a degree of truth (T), indeterminacy (I) and falsity (F), where T, I, F are standard or non–standard subsets of ]^-0, 1⁺[. In 1995, Smarandache talked for the first time about neutrosophy and in 1999 and 2005 [14] he initiated the theory of neutrosophic set as a new mathematical tool for handling problems involving imprecise, indeterminacy, and inconsistent data. Alkhazaleh et al. generalized the concept of fuzzy soft set to neutrosophic soft set and they gave some applications of this concept in decision making and medical diagnosis [1] and Borumand et al. applied the concept of neutrosophic logic in filters in BE-algebras [13].

Regarding these points, this paper aims to introduce the notation of single–valued neutrosophic EQ–subalgebras and single–valued neutrosophic EQ–filters. We investigate some properties of single–valued neutrosophic EQ–subalgebras and single–valued neutrosophic EQ–filters and prove them. Indeed show that how to construct single–valued neutrosophic EQ–subalgebras and single–valued neutrosophic EQ–filters. We applied the concept of homomorphisms in EQ–algebras and with this regard, new single–valued neutrosophic EQ–subalgebras and single–valued neutrosophic EQ–filters are generated.

2 Preliminaries

In this section, we recall some definitions and results are indispensable to our research paper.

Definition 2.1. [8] An algebra $E = (E, \land, \otimes, \sim, 1)$ of type (2, 2, 2, 0) is called an EQ–algebra, if for all x, y, z, t ∈ E:

(E, ∧ , 1) is a commutative idempotent monoid (i.e. ∧–semilattice with top element “1”);

(E, ⊗ , 1) is a monoid and ⊗ is isotone w.r.t. “≤” (where x ≤ y is defined as x ∧ y = x);

x ∼ x = 1; (reflexivity axiom)

((x ∧ y) ∼ z) ⊗ (t ∼ x) ≤ z ∼ (t ∧ y); (substitution axiom)

(x ∼ y) ⊗ (z ∼ t) ≤ (x ∼ z) ∼ (y ∼ t); (congruence axiom)

(x ∧ y ∧ z) ∼ x ≤ (x ∧ y) ∼ x; (monotonicity axiom)

x ⊗ y ≤ x ∼ y, (boundedness axiom).

The binary operation “∧” is called meet (infimum), “⊗” is called multiplication and “∼” is called fuzzy equality. (E, ∧ , ⊗ , ∼ , 1) is called a separated EQ–algebra if 1 = x ∼ y, implies that x = y.

Proposition 2.2. [8] Let $E$ be an EQ–algebra, x → y : = (x ∧ y) ∼ x and $\tilde{x} = x \sim 1$ . Then for all x, y, z ∈ E, the following properties hold:

x ⊗ y ≤ x, y, x ⊗ y ≤ x ∧ y;

x ∼ y = y ∼ x;

(x ∧ y) ∼ x ≤ (x ∧ y ∧ z) ∼ (x ∧ z);

x → x = 1;

(x ∼ y) ⊗ (y ∼ z) ≤ x ∼ z;

(x → y) ⊗ (y → z) ≤ x → z;

$x \leq \tilde{x}, \tilde{1} = 1$ .

Proposition 2.3. [8] Let $E$ be an EQ–algebra. Then for all x, y, z ∈ E, the following properties hold:

$x \otimes (x \sim y) \leq \bar{y}$ ;

(z → (x ∧ y)) ⊗ (x ∼ t) ≤ z → (t ∧ y);

(y → z) ⊗ (x → y) ≤ x → z;

(x → y) ⊗ (y → x) ≤ x ∼ y;

if x ≤ y → z, then $x \otimes y \leq \bar{z}$ ;

if x ≤ y ≤ z, then z ∼ x ≤ z ∼ y and x ∼ z ≤ x ∼ y;

x → (y → x) =1.

Definition 2.4. [8] Let $E = (E, \land, \otimes, \sim, 1)$ be a separated EQ–algebra. A subset F of E is called an EQ–filter of E if for all a, b, c ∈ E it holds that

1 ∈ F,

if a, a → b ∈ F, then b ∈ F,

if a → b ∈ F, then a ⊗ c → b ⊗ c ∈ F and c ⊗ a → c ⊗ b ∈ F.

Theorem 2.5. [8] Let F be a prefilter of separated EQ–algebra $E$ . Then for all a, b, c ∈ E it holds that

if a ∈ F and a ≤ b, then b∈ F ;

if a, a ∼ b ∈ F, then b ∈ F;

If a, b ∈ F, then a ∧ b ∈ F;

If a ∼ b ∈ F and b ∼ c ∈ F then a ∼ c ∈ F.

Definition 2.6. [17] Let $E$ be an EQ–algebras. A fuzzy subset μ of E is called a fuzzy prefilter of $E$ , if for all x, y, z ∈ E:

ν (1) ≥ ν (x);

ν (y) ≥ ν ((x ∧ y) ∼ y) ∧ ν (x).

A fuzzy EQ–prefilter is called a fuzzy EQ–filter if it satisfies:

ν ((x ∧ y) ∼ y) ≤ ν ((x ⊗ z) ∧ (y ⊗ z)) ∼ (y ⊗ z)).

Definition 2.7. [16] Let X be a set. A single valued neutrosophic set A in X (SVN–S A) is a function A : X → [0, 1] × [0, 1] × [0, 1] with the form A = {(x, T_A (x) , I_A (x) , F_A (x)) | x ∈ X} where the functions T_A, I_A, F_A define respectively the truth–membership function, an indeterminacy–membership function, and a falsity–membership function of the element x ∈ X to the set A such that 0 ≤ T_A (x) + I_A (x) + F_A (x) ≤3. Moreover, Supp (A) = {x | T_A (x) ≠0, I_A (x) ≠0, F_A (x) ≠0} is a crisp set.

3 Single–Valued Neutrosophic EQ–subalgebras

In this section, we introduce the concept of single–valued neutrosophic EQ–subalgebra and prove some their properties.

Definition 3.1. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra. A map A in E, is called a single–valued neutrosophic EQ–subalgebra of $E$ , if for all x, y ∈ E,

T_A (x ∧ y) = T_A (x) ∧ T_A (y) , I_A (x ∧ y) = I_A (x) ∧ I_A (y) and F_A (x ∧ y) = F_A (x) ∨ F_A (y) ,

T_A (x ∼ y) ≥ T_A (x) ∧ T_A (y) , I_A (x ∼ y) ≥ I_A (x) ∧ I_A (y) and F_A (x ∼ y) ≤ F_A (x) ∨ F_A (y) .

From now on, when we say $(E, A)$ is a single–valued neutrosophic EQ–subalgebra, means that $E = (E, \land, \otimes, \sim, 1)$ is an EQ–algebra and A is a single–valued neutrosophic EQ–subalgebra of $E$ .

Theorem 3.2. Let $(E, A)$ be a single–valued neutrosophic EQ–subalgebra. Then for all x, y ∈ H,

if x ≤ y, then T_A (x) ≤ T_A (y),

if x ≤ y, then I_A (x) ≤ I_A (y) ,

if x ≤ y, then F_A (x) ≥ F_A (y),

T_A (x) ≤ T_A (1) , I_A (x) ≤ I_A (1) and F_A (x) ≥ F_A (1),

T_A (x ⊗ y) ≤ T_A (x) ∧ T_A (y),

I_A (x ⊗ y) ≤ I_A (x) ∧ T_A (y),

F_A (x ⊗ y) ≥ F_A (x) ∨ F_A (y),

T_A (x → y) ≥ T_A (x) ∧ T_A (y),

I_A (x → y) ≥ I_A (x) ∧ I_A (y),

F_A (x → y) ≤ F_A (x) ∨ F_A (y).

Proof. (i), (ii), (iii), (iv) Let x, y ∈ E. Since x ≤ y, we get that x ∧ y = x and so T_A (x) ∧ T_A (y) = T_A (x ∧ y) = T_A (x). It follows that T_A (x) ≤ T_A (y). In a similar way I_A (x) ≤ I_A (y) and F_A (x) ≥ F_A (y) are obtained.

(v) , (vi) , (vii) By the previous items, for all x, y ∈ E, x ⊗ y ≤ x ∧ y implies that T_A (x ⊗ y) ≤ T_A (x) ∧ T_A (y), I_A (x ⊗ y) ≤ I_A (x) ∧ I_A (y) and F_A (x ⊗ y) ≥ F_A (x) ∨ F_A (y).

(viii) , (ix) , (x) Since (x ∼ y) ≤ (x → y), by the previous items we get that T_A (x → y) ≥ T_A (x) ∧ T_A (y), I_A (x → y) ≥ I_A (x) ∧ T_A (y) and F_A (x → y) ≤ F_A (x) ∨ F_A (y).

Example 3.3. Let E = {a₁, a₂, a₃, a₄, a₅, a₆}. Define operations “⊗, ∼” and “∧” on E as follows: $\begin{matrix} \begin{matrix} \land | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ a_{1} | & a_{1} & a_{1} & a_{1} & a_{1} & a_{1} & a_{1} \\ a_{2} | & a_{1} & a_{2} & a_{2} & a_{2} & a_{2} & a_{2} \\ a_{3} | & a_{1} & a_{2} & a_{3} & a_{3} & a_{3} & a_{3} \\ a_{4} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{4} & a_{4} \\ a_{5} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{5} \\ a_{6} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \end{matrix}, \end{matrix} \begin{matrix} \otimes | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ a_{1} | & a_{1} & a_{1} & a_{1} & a_{1} & a_{1} & a_{1} \\ a_{2} | & a_{1} & a_{1} & a_{1} & a_{1} & a_{1} & a_{2} \\ a_{3} | & a_{1} & a_{1} & a_{1} & a_{1} & a_{2} & a_{3} \\ a_{4} | & a_{1} & a_{1} & a_{1} & a_{2} & a_{2} & a_{4} \\ a_{5} | & a_{1} & a_{1} & a_{2} & a_{2} & a_{2} & a_{5} \\ a_{6} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \end{matrix} and$ $\begin{matrix} \begin{matrix} \sim | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ a_{1} | & a_{6} & a_{4} & a_{3} & a_{2} & a_{1} & a_{1} \\ a_{2} | & a_{4} & a_{6} & a_{3} & a_{2} & a_{2} & a_{2} \\ a_{3} | & a_{3} & a_{3} & a_{6} & a_{3} & a_{3} & a_{3} \\ a_{4} | & a_{2} & a_{2} & a_{3} & a_{6} & a_{4} & a_{4} \\ a_{5} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{6} & a_{5} \\ a_{6} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \end{matrix} . \end{matrix}$ Then $E = (E, \land, \otimes, \sim, a_{6})$ is an EQ–algebra.

Define a single valued neutrosophic set map A in E as follows: $\begin{matrix} \begin{matrix} T_{A} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ | & 0.22 & 0.33 & 0.44 & 0.55 & 0.66 & 0.77 \end{matrix}, \end{matrix}$ $\begin{matrix} \begin{matrix} I_{A} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ | & 0.21 & 0.31 & 0.41 & 0.51 & 0.61 & 0.71 \end{matrix} \end{matrix}$ and $\begin{matrix} \begin{matrix} F_{A} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ | & 0.98 & 0.88 & 0.78 & 0.68 & 0.58 & 0.48 \end{matrix} \end{matrix}$ Hence $(A, E)$ is a single–valued neutrosophic EQ–subalgebra.

Corollary 3.4. Let $(E, A)$ be a single–valued neutrosophic EQ–subalgebra. Then for all x, y ∈ H,

if x ≤ y, then T_A (y → x) = T_A (x ∼ y) ,

if x ≤ y, then I_A (y → x) = I_A (x ∼ y) ,

if x ≤ y, then F_A (y → x) = F_A (x ∼ y) .

3.1 Single–Valued Neutrosophic EQ–prefilters

In this section, we introduce the concept of single–valued neutrosophic EQ–prefilters and show how to construct of single–valued neutrosophic EQ–prefilters.

Definition 3.5. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra. A map A in E, is called a single–valued neutrosophic EQ–prefilter of $E$ , if for all x, y ∈ E,

T_A (x) ≤ T_A (1) , I_A (x) ≥ I_A (1) and F_A (x) ≤ F_A (1) ,

∧ {T_A (x) , T_A (x → y)} ≤ T_A (y) , ∨ {I_A (x) , I_A (x → y)} ≥ I_A (y) and ∧ {F_A (x) , F_A (x → y)} ≤ F_A (y).

In the following theorem, we will show that how to construct of single–valued neutrosophic EQ–prefilters in EQ–algebras.

Theorem 3.6. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra, A be a single–valued neutrosophic EQ–prefilter of $E$ and x, y ∈ E.

If x ≤ y, then ∧ {T_A (x) , T_A (x → y)} = T_A (x),

If x ≤ y, then ∨ {I_A (x) , I_A (x → y)} = I_A (x),

If x ≤ y, then ∧ {F_A (x) , F_A (x → y)} = F_A (x),

If x ≤ y, then T_A (x) ≤ T_A (y) and F_A (x) ≤ F_A (y),

If x ≤ y, then I_A (y) ≤ I_A (x).

Proof. (i) , (ii) , (iii) Since x ≤ y we get that x → y = 1, so by definition, ∧ {T_A (x) , T_A (x → y)} = T_A (x) , ∨ {I_A (x) , I_A (x → y)} = I_A (x) and ∧ {F_A (x) , F_A (x → y)} = F_A (x).(iv) Since x ≤ y, by (i) we have ∧ {T_A (x) , T_A (x → y)} = T_A (x). So by definition we get T_A (x) = ∧ {T_A (x) , T_A (x → y)} ≤ T_A (y). In a similar way x ≤ y implies that F_A (x) ≤ F_A (x). (v) Since x ≤ y, by (ii) we have ∨ {I_A (x) , I_A (x → y)} = I_A (x). Thus by definition we get I_A (y) ≤ ∨ {I_A (x) , I_A (x → y)} = I_A (x) and it follows that I_A (x) ≥ I_A (y).□

Corollary 3.7. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra, A be a single–valued neutrosophic EQ–prefilter of $E$ and 0 ∈ E. If for every y ∈ E, 0 ∧ y = 0, then

∧ {T_A (0) , T_A (0 → y)} = T_A (0), ∨ {I_A (0) , I_A (0 → y)} = I_A (0),

$\land {T_{A} (1), T_{A} (1 \to y)} = T_{A} (\bar{y})$ , $\lor {I_{A} (1), I_{A} (1 \to y)} = I_{A} (\bar{y})$ ,

∧ {T_A (y) , T_A (y → 1)} = T_A (y), ∨ {I_A (y) , I_A (y → 1)} = I_A (y),

∧ {T_A (y) , T_A (y → y)} = T_A (y), ∨ {I_A (y) , I_A (y → y)} = I_A (y),

T_A (0) ≤ T_A (1) and I_A (1) ≤ I_A (0),

T_A (x) ≤ T_A (y → x) and I_A (x → y) ≥ I_A (y),

T_A (x ⊗ y) ≤ T_A (y ∼ x) and I_A (x ⊗ y) ≥ I_A (y ∼ x).

Example 3.8. Let E = {a, b, c, d, 1}. Define operations “⊗, ∼” and an operation “∧” on E as follows:

\begin{matrix} \land | & a & b & c & d & 1 \\ a | & a & a & a & a & a \\ b | & a & b & b & b & b \\ c | & a & b & c & c & c \\ d | & a & b & c & d & d \\ 1 | & a & b & c & d & 1 \end{matrix}, \begin{matrix} \otimes | & a & b & c & d & 1 \\ a | & a & a & a & a & a \\ b | & a & a & a & a & b \\ c | & a & a & a & c & c \\ d | & a & a & a & d & d \\ 1 | & a & b & c & d & 1 \end{matrix} and \begin{matrix} \sim | & a & b & c & d & 1 \\ a | & 1 & b & a & a & a \\ b | & b & 1 & b & b & b \\ c | & a & b & 1 & c & c \\ d | & a & b & c & 1 & d \\ 1 | & a & b & c & d & 1 \end{matrix} .

Then

E = (E, \land, \otimes, \sim, 1)

is an EQ–algebra and obtain the operation “→” as follows: Define a single valued neutrosophic set map A in E as follows:

\begin{matrix} \begin{matrix} \to | & a & b & c & d & 1 \\ a | & 1 & 1 & 1 & 1 & 1 \\ b | & b & 1 & 1 & 1 & 1 \\ c | & a & b & 1 & 1 & 1 \\ d | & a & b & c & 1 & 1 \\ 1 | & a & b & c & d & 1 \end{matrix} \end{matrix}

\begin{matrix} \begin{matrix} T_{A} | & a & b & c & d & 1 \\ | & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 \end{matrix}, \end{matrix}

\begin{matrix} \begin{matrix} F_{A} | & a & b & c & d & 1 \\ | & 0.55 & 0.45 & 0.35 & 0.25 & 0.15 \end{matrix} \end{matrix}

and

\begin{matrix} \begin{matrix} I_{A} | & a & b & c & d & 1 \\ | & 0.17 & 0.27 & 0.37 & 0.47 & 0.57 \end{matrix} \end{matrix}

Hence A is a single–valued neutrosophic EQ–prefilter of

E

Theorem 3.9. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra, A be a single–valued neutrosophic EQ–prefilter of $E$ and x, y ∈ E. Then

∧ {T_A (x) , T_A (x ∼ y)} ≤ T_A (y) and (I_A (x) ∨ I_A (x ∼ y)) ≥ I_A (y),

∧ {T_A (x) , T_A (x ⊗ y)} ≤ T_A (y) and (I_A (x) ∨ I_A (x ⊗ y)) ≥ I_A (y),

∧ {T_A (x) , T_A (x ∧ y)} ≤ T_A (y) and (I_A (x) ∨ I_A (x ∧ y)) ≥ I_A (y),

T_A (x) ∧ T_A (y) ≤ T_A (x) ∧ T_A (x → y),

I_A (x) ∨ I_A (x → y) ≤ I_A (x) ∨ I_A (y),

T_A (x ⊗ y) ≤ T_A (x) ∧ T_A (x),

I_A (x ⊗ y) ≥ I_A (x) ∨ I_A (x).

Proof. (i) , (ii) , (iii) Let x, y ∈ E. Since x ∼ y ≤ x → y and T_A ia a monotone map, we get that T_A (x ∼ y) ≤ T_A (x → y). Hence

\begin{matrix} \land {T_{A} (x), T_{A} (x \sim y)} & \leq & \land {T_{A} (x), T_{A} (x \to y)} \\ \leq & T_{A} (y) . \end{matrix}

In addition, since I_A is an antimonotone map, x ∼ y ≤ x → y concludes that I_A (x ∼ y) ≥ I_A (x → y). Hence ∨ {I_A (x) , I_A (x ∼ y)} ≥ ∨ {I_A (x) , I_A (x → y)} ≥ I_A (y) . In a similar way x ∧ y ≤ y and x ⊗ y ≤ x → y, imply that ∧ {T_A (x) , T_A (x ⊗ y)} ≤ T_A (y), ∧ {T_A (x) , T_A (x ∧ y)} ≤ T_A (y), (I_A (x) ∨ I_A (x ⊗ y)) ≥ I_A (y) and (I_A (x) ∨ I_A (x ∧ y)) ≥ I_A (y). (iv) , (v) Let x, y ∈ E. Since y ≤ (x → y), we get that

(T_{A} (x) \land T_{A} (y)) \leq (T_{A} (x) \land T_{A} (x \to y)) \leq T_{A} (y) .

In a similar way we conclude that I_A (y) ≤ (I_A (x) ∨ I_A (x → y)) ≤ I_A (x) ∨ I_A (y).(vi) , (vii) Since x ⊗ y ≤ (x ∧ y) and T_A is a monotone map, then we get that T_A (x ⊗ y) ≤ T_A (x ∧ y) ≤ T_A (x) ∧ T_A (y). In a similar way since I_A is an antimonotone map, then we get that I_A (x ⊗ y) ≥ T_A (x ∧ y) ≥ I_A (x) ∨ I_A (y).□

Corollary 3.10. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra, A be a single–valued neutrosophic EQ–prefilter of $E$ and x, y ∈ E. Then

∧ {F_A (x) , F_A (x ∼ y)} ≤ F_A (y),

∧ {F_A (x) , F_A (x ⊗ y)} ≤ F_A (y),

∧ {F_A (x) , F_A (x ∧ y)} ≤ F_A (y),

F_A (x) ∧ F_A (y) ≤ F_A (x) ∧ F_A (x → y),

F_A (x ⊗ y) ≤ F_A (x) ∧ F_A (x).

Theorem 3.11. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra, A be a single–valued neutrosophic EQ–prefilter of $E$ and x, y, z ∈ E.

If x ≤ y, then T_A (x) ∧ T_A (x ∼ y) = T_A (x) ∧ T_A (y → x),

If x ≤ y, then T_A (z) ∧ T_A (z → x) ≤ T_A (y),

If x ≤ y, then T_A (x) ∧ T_A (y → z) = T_A (x) ∧ T_A (z),

If x ≤ y, then I_A (x) ∨ I_A (x ∼ y) = I_A (x) ∨ I_A (y → x),

If x ≤ y, then I_A (z) ∨ I_A (z → x) = I_A (x) ∨ I_A (z),

If x ≤ y, then I_A (x) ∨ I_A (y → z) = I_A (x) ∨ I_A (z).

Proof. (i) Let x, y ∈ E. Then x ≤ y follows that x ∼ y = y → x and so T_A (x) ∧ T_A (x ∼ y) = T_A (x) ∧ T_A (y → x). (ii) Let x, y, z ∈ E. Since z → x ≤ z → y, we get that T_A (z → x) ≤ T_A (z → y) and so T_A (z) ∧ T_A (z → x) ≤ T_A (z) ∧ T_A (z → y) ≤ T_A (y). (iii) Let x, y, z ∈ E. Since y → z ≤ x → z, we get that T_A (y → z) ≤ T_A (x → z) and so T_A (x) ∧ T_A (y → z) ≤ T_A (x) ∧ T_A (x → z) ≤ T_A (z). Moreover, z ≤ y → z implies that T_A (z) ≤ T_A (y → z), hence T_A (z) ∧ T_A (x) ≤ T_A (x) ∧ T_A (y → z) ≤ T_A (z) ∧ T_A (x) and so T_A (x) ∧ T_A (y → z) = T_A (z) ∧ T_A (x). (v) Let x, y, z ∈ E. Since z → x ≤ z → y, we get that I_A (z → y) ≤ I_A (z → x) and so I_A (z) ∨ I_A (z → y) ≤ I_A (z) ∨ I_A (z → x). Moreover, x ≤ y implies that I_A (x) ∨ I_A (y) = I_A (x), hence by Theorem 3.9, I_A (z) ∨ I_A (x) ∨ I_A (y) ≤ I_A (z) ∨ I_A (z → x) ≤ T_A (x) ∨ I_A (z) and so T_A (z) ∧ I_A (z → x) = I_A (z) ∨ I_A (x). (iv) and (vi) in a similar way are obtained.□

Corollary 3.12. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra, A be a single–valued neutrosophic EQ–prefilter of $E$ and x, y, z ∈ E.

If x ≤ y, then F_A (x) ∧ F_A (x ∼ y) = F_A (x) ∧ F_A (y → x),

If x ≤ y, then F_A (z) ∧ F_A (z → x) = F_A (x) ∧ F_A (z),

If x ≤ y, then F_A (x) ∧ F_A (y → z) = F_A (x) ∧ F_A (z).

Theorem 3.13. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra, A be a single–valued neutrosophic EQ–prefilter of $E$ and x, y, z ∈ E. Then

T_A (x ∧ y) = T_A (x) ∧ T_A (y),

T_A (x) ∧ T_A (x ∼ y) ≤ T_A (x) ∧ T_A (y),

Proof. (i) Since T_A is a monotone map, x ∧ y ≤ x and x ∧ y ≤ y, we obtain T_A (x ∧ y) ≤ T_A (x) ∧ T_A (y). In addition from y ≤ x → (x ∧ y) and Theorem 3.9, we conclude that T_A (x) ∧ T_A (y) ≤ (T_A (x) ∧ T_A (x → (x ∧ y))) ≤ T_A (x ∧ y) . Hence T_A (x ∧ y) = T_A (x) ∧ T_A (y). (ii) Let x, y ∈ E. Then by Theorem 3.9, T_A (x) ∧ T_A (x ∼ y) ≤ T_A (y). Since x ∼ y = y ∼ x, we obtain T_A (x) ∧ T_A (x ∼ y) = T_A (x) ∧ T_A (y ∼ x) ≤ T_A (x). So T_A (x) ∧ T_A (x ∼ y) ≤ T_A (x) ∧ T_A (y).□

Corollary 3.14. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra, A be a single–valued neutrosophic EQ–prefilter of $E$ and x, y, z ∈ E. Then

F_A (x ∧ y) = F_A (x) ∧ F_A (y),

F_A (x) ∧ F_A (x ∼ y) ≤ F_A (x) ∧ F_A (y),

Theorem 3.15. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra, A be a single–valued neutrosophic EQ–prefilter of $E$ and x, y ∈ E. Then

I_A (x ∧ y) = I_A (x) ∨ I_A (y),

I_A (x) ∨ I_A (x ∼ y) ≥ I_A (x ∧ y),

Proof. (i) Since I_A is an antimonotone map, x ∧ y ≤ x and x ∧ y ≤ y, we obtain I_A (x ∧ y) ≥ I_A (x) ∨ I_A (y). In addition from y ≤ x → (x ∧ y), we conclude that

I_{A} (x) \lor I_{A} (y) \geq (I_{A} (x) \lor I_{A} (x \to (x \land y))) \geq I_{A} (x \land y) .

Hence I_A (x ∧ y) = I_A (x) ∨ I_A (y).

(ii) Let x, y ∈ E. Then, I_A (x) ∨ I_A (x ∼ y) ≥ I_A (y). Since x ∼ y = y ∼ x, we obtain I_A (x) ∨ I_A (x ∼ y) = I_A (x) ∨ I_A (y ∼ x) ≥ I_A (x).

So I_A (x) ∨ I_A (x ∼ y) ≥ I_A (x) ∨ I_A (y).□

Corollary 3.16. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra, A be a single–valued neutrosophic EQ–prefilter of $E$ and x, y ∈ E. Then x = y, implies that I_A (x) ∨ I_A (x ∼ y) = I_A (x ∧ y).

In Example 3.8, for x = a and y = d, we have I_A (x) ∨ I_A (x ∼ y) = I_A (x ∧ y), while x ≠ y.

4 Single–Valued Neutrosophic EQ–filters

In this section, we introduce the concept of single–valued neutrosophic EQ–filters as generalization of single–valued neutrosophic EQ–prefilters and prove some their properties.

Definition 4.1. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra. A map A in E, is called a single–valued neutrosophic EQ–filter of $E$ , if for all x, y, z ∈ E,

T_A (x) ≤ T_A (1) , I_A (x) ≥ I_A (1) and F_A (x) ≤ F_A (1) ,

∧ {T_A (x) , T_A (x → y)} ≤ T_A (y) , ∨ {I_A (x) , I_A (x → y)} ≥ I_A (y) and ∧ {F_A (x) , F_A (x → y)} ≤ F_A (y),

T_A (x → y) ≤ T_A ((x ⊗ z) → (y ⊗ z)) , I_A (x → y) ≥ I_A ((x ⊗ z) → (y ⊗ z)) , and F_A (x → y) ≤ F_A ((x ⊗ z) → (y ⊗ z)).

In the following theorem, we will show that how to construct of single–valued neutrosophic EQ–prefilters in EQ–algebras.

Theorem 4.2. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra, A be a single–valued neutrosophic EQ–filter of $E$ and x, y ∈ E.

If T_A (x → y) = T_A (1), then for every z ∈ E, T_A ((x ⊗ z) → (y ⊗ z)) = T_A (x → y).

If x ≤ y, then for every z ∈ E, T_A ((x ⊗ z) → (y ⊗ z)) = T_A (x → y).

If T_A (x → y) = T_A (0), then for every z ∈ E, T_A ((x ⊗ z) → (y ⊗ z)) ≥ T_A (x → y).

If I_A (x → y) = I_A (1), then for every z ∈ E, I_A ((x ⊗ z) → (y ⊗ z)) = I_A (x → y).

If x ≤ y, then for every z ∈ E, I_A ((x ⊗ z) → (y ⊗ z)) = I_A (x → y).

If I_A (x → y) = I_A (0), then for every z ∈ E, I_A ((x ⊗ z) → (y ⊗ z)) ≤ I_A (x → y).

Proof. (i) , (iii) , (iv) and (vi) by definition are obtained.

(ii) Since x ≤ y we get that x → y = 1 and by definition x ⊗ z ≤ y ⊗ z. Hence by item (i), we have T_A ((x ⊗ z) → (y ⊗ z)) = T_A (x → y). (v) It is similar to the item (ii) .□

Corollary 4.3. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra, A be a single–valued neutrosophic EQ–prefilter of $E$ and 0, x, y, z ∈ E. If for every y ∈ E, 0 ∧ y = 0, Then

T_A (0 → y) = T_A ((x ⊗ z) → (y ⊗ z)),

T_A (x → x) = T_A ((x ⊗ z) → (y ⊗ z)),

T_A (x → 1) = T_A ((x ⊗ z) → (y ⊗ z)),

I_A (0 → y) = I_A ((x ⊗ z) → (y ⊗ z)),

I_A (x → x) = I_A ((x ⊗ z) → (y ⊗ z)),

I_A (x → 1) = I_A ((x ⊗ z) → (y ⊗ z)).

Corollary 4.4. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra, A be a single–valued neutrosophic EQ–filter of $E$ and x, y ∈ E.

If F_A (x → y) = F_A (1), then for every z ∈ E, F_A ((x ⊗ z) → (y ⊗ z)) = F_A (x → y),

If x ≤ y, then for every z ∈ E, F_A ((x ⊗ z) → (y ⊗ z)) = F_A (x → y).

If F_A (x → y) = F_A (0), then for every z ∈ E, F_A ((x ⊗ z) → (y ⊗ z)) ≥ F_A (x → y).

Example 4.5. Let E = {0, a, b, c, 1}. Define operations “⊗, ∼” and an operation “∧” on E as follows: $\begin{matrix} \begin{matrix} \land | & 0 & a & b & c & 1 \\ 0 | & 0 & 0 & 0 & 0 & 0 \\ a | & 0 & a & a & a & a \\ b | & 0 & a & b & - & b \\ c | & 0 & a & - & c & c \\ 1 | & 0 & a & b & c & 1 \end{matrix}, \end{matrix} \begin{matrix} \otimes | & 0 & a & b & c & 1 \\ 0 | & 0 & 0 & 0 & 0 & 0 \\ a | & 0 & 0 & 0 & a & a \\ b | & 0 & a & b & a & b \\ c | & 0 & 0 & 0 & c & c \\ 1 | & 0 & a & b & c & 1 \end{matrix} and \begin{matrix} \sim | & 0 & a & b & c & 1 \\ 0 | & 1 & 0 & 0 & 0 & 0 \\ a | & 0 & 1 & a & a & a \\ b | & 0 & a & 1 & a & b \\ c | & 0 & a & a & 1 & c \\ 1 | & 0 & a & b & c & 1 \end{matrix} .$ Then $E = (E, \land, \otimes, \sim, 1)$ is an EQ–algebra, where b and c are non-comparable. Now, obtain the operation “→” as follows: $\begin{matrix} \begin{matrix} \to | & 0 & a & b & c & 1 \\ 0 | & 1 & 1 & 1 & 1 & 1 \\ a | & 0 & 1 & 1 & 1 & 1 \\ b | & 0 & a & 1 & c & 1 \\ c | & 0 & a & b & 1 & 1 \\ 1 | & 0 & a & b & c & 1 \end{matrix} . \end{matrix}$ Define a single valued neutrosophic set map A in E as follows: $\begin{matrix} \begin{matrix} T_{A} | & 0 & a & b & c & 1 \\ | & 0.4 & 0.4 & 0.4 & 0.4 & 0.5 \end{matrix}, \end{matrix}$ $\begin{matrix} \begin{matrix} I_{A} | & 0 & a & b & c & 1 \\ | & 0.62 & 0.62 & 0.62 & 0.62 & 0.11 \end{matrix} \end{matrix}$ and $\begin{matrix} \begin{matrix} F_{A} | & 0 & a & b & c & 1 \\ | & 0.2 & 0.2 & 0.2 & 0.2 & 0.6 \end{matrix} \end{matrix}$ Hence A is a single–valued neutrosophic EQ–filter of $E$ .

Theorem 4.6. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra, A be a single–valued neutrosophic EQ–filter of $E$ and x, y ∈ E. Then

T_A (x ⊗ y) = T_A (x) ∧ T_A (y),

I_A (x ⊗ y) = I_A (x) ∨ I_A (y),

T_A (x ∼ y) ≤ T_A (y → x),

T_A (z) ∧ T_A (y) ≤ T_A (x → z),

T_A (x ∼ y) ∧ T_A (y ∼ z) ≤ T_A (x ∼ z),

I_A (x ∼ y) ≥ I_A (y → x),

I_A (z) ∨ I_A (y) ≥ I_A (x → z),

I_A (x ∼ y) ∨ I_A (y ∼ z) ≥ I_A (x ∼ z).

Proof. (i) Let x, y ∈ E. Since A is a single–valued neutrosophic EQ–filter of

E

, we get that

\begin{matrix} T_{A} (1 \to y) & \leq & T_{A} ((1 \otimes x) \to (y \otimes x)) \\ = & T_{A} (x \to (y \otimes x)) . \end{matrix}

In addition by the item (SVNF2), we have

T_{A} (x) \land T_{A} (x \to (y \otimes x)) \leq T_{A} (y \otimes x) .

Hence

\begin{matrix} T_{A} (x) \land T_{A} (y) \leq T_{A} (x) \land T_{A} (1 \to y) \leq T_{A} (y \otimes x) . \end{matrix}

We apply Theorem 3.9 and obtain T_A (x) ∧ T_A (y) = T_A (y ⊗ x) . (ii) Let x, y ∈ E. By item (SVNF2), we have

I_{A} (1 \to y) \geq I_{A} (1 \otimes x) \to (y \otimes x) .

Then I_A (x) ∨ I_A (1 → y) ≥ I_A (x) ∨ I_A (x → (y ⊗ x)) ≥ I_A (y ⊗ x) . It follows that I_A (x) ∨ I_A (y) ≥ I_A (x) ∨ I_A (1 → y) ≥ I_A (y ⊗ x). Therefore, Theorem 3.9 implies that I_A (x) ∨ I_A (y) = I_A (y ⊗ x) . (iii) Let x, y ∈ E. Then x ∼ y ≤ (x → y) ∧ (y → x) implies that T_A (x ∼ y) ≤ T_A (y → x). (iv) Let x, y, z ∈ E. Since (x → y) ⊗ (y → z) ≤ (x → z), by item (i), we get that

\begin{matrix} T_{A} (y) \land T_{A} (z) & \leq & T_{A} (x \to y) \land T_{A} (y \to z) \\ = & T_{A} ((x \to y) \otimes (y \to z)) \\ \leq & T_{A} (x \to z) . \end{matrix}

(v) Let x, y, z ∈ E. Since (x ∼ y) ⊗ (y ∼ z) ≤ x ∼ z, we get that T_A ((x ∼ y) ⊗ (y ∼ z)) ≤ T_A (x ∼ z). Now by item (i), we get that T_A (x ∼ y) ∧ T_A (y ∼ z) = T_A ((x ∼ y) ⊗ (y ∼ z)) ≤ T_A (x ∼ z) . (vi) , (vii) and (viii) in a similar way are obtained.□

Example 4.7. Consider the EQ–algebra and the single–valued neutrosophic EQ–prefilter A of $E$ which are defined in Example 3.8. Since 0.1 = T_A (a) = T_A (d ⊗ c) ≠0.3 = 0.4 ∧ 0.3 = T_A (d) ∧ T_A (c) , we conclude that A is not a single–valued neutrosophic EQ–filter A of $E$ .

Corollary 4.8. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra, A be a single–valued neutrosophic EQ–filter of $E$ and x, y, z ∈ E. Then

F (x ⊗ y) = F_A (x) ∧ F_A (y),

F_A (x ∼ y) ≤ F_A (y → x),

F_A (z) ∧ F_A (y) ≤ F_A (x → z),

F_A (x ∼ y) ∧ F_A (y ∼ z) ≤ F_A (x ∼ z).

4.1 Special single–valued neutrosophic EQ–filters

In this section, we apply the concept of homomorphisms and (α, β, γ)–level sets to construct of single–valued neutrosophic EQ–filters.

Theorem 4.9. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra and {A_i = (T_{A
_i}, F_{A
_i}, I_{A
_i})} _i∈I be a family of single–valued neutrosophic EQ–filters of $E$ . Then ⋂_i∈IA_i is a single–valued neutrosophic EQ–filter of $E$ .

Proof. Let x ∈ E, then for any i ∈ I, T_{A
_i} (x) ≤ T_{A
_i} (1) , F_{A
_i} (x) ≤ F_{A
_i} (1) , I_{A
_i} (x) ≥ I_{A
_i} (1) and so(⋂ _i∈IT_{A
_i}) (x) = ⋀ _i∈IT_{A
_i} (x) ≤ T_{A
_i} (1) , (⋂ _i∈IF_{A
_i}) (x) = ⋀ _i∈IF_{A
_i} (x) ≤ F_{A
_i} (1) and (⋂ _i∈II_{A
_i}) (x) = ⋀ _i∈II_{A
_i} (x) ≥ I_{A
_i} (1) . Let x, y ∈ E. Then $\begin{matrix} (⋂_{i \in I} T_{A_{i}}) (x) \land (⋂_{i \in I} T_{A_{i}}) (x \to y) \\ = & ⋀_{i \in I} T_{A_{i}} (x) \land ⋀_{i \in I} T_{A_{i}} (x \to y) \leq ⋀_{i \in I} T_{A_{i}} (y) \\ = & ⋂_{i \in I} T_{A_{i}} (y), \\ (⋂_{i \in I} F_{A_{i}}) (x) \land (⋂_{i \in I} F_{A_{i}}) (x \to y) \\ = & ⋀_{i \in I} F_{A_{i}} (x) \land ⋀_{i \in I} F_{A_{i}} (x \to y) \leq ⋀_{i \in I} F_{A_{i}} (y) \\ = & ⋂_{i \in I} F_{A_{i}} (y) and \\ (⋂_{i \in I} I_{A_{i}}) (x) \lor (⋂_{i \in I} I_{A_{i}}) (x \to y) \\ = & ⋀_{i \in I} I_{A_{i}} (x) \lor ⋀_{i \in I} I_{A_{i}} (x \to y) \geq ⋀_{i \in I} I_{A_{i}} (y) \\ = & ⋂_{i \in I} I_{A_{i}} (y) . \end{matrix}$ Let x, y, z ∈ E. Then $\begin{matrix} (⋂_{i \in I} T_{A_{i}}) (x \to y) = ⋀_{i \in I} T_{A_{i}} (x \to y) \\ \leq & ⋀_{i \in I} T_{A_{i}} (x \otimes z \to y \otimes z) \\ = & ⋂_{i \in I} T_{A_{i}} (x \otimes z \to y \otimes z), \\ (⋂_{i \in I} F_{A_{i}}) (x \to y) = ⋀_{i \in I} F_{A_{i}} (x \to y) \\ \leq & ⋀_{i \in I} F_{A_{i}} (x \otimes z \to y \otimes z) \\ = & ⋂_{i \in I} I_{A_{i}} (x \otimes z \to y \otimes z) and \\ (⋂_{i \in I} I_{A_{i}}) (x \to y) = ⋀_{i \in I} I_{A_{i}} (x \to y) \\ \leq & ⋀_{i \in I} I_{A_{i}} (x \otimes z \to y \otimes z) \\ = & ⋂_{i \in I} I_{A_{i}} (x \otimes z \to y \otimes z) . \end{matrix}$ Thus ⋂_i∈I A_i is a single–valued neutrosophic EQ–filter of $E$ .□

Definition 4.10. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra, A be a single–valued neutrosophic EQ–filter of $E$ and α, β, γ ∈ [0, 1]. Consider $T_{A}^{α} = {x \in E | T_{A} (x) \geq α}, F_{A}^{β} = {x \in E | F_{A} (x) \geq β},$ $I_{A}^{γ} = {x \in E | T_{A} (x) \leq γ}$ and define A^(α,β,γ) = {x ∈ E | T_A (x) ≥ α, F_A (x) ≥ β, I_A (x) ≤ γ} . For any α, β, γ ∈ [0, 1] the set A^(α,β,γ) is called an (α, β, γ)-level set.

Example 4.11. Consider the EQ–algebra $E = (E, \land, \otimes, \sim, 1)$ , single–valued neutrosophic EQ–filter A of $E$ which are defind in Example 4.5. If α = 0.3, β = 0.4 and γ = 0.5, then $T_{A}^{α} = E, F_{A}^{β} = {1}, I_{A}^{γ} = {1}$ and A^(α,β,γ) = {1} .

Theorem 4.12. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra, A be a single–valued neutrosophic EQ–filter of $E$ and α, β, γ ∈ [0, 1]. Then

$A^{(α, β, γ)} = T_{A}^{α} \cap I_{A}^{β} \cap F_{A}^{γ},$

if ∅ ≠ A^(α,β,γ), then A^(α,β,γ) is an EQ–filter of $E$ ,

if A^(α,β,γ) is an EQ–filter of $E$ , then A is a single–valued neutrosophic EQ–filter in $E$ .

Proof. (i) It is obtained by definition.

(ii) ∅ ≠ A^(α,β,γ), implies that there exists x ∈ A^(α,β,γ). By Theorem 3.6, we conclude that α ≤ T_A (x) ≤ T_A (1) , β ≤ F_A (x) ≤ F_A (1) and γ ≥ I_A (x) ≥ I_A (1). Therefore, 1 ∈ A^(α,β,γ).

Let x ∈ A^(α,β,γ) and x ≤ y. Since T_A and F_A are monotone maps and I_A is an antimonotone map, we get that α ≤ T_A (x) ≤ T_A (y) , β ≤ F_A (x) ≤ F_A (y) and γ ≥ I_A (x) ≥ I_A (y). Hence y ∈ A^(α,β,γ).

Let x ∈ A^(α,β,γ) and x → y ∈ A^(α,β,γ). Since A is a single–valued neutrosophic EQ–filter of $E$ , by definition we get that α ≤ T_A (x) ∧ T_A (x → y) ≤ T_A (y) , β ≤ F_A (x) ∧ F_A (x → y) ≤ F_A (y) and γ ≥ I_A (x) ∨ I_A (x → y) ≥ I_A (y). So y ∈ A^(α,β,γ).

Let x → y ∈ A^(α,β,γ) and z ∈ E. Since A is a single–valued neutrosophic EQ–filter of $E$ , by definition we get that α ≤ T_A (x → y) ≤ T_A ((x ⊗ z) → (y ⊗ z)) , γ ≥ I_A (x → y) ≥ I_A ((x ⊗ z) → (y ⊗ z)) and β ≤ F_A (x → y) ≤ F_A ((x ⊗ z) → (y ⊗ z)). It follows that (x ⊗ z) → (y ⊗ z) ∈ A^(α,β,γ) and so A^(α,β,γ) is an EQ–filter of $E$ .

(iii) Let x, y, z ∈ E. Consider α_x = T_A (x) , β_x = F_A (x) and γ_x = I_A (x) . Since A^(α,β,γ) is an EQ–filter of $E$ , then 1 ∈ A^(α,β,γ) implies that $T_{A} (1) \geq α_{x} = T_{A} (x), F_{A} (1) \geq β_{x} = F_{A} (x),$ I_A (1) ≤ γ_x = I_A (x) .

Let α_x→y = T_A (x → y) , β_x→y = F_A (x → y), γ_x→y = I_A (x → y), α = α_x ∧ α_x→y, β = β_x ∧ β_x→y and γ = γ_x ∨ γ_x→y . We have T_A (x) = α_x ≥ α, T_A (x → y) = α_x→y ≥ α, F_A (x) = β_x ≥ β, F_A (x → y) = β_x→y ≥ β and I_A (x) = γ_x ≤ γ, I_A (x → y) = γ_x→y ≤ γ, so x, x → y ∈ A^(α,β,γ). Since A^(α,β,γ) is an EQ–filter of $E$ we get y ∈ A^(α,β,γ). Thus we conclude that $\begin{matrix} T_{A} (y) \geq α = α_{x} \land α_{x \to y} = T_{A} (x) \land T_{A} (x \to y), \\ F_{A} (y) \geq β = β_{x} \land β_{x \to y} = F_{A} (x) \land F_{A} (x \to y) \end{matrix}$ and I_A (y) ≤ γ = γ_x ∨ γ_x→y = I_A (x) ∨ I_A (x → y) . We have T_A (x → y) = α_x→y ≥ α_x→y, F_A (x → y) = β_x→y ≥ β_x→y and I_A (x → y) = γ_x→y ≤ γ_x→y, so x → y ∈ A^{(α_x→y,β_x→y,γ_x→y)}. Since A^{(α_x→y,β_x→y,γ_x→y)} is an EQ–filter of $E$ we get x ⊗ z → y ⊗ z ∈ A^{(α_x→y,β_x→y,γ_x→y)}. Thus we conclude that $\begin{matrix} T_{A} ((x \otimes z) \to (y \otimes z)) \geq α_{x \to y} = T_{A} (x \to y), \\ F_{A} ((x \otimes z) \to (y \otimes z)) \geq β_{x \to y} = F_{A} (x \to y) \end{matrix}$ and I_A ((x ⊗ z) → (y ⊗ z)) ≥ γ_x→y = I_A (x → y) . It follows that A is a single–valued neutrosophic EQ–filter $E$ .□

Corollary 4.13. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra, A be a single–valued neutrosophic EQ–filter of $E$ , α, β, γ ∈ [0, 1] and ∅ ≠ A^(α,β,γ).

A^(α,β,γ) is an EQ–filter of $E$ if and only if A is a single–valued neutrosophic EQ–filter in $E$ .

If G_A = {x ∈ E | T_A (1) = F_A (1) =1, I_A (0) =1}, then G_A is an EQ–filter in $E$

Let A = (T_A, F_A, I_A) be a single–valued neutrosophic EQ–filter in

E, α, α^{'}, β, β^{'}, γ, γ^{'} \in [0, 1]

and

\emptyset \neq H \subseteq E

. Consider

T_{A, H}^{[α, α^{'}]} = {\begin{matrix} α & if x \in H, \\ α^{'} & otherwise, \end{matrix}

F_{A, H}^{[α, α^{'}]} = {\begin{matrix} β & if x \in H \\ β^{'} & o . w, \end{matrix}

and

I_{A, H}^{[α, α^{'}]} = {\begin{matrix} γ & if x \in H, \\ γ^{'} & otherwise . \end{matrix}

Then we have the following corollary.

Corollary 4.14. Let A = (T_A, F_A, I_A) be a single–valued neutrosophic EQ–filter in $E$ . Then

$T_{A, H}^{[α, α^{'}]}, F_{A, G}^{[α, α^{'}]}$ and $I_{A, G}^{[α, α^{'}]}$ are fuzzy subsets,

$T_{A, H}^{[α, α^{'}]}$ is a fuzzy filter in E if and only if G is an EQ–filter of $E$ ,

$F_{A, H}^{[α, α^{'}]}$ is a fuzzy filter in E if and only if G is an EQ–filter of $E$ ,

$I_{A, H}^{[α, α^{'}]}$ is a fuzzy filter in E if and only if G is an EQ–filter of $E$ .

Definition 4.15. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra, A be a single–valued neutrosophic EQ–filter of $E$ . Then A is said to be a normal single–valued neutrosophic EQ–filter of $E$ if there exists x, y, z ∈ E such that T_A (x) =1, I_A (y) =1 and F_A (z) =1.

Example 4.16. Consider the EQ–algebra $E = (E, \land, \otimes, \sim, 1)$ , which is defind in Example 4.5. If Define a single valued neutrosophic set map A in E as follows: $\begin{matrix} \begin{matrix} T_{A} | & 0 & a & b & c & 1 \\ | & 0.4 & 0.4 & 0.4 & 0.4 & 1 \end{matrix}, \end{matrix}$ $\begin{matrix} \begin{matrix} I_{A} | & 0 & a & b & c & 1 \\ | & 1 & 1 & 1 & 1 & 0.11 \end{matrix} \end{matrix}$ and $\begin{matrix} \begin{matrix} F_{A} | & 0 & a & b & c & 1 \\ | & 0.2 & 0.2 & 0.2 & 0.2 & 1 \end{matrix} \end{matrix}$ Hence A is a normal single–valued neutrosophic EQ–filter of $E$ .

Theorem 4.17. Let $E = (E, \land, \otimes, \sim, 1)$ be an EQ–algebra and A be a single–valued neutrosophic EQ–filter of $E$ . Then A is a normal single–valued neutrosophic EQ–filter of $E$ if and only if T_A (1) =1, F_A (1) =1 and I_A (0) =1.

Proof. By Corollary 3.7, it is straightforward.□

Corollary 4.18. Let A = (T_A, I_A, F_A) be a single–valued neutrosophic EQ–filter of $E$ . Then

A is a normal single–valued neutrosophic EQ–filter of $E$ if and only if T_A, F_A and I_A are normal fuzzy subset.

If there exists a sequence ${(x_{n}, y_{n}, z_{n})}_{n = 1}^{\infty}$ of elements E in such a way that ${(T_{A} (x_{n}), I_{A} (y_{n}), F_{A} (z_{n}))} \to (1, 1, 1),$ then A (1, 0, 1) = (1, 1, 1) .

Corollary 4.19. Let {A_i = (T_{A
_i}, F_{A
_i}, I_{A
_i})} _i∈I be a family of normal single–valued neutrosophic EQ–filters of $E$ . Then ⋂_i∈IA_i is a normal single–valued neutrosophic EQ–filter of $E$ .

Let A = (T_A, I_A, F_A) be a single–valued neutrosophic EQ–filter of $E, x \in E$ and p ∈ [1, + ∞). Consider $T_{A}^{+ p} (x) = \frac{1}{p} (p + T_{A} (x) - T_{A} (1)), F_{A}^{+ p} (x) = \frac{1}{p} (p + F_{A} (x) - F_{A} (1))$ and $I_{A}^{+ p} (x) = \frac{1}{p} (p + I_{A} (x) - I_{A} (0)) .$

Theorem 4.20. Let A = (T_A, I_A, F_A) be a single–valued neutrosophic EQ–filter of $E$ . Then

$T_{A}^{+ p}$ is a normal EQ–filter of $E$ ,

$I_{A}^{+ p}$ is a normal EQ–filter of $E$ ,

$(T_{A}^{+ p})^{+ p} = T_{A}^{+ p}$ if and only if p = 1,

$(I_{A}^{+ p})^{+ p} = I_{A}^{+ p}$ if and only if p = 1,

$(T_{A}^{+ p})^{+ p} = T_{A}$ if and only if T_A is normal EQ–filter,

$(I_{A}^{+ p})^{+ p} = I_{A}$ if and only if I_A (0) =1.

Proof. (i) Let x ∈ E. Because T_A (x) ≤ T_A (1), then we have

T_{A}^{+ p} (x) = \frac{1}{p} (p + T_{A} (x) - T_{A} (1)) \leq 1

. Assume that x, y ∈ E. Using (SVNF2), we get that

\begin{matrix} T_{A}^{+ p} (x) \land T_{A}^{+ p} (x \to y) = \frac{1}{p} (p + T_{A} (x) - T_{A} (1)) \\ \land & \frac{1}{p} (p + T_{A} (x \to y) - T_{A} (1)) \\ = & \frac{1}{p} [(p + T_{A} (x) - T_{A} (1)) \\ \land & (p + T_{A} (x \to y) - T_{A} (1))] \\ = & \frac{1}{p} [((p \land p) + (T_{A} (x) \land T_{A} (x \to y)) \\ - & (T_{A} (1) \land T_{A} (1))] \\ \leq & \frac{1}{p} (p + T_{A} (y) - T_{A} (1)) = T_{A}^{+ p} (y) . \end{matrix}

Suppose that x, y, z ∈ E. Using (SVNF3), we get that

\begin{matrix} T_{A}^{+ p} (x \to y) & = & \frac{1}{p} (p + T_{A} (x \to y) - T_{A} (1)) \\ \leq & \frac{1}{p} (p + T_{A} (x \otimes z \to y \otimes z) \\ - & T_{A} (1)) = T_{A}^{+ p} (x \otimes z \to y \otimes z) . \end{matrix}

Thus

T_{A}^{+ p}

is an EQ–filter of

E

. In addition the equality

T_{A}^{+ p} (1) = \frac{1}{p} (p + T_{A} (1) - T_{A} (1)) = 1

, implies that

T_{A}^{+ p}

is a normal EQ–filter of

E

(ii) Let x ∈ E. Since I_A (x) ≤ I_A (0) we get that $I_{A}^{+ p} (x) = \frac{1}{p} (p + I_{A} (x) - I_{A} (0)) \leq 1$ . Items (SVNF2) and (SVNF3) are obtained similar to the item (i).

(iii) Assume that x ∈ E. Then $\begin{matrix} {(T_{A}^{+ p})}^{+ p} (x) = & {[\frac{1}{p} (p + T_{A} (x) - T_{A} (1))]}^{+ p} \\ = & \frac{1}{p} [p + \frac{1}{p} (p + T_{A} (x) - T_{A} (1)) \\ - & \frac{1}{p} (p + T_{A} (1) - T_{A} (1))] \\ = & \frac{1}{p} (p + \frac{1}{p} (T_{A} (x) - T_{A} (1))) . \end{matrix}$ So $\begin{matrix} (T_{A}^{+ p})^{+ p} (x) & = & T_{A}^{+ p} (x) \\ \Leftrightarrow & \frac{1}{p} (p + \frac{1}{p} (T_{A} (x) - T_{A} (1))) \\ = & \frac{1}{p} (p + T_{A} (x) - T_{A} (1)) \\ \Leftrightarrow & p = 1 . \end{matrix}$ (iv) It is similar to (iii).(v) Let x ∈ E. $(T_{A}^{+ p})^{+ p} = T_{A}$ if and only if $\begin{matrix} \frac{1}{p} (p + \frac{1}{p} (T_{A} (x) - T_{A} (1))) = T_{A} (x) \\ \Leftrightarrow & T_{A} (1) = (1 - p^{2}) T_{A} (x) + p^{2} \\ \Leftrightarrow & p = 1 \Leftrightarrow T_{A} (1) = 1 . \end{matrix}$ (vi) It is similar to (v).□

Example 4.21. Let E = {0, a, b, c, 1}. Define operations “⊗, ∼” and an operation “∧” on E as follows: $\begin{matrix} \begin{matrix} \land | & 0 & a & b & c & 1 \\ 0 | & 0 & 0 & 0 & 0 & 0 \\ a | & 0 & a & a & a & a \\ b | & 0 & a & b & b & b \\ c | & 0 & a & b & c & c \\ 1 | & 0 & a & b & c & 1 \end{matrix}, \end{matrix} \begin{matrix} \otimes | & 0 & a & b & c & 1 \\ 0 | & 0 & 0 & 0 & 0 & 0 \\ a | & 0 & 0 & 0 & 0 & a \\ b | & 0 & 0 & 0 & a & b \\ c | & 0 & 0 & 0 & a & c \\ 1 | & 0 & a & b & c & 1 \end{matrix} and \begin{matrix} \sim | & 0 & a & b & c & 1 \\ 0 | & 1 & c & b & a & 0 \\ a | & c & 1 & c & b & a \\ b | & b & c & 1 & c & b \\ c | & a & b & c & 1 & c \\ 1 | & 0 & a & b & c & 1 \end{matrix} .$

Then $E = (E, \land, \otimes, \sim, 1)$ is an EQ–algebra, where b and c are non-comparable. Now, obtain the operation “→” as follows: $\begin{matrix} \begin{matrix} \to | & 0 & a & b & c & 1 \\ 0 | & 1 & 1 & 1 & 1 & 1 \\ a | & c & 1 & 1 & 1 & 1 \\ b | & b & c & 1 & 1 & 1 \\ c | & a & b & c & 1 & 1 \\ 1 | & 0 & a & b & c & 1 \end{matrix} . \end{matrix}$

Define a single valued neutrosophic set map A in E as follows: $\begin{matrix} \begin{matrix} T_{A} | & 0 & a & b & c & 1 \\ | & 0.41 & 0.42 & 0.43 & 0.44 & 0.5 \end{matrix}, \end{matrix}$ $\begin{matrix} \begin{matrix} I_{A} | & 0 & a & b & c & 1 \\ | & 0.69 & 0.68 & 0.67 & 0.66 & 0.65 \end{matrix} \end{matrix}$ and $\begin{matrix} \begin{matrix} F_{A} | & 0 & a & b & c & 1 \\ | & 0.21 & 0.22 & 0.23 & 0.24 & 0.25 \end{matrix} \end{matrix}$ Hence A is a single–valued neutrosophic EQ–prefilter of $E$ . Consider p = 3, then we obtain a single–valued neutrosophic EQ–prefilter A⁺³ in E as follows: $\begin{matrix} \begin{matrix} T_{A}^{+ 3} | & 0 & a & b & c & 1 \\ | & 0.97 & 0.973 & 0.977 & 0.98 & 1 \end{matrix}, \end{matrix}$ $\begin{matrix} \begin{matrix} I_{A}^{+ 3} | & 0 & a & b & c & 1 \\ | & 1 & 0.977 & 0.973 & 0.99 & 0.987 \end{matrix} \end{matrix}$ and $\begin{matrix} \begin{matrix} F_{A}^{+ 3} | & 0 & a & b & c & 1 \\ | & 0.987 & 0.99 & 0.993 & 0.997 & 1 \end{matrix} . \end{matrix}$ Corollary 4.22. Let A = (T_A, I_A, F_A) be a single–valued neutrosophic EQ–filter of $E$ . Then

$F_{A}^{+ p}$ is a normal EQ–filter of $E$ ,

$(F_{A}^{+ p})^{+ p} = F_{A}^{+ p}$ if and only if p = 1,

$(F_{A}^{+ p})^{+ p} = F_{A}$ if and only if F_A is normal EQ–filter.

Corollary 4.23. Let A = (T_A, I_A, F_A) be a single–valued neutrosophic EQ–filter of $E$ . Then

$A^{+ p} = (T_{A}^{+ p}, I_{A}^{+ p}, F_{A}^{+ p})$ is a normal single–valued neutrosophic EQ–filter of $E$ ,

(A^+p) ^+p = A^+p if and only if p = 1,

(A^+p) ^+p = A if and only if A is a normal single–valued neutrosophic EQ–filter.

Proof. It is trivial by Theorem 4.20 and Corollary 4.22.□

Let A = (T_A, I_A, F_A) be a single–valued neutrosophic EQ–filter of $E$ and g be an endomorphism on $E$ . Now we define $A^{g} = (T_{A}^{g}, I_{A}^{g}, F_{A}^{g})$ by $T_{A}^{g} (x) = T_{A} (g (x)), F_{A}^{g} (x) = F_{A} (g (x))$ and $I_{A}^{g} (x) = I_{A} (g (x)) .$

Theorem 4.24. Let A = (T_A, I_A, F_A) be a single–valued neutrosophic EQ–filter of $E$ and x, y ∈ E. Then

if x ≤ y, then $T_{A}^{g} (x) \leq T_{A}^{g} (y), F_{A}^{g} (x) \leq F_{A}^{g} (y)$ and $I_{A}^{g} (x) \geq I_{A}^{g} (y),$

A^g is a single–valued neutrosophic EQ–filter of $E$ ,

$T_{A}^{'} (x) = \frac{1}{2} (T_{A}^{g} (x) + T_{A} (x))$ is a fuzzy filter in E,

$F_{A}^{'} (x) = \frac{1}{2} (F_{A}^{g} (x) + F_{A} (x))$ is a fuzzy filter in E,

${A^{'}}^{g} = (T_{A}^{'}, I_{A}^{'}, F_{A}^{'})$ is a single–valued neutrosophic EQ–filter of $E$ .

Proof. (i) Let x, y ∈ E. If x ≤ y, then g (x) ≤ g (y). It follows that $T_{A}^{g} (x) = T_{A} (g (x)) \leq T_{A} (g (y)), F_{A}^{g} (x) = F_{A} (g (x)) \leq F_{A} (g (y))$ and $I_{A}^{g} (x) = I_{A} (g (x)) \geq I_{A} (g (y))$ . (ii) Since g (x → y) = g (x) → g (y), we get that $\begin{matrix} T_{A}^{g} (x) & \land & T_{A}^{g} (x \to y) \\ = & T_{A} (g (x)) \land T_{A} (g (x) \to g (y)) \\ \leq & T_{A} (g (y)) = T_{A}^{g} (y), F_{A}^{g} (x) \land F_{A}^{g} (x \to y) \\ = & F_{A} (g (x)) \land F_{A} (g (x) \to g (y)) \\ \leq & F_{A} (g (y)) = F_{A}^{g} (y) \end{matrix}$ and $I_{A}^{g} (x) \lor I_{A}^{g} (x \to y) = I_{A} (g (x)) \lor I_{A} (g (x) \to g (y)) \leq I_{A} (g (y)) = I_{A}^{g} (y) .$

Let z ∈ E. Since g (x ⊗ z → y ⊗ z) = g (x ⊗ z) → g (y ⊗ z), we get that $\begin{matrix} T_{A}^{g} (x \to y) & = & T_{A} (g (x) \to g (y)) \\ \leq & T_{A} (g (x \otimes z \to y \otimes z)) \\ = & T_{A} (g (x \otimes z) \to (y \otimes z)) \\ = & T_{A}^{g} (x \otimes z \to y \otimes z), \\ F_{A}^{g} (x \to y) & = & F_{A} (g (x) \to g (y)) \\ \leq & F_{A} (g (x \otimes z \to y \otimes z)) \\ = & F_{A} (g (x \otimes z) \to (y \otimes z)) \\ = & F_{A}^{g} (x \otimes z \to y \otimes z), \\ I_{A}^{g} (x \to y) & = & I_{A} (g (x) \to g (y)) \\ \geq & I_{A} (g (x \otimes z \to y \otimes z)) \\ = & I_{A} (g (x \otimes z) \to (y \otimes z)) \\ = & I_{A}^{g} (x \otimes z \to y \otimes z) . \end{matrix}$ So by the item (i) , A^g is a single–valued neutrosophic EQ–filter of $E$ . (iii) , (iv) Let x ∈ E. Since g (1) =1, so T_A (x) + T_A (g (x)) ≤2 implies that $T_{A}^{'} (x) = \frac{1}{2} (T_{A}^{g} (x) + T_{A} (x)) \leq T_{A}^{'} (1)$ . In a similar way $F_{A}^{'} (x) \leq F_{A}^{'} (1)$ and $I_{A}^{'} (x) \geq I_{A}^{'} (1)$ are obtained. Suppose that x, y ∈ E. Then we have $\begin{matrix} T_{A}^{'} (x) \land T_{A}^{'} (x \to y) \\ = & \frac{1}{2} (T_{A}^{g} (x) + T_{A} (x)) \\ \land & \frac{1}{2} (T_{A}^{g} (x \to y) + T_{A} (x \to y)) \\ = & \frac{1}{2} (T_{A}^{g} (x) \land T_{A}^{g} (x \to y)) \\ + & \frac{1}{2} (T_{A} (x) + T_{A} (x \to y)) \\ \leq & \frac{1}{2} (T_{A}^{g} (y) + T_{A} (y)) = T_{A}^{'} (y) . \end{matrix}$ We can show that $F_{A}^{'} (x) \land F_{A}^{'} (x \to y) \leq F_{A}^{'} (y)$ and $I_{A}^{'} (x) \lor I_{A}^{'} (x \to y) \geq I_{A}^{'} (y)$ . Let x, y, z ∈ E. Then $\begin{matrix} {T^{'}}_{A} (x \to y) = \frac{1}{2} (T_{A}^{g} (x \to y) + T_{A} (x \to y)) \\ = \frac{1}{2} (T_{A} (g (x \to y)) + T_{A} (x \to y)) \\ \leq \frac{1}{2} (T_{A} (g (x \otimes z \to y \otimes z)) + T_{A} (x \otimes z \to y \otimes z)) \\ = \frac{1}{2} (T_{A}^{g} ((x \otimes z \to y \otimes z)) + T_{A} (x \otimes z \to y \otimes z)) \\ = T_{A}^{'} (x \otimes z \to y \otimes z) . \end{matrix}$ In a similar way can see that $F_{A}^{'} (x \to y) \leq F_{A}^{'} (x \otimes z \to y \otimes z)$ and $I_{A}^{'} (x \to y) \geq I_{A}^{'} (x \otimes z \to y \otimes z) .$ (v) It is obtained from previous items.□

Example 4.25. Let E = {a₁, a₂, a₃, a₄, a₅, a₆}. Define operations “⊗, ∼” and “∧” on E as follows: $\begin{matrix} \begin{matrix} \land | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ a_{1} | & a_{1} & a_{1} & a_{1} & a_{1} & a_{1} & a_{1} \\ a_{2} | & a_{2} & a_{2} & a_{2} & a_{2} & a_{2} & a_{2} \\ a_{3} | & a_{3} & a_{3} & a_{3} & a_{3} & a_{3} & a_{3} \\ a_{4} | & a_{4} & a_{4} & a_{4} & a_{4} & a_{4} & a_{4} \\ a_{5} | & a_{5} & a_{5} & a_{5} & a_{5} & a_{5} & a_{5} \\ a_{6} | & a_{6} & a_{6} & a_{6} & a_{6} & a_{6} & a_{6} \end{matrix}, \end{matrix} \begin{matrix} \otimes | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ a_{1} | & a_{1} & a_{1} & a_{1} & a_{1} & a_{1} & a_{1} \\ a_{2} | & a_{1} & a_{1} & a_{1} & a_{1} & a_{1} & a_{1} \\ a_{3} | & a_{1} & a_{1} & a_{1} & a_{1} & a_{1} & a_{1} \\ a_{4} | & a_{1} & a_{1} & a_{1} & a_{1} & a_{1} & a_{1} \\ a_{5} | & a_{1} & a_{1} & a_{1} & a_{1} & a_{5} & a_{5} \\ a_{6} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \end{matrix} and$ $\begin{matrix} \begin{matrix} \sim | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ a_{1} | & a_{6} & a_{6} & a_{1} & a_{1} & a_{1} & a_{1} \\ a_{2} | & a_{6} & a_{6} & a_{1} & a_{1} & a_{1} & a_{1} \\ a_{3} | & a_{1} & a_{1} & a_{6} & a_{4} & a_{4} & a_{4} \\ a_{4} | & a_{1} & a_{1} & a_{4} & a_{6} & a_{4} & a_{4} \\ a_{5} | & a_{1} & a_{1} & a_{4} & a_{4} & a_{6} & a_{5} \\ a_{6} | & a_{1} & a_{1} & a_{4} & a_{4} & a_{5} & a_{6} \end{matrix} . \end{matrix}$ Now, we obtain the operation “→” as follows: $\begin{matrix} \begin{matrix} \to | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ a_{1} | & a_{6} & a_{6} & a_{6} & a_{6} & a_{6} & a_{6} \\ a_{2} | & a_{6} & a_{6} & a_{6} & a_{6} & a_{6} & a_{6} \\ a_{3} | & a_{1} & a_{1} & a_{6} & a_{6} & a_{6} & a_{6} \\ a_{4} | & a_{1} & a_{1} & a_{4} & a_{4} & a_{6} & a_{6} \\ a_{5} | & a_{1} & a_{1} & a_{4} & a_{4} & a_{6} & a_{6} \\ a_{6} | & a_{1} & a_{1} & a_{4} & a_{4} & a_{5} & a_{6} \end{matrix} . \end{matrix}$

Then $E = (E, \land, \otimes, \sim, a_{6})$ is an EQ–algebra. Let g ∈ End (E). Clearly g (a₆) = a₆. Since for any 1 ≤ i ≤ 4, 1 ≤ j ≤ 6, g (a₁) = g (a_i ⊗ a_j) = g (a_i) ⊗ g (a_j). So a₁ = g (a₁) = g (a₅ ∼ a₂) = g (a₅) ∼ g (a₂) = g (a₅) ∼ a₁ = a₁ implies that g (a₅) = a₁. Hence define a single valued neutrosophic set map A in E and a map g on E as follows: $\begin{matrix} \begin{matrix} T_{A} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ | & 0.01 & 0.02 & 0.03 & 0.04 & 0.05 & 0.06 \end{matrix}, \end{matrix}$ $\begin{matrix} \begin{matrix} F_{A} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ | & 0.11 & 0.12 & 0.13 & 0.14 & 0.15 & 0.16 \end{matrix}, \end{matrix}$ $\begin{matrix} \begin{matrix} I_{A} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ | & 0.61 & 0.52 & 0.43 & 0.34 & 0.25 & 0.16 \end{matrix} \end{matrix}$ and $\begin{matrix} \begin{matrix} g_{A} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ | & a_{1} & a_{1} & a_{1} & a_{1} & a_{5} & a_{6} \end{matrix} . \end{matrix}$

Hence $(A, E)$ is a single–valued neutrosophic EQ–prefilter. Now, we obtain a single valued neutrosophic EQ–prefilter A^g in E follows: $\begin{matrix} \begin{matrix} T_{A}^{g} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ | & 0.01 & 0.01 & 0.01 & 0.01 & 0.05 & 0.06 \end{matrix}, \end{matrix}$ $\begin{matrix} \begin{matrix} F_{A}^{g} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ | & 0.11 & 0.11 & 0.11 & 0.11 & 0.15 & 0.16 \end{matrix} \end{matrix}$ and $\begin{matrix} \begin{matrix} I_{A}^{g} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ | & 0.61 & 0.61 & 0.61 & 0.61 & 0.25 & 0.16 \end{matrix} . \end{matrix}$ and obtain a single valued neutrosophic EQ–prefilter A^′g in E follows: $\begin{matrix} \begin{matrix} T_{A}^{' g} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ | & 0.01 & 0.015 & 0.02 & 0.025 & 0.05 & 0.06 \end{matrix}, \end{matrix}$ $\begin{matrix} \begin{matrix} F_{A}^{' g} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ | & 0.11 & 0.115 & 0.12 & 0.125 & 0.15 & 0.16 \end{matrix} \end{matrix}$

and

$\begin{matrix} \begin{matrix} I_{A}^{' g} | & a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \\ | & 0.61 & 0.565 & 0.52 & 0.475 & 0.25 & 0.16 \end{matrix} . \end{matrix}$

5 Conclusion

The current paper considered the concept of single–valued neutrosophic EQ–algebras and introduce the concepts single–valued neutrosophic EQ–subalgebras, single–valued neutrosophic EQ–prefilters and single–valued neutrosophic EQ–filters.

It is showed that single–valued neutrosophic EQ–subalgebras preserve some binary relation on EQ–algebras under some conditions.

Using the some properties of single–valued neutrosophic EQ–prefilters, we construct new single–valued neutrosophic EQ–prefilters.

We considered that single–valued neutrosophic EQ–filters as generalisation of single–valued neutrosophic EQ–prefilters and constructed them.

We connected the concept of EQ–prefilters to single–valued neutrosophic EQ–prefilters and the concept of EQ–filters to single–valued neutrosophic EQ–filters, so we obtained such structures from this connection.

Footnotes

Acknowledgments

The authors would like to express their gratitude to anonymous referees for their comments and suggestions which improved the paper.

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Single–valued neutrosophic filters in EQ –algebras

Abstract

1 Introduction

2 Preliminaries

3 Single–Valued Neutrosophic EQ–subalgebras

3.1 Single–Valued Neutrosophic EQ–prefilters

4 Single–Valued Neutrosophic EQ–filters

4.1 Special single–valued neutrosophic EQ–filters

5 Conclusion

Footnotes

Acknowledgments

References