Intuitionistic N-fuzzy set and its application in biΓ-ternary semigroups

Abstract

This paper aims at introducing the notion of intuitionistic N-fuzzy set (INFS) and its application along with its examples. As the application of this set, its idea has been applied to a newly defined algebraic structure “BiΓ-Ternary Semigroup”. The notions of intuitionistic N-fuzzy biΓ-ternary subsemigroup and intuitionistic N-fuzzy biΓ-left (right, lateral and bi) ideals have been defined and related properties have been investigated here. The characterization of biΓ-ternary semigroup under these ideals has been established.

Keywords

BiΓ-ternary smigroup intuitionistic N-fuzzy set intuitionistic N-fuzzy biΓ-ternary subsmigroup intuitionistic N-fuzzy biΓ-ideal

1 Introduction

In 1932 Lehmer [9] introduced the concept of ternary semigroup. The notion of Γ-semigroup was introduced by Sen [11] in 1981, which was modified by Sen and Saha [12] in 1986. As a generalization of Γ-semigroup and ternary semigroup, Akram et al. [2] proposed a new algebraic structure called biΓ-ternary semigroup just recently. They introduced the notions of biΓ-ternary subsemigroup, biΓ-left (right, lateral) ideal, biΓ-quasi ideal and biΓ-bi-ideals for this structure and discussed the relationship between these substructures. They also defined the regular biΓ-ternary semigroup and characterized it by these ideals.

In 1965 Zadeh [13] gave the concept of fuzzy set. The fuzzy set theories developed by Zadeh and others have been found many applications in the domain of mathematics and elsewhere. The study of fuzzy algebraic structures started with the introduction of the concepts of fuzzy subgroup (subgroupoid) and fuzzy (left, right)ideals in the pioneering paper of Rosenfeld [10] in 1971.

The concept of intuitionistic fuzzy set was introduced by Atanassov [3] as a generalization of fuzzy set in 1986. Biswas [4] introduced the concept of intuitionistic fuzzy subgroupoids. Kim and Jun [8] applied the concept of intuitionistic fuzzy sets to the ideal theory of semigroup and defined several ideals of semigroup. Many other authors applied the concept of fuzzy set and intuitionistic fuzzy set to the algebraic structures.

A crisp set A in a universe X can be defined in the form of its characteristics function μ_A : X → {0, 1} yielding the value 1 for elements belonging to the set A and the value 0 for elements excluded from the set A. So far most of the generalizations of the crisp set have been conducted on the unit interval [0, 1] and they are consistent with the asymmetry observation. the generalization of the crisp set to fuzzy set and fuzzy set to intuitionistic fuzzy set relied on spreading positive information that fit the crisp point {1} into the interval [0, 1] and [0, 1] into [0, 1] ². Jun et al. [5] introduced and used a new function which is called negative-valued function (or negative fuzzy set, briefly, N-fuzzy set) and constructed N-structures. In 2013, Jun et al. [6] applied the concept of coupled N-structures to BCK/BCI -algebras which was extended to d-algebras by Ahan et al. [1]. But no negative meaning of information is suggested for intuitionistic fuzzy set, we now feel a need to deal with intuitionistic negative information.

In this paper the concept of intuitionistic N-fuzzy set has been introduced and applied to biΓ-ternary semigroup. The notions of intuitionistic N-fuzzy biΓ-ternary subsemigroup, intuitionistic N-fuzzy biΓ-left (right, lateral and bi) ideals have been defined and the relationship between them have been investigated. The characterizations of biΓ-ternary semigroup by these ideals have been discussed here.

2 Preliminary concepts

Definition 1. [2] Let T = {x, y, z, . . .} and Γ = {α, β, γ, . . .} be two non-empty sets. Then T is called a biΓ-ternary semigroup if for all x, y, z, u, v ∈ S, α, β, γ ∈ Γ, it satisfies,

(xαy) βz ∈ T

((xαyβz) γu) δv = xα (yβzγu) δv = xα (yβ (zγuδv)) ,

Example 1. [2] Let T = Z^- and Γ = Z⁺ . Define (xγyδz) = xγyδz, for x, y, z ∈ T and γ, δ ∈ Γ as the usual multiplication of integers. Then T is a biΓ-ternary semigroup but not a Γ-semigroup.

Example 2. [2] Let T = iR, where, $i = \sqrt{- 1}$ and R is the set of real numbers. If Γ ⊆ R and (xαyβz) is defined as the usual multiplication of complex numbers T is a biΓ-ternary semigroup but not a Γ-semigroup.

Definition 2. [2] A non empty subset A of a biΓ-ternary semigroup T is called a biΓ-ternary subsemigroup of T if, AΓAΓA ⊆ A .

Example 3. [2] Let T = N = {1, 2, 3, . . .} and Γ = {4n + 2, n ∈ N} . Define (xαy) βz = x + α + y + β + z . Then T is a biΓ-ternary semigroup. Let A = {4n, n ∈ N} be a non empty subset of T . Then A is a biΓ-ternary subsemigroup.

Definition 3. [2] A non empty subset A of a biΓ-ternary semigroup T is called a biΓ-left ideal of T if, TΓTΓA ⊆ A .

Definition 4. [2] A non empty subset A of a biΓ-ternary semigroup T is called a biΓ-right ideal of T if, AΓTΓT ⊆ A .

Definition 5. [2] A non empty subset A of a biΓ-ternary semigroup T is called a biΓ-lateral ideal of T if, TΓAΓT ⊆ A .

Definition 6. [2] A non empty subset A of a biΓ-ternary semigroup T is called a biΓ-ideal of T if it is a biΓ-left, a biΓ-right and a biΓ-lateral ideal of T .

Definition 7. [2] Let T be a biΓ-ternary semigroup. A biΓ-ternary subsemigroup B of T is called a biΓ-bi-ideal of T if BΓTΓBΓTΓB ⊆ B .

Example 4. [2] Let T = {2n, n ∈ N} , Γ = {α, β, γ, . . .} and A = {4n, n ∈ N} . Define, (xαyβz) = (2x + 2y) + z, for x, y, z ∈ T and α, β ∈ Γ . Then T is a biΓ-ternary semigroup and A is a biΓ-left ideal of T but neither a biΓ-right nor a biΓ-lateral ideal of T . If we define, (xαyβz) = x + 2y + 2z and (xαyβz) =2x + y + 2z respectively, then A is a biΓ-right and a biΓ-lateral ideal of T .

Proposition 1. [2] Let T be a biΓ-ternary semigroup and φ ≠ X ⊆ T, then

TΓTΓX is a biΓ-left ideal of T .

XΓTΓT is a biΓ-right ideal of T .

TΓXΓT ∪ TΓTΓXΓT ΓT is a biΓ-lateral ideal of T .

3 Institutionistic N-fuzzy sets

Definition 8. [7] A negative fuzzy set (briefly, N-fuzzy set) in a nonempty set X is a function $\bar{μ} : X \to [- 1, 0] .$ Here we are using “^-” for the negative fuzzy function.

Jun et al. [5] used the term negative-valued function and N-function for negative fuzzy set and N-fuzzy set.

Definition 9. An intuitionistic N-fuzzy set (briefly, INFS) A in a nonempty set X is an object of the form $A = {〈 x; {\bar{μ}}_{A}, {\bar{γ}}_{A}$ : x ∈ X〉} , where ${\bar{μ}}_{A} : X \to [- 1, 0]$ and ${\bar{γ}}_{A} : X \to [- 1, 0]$ such that $- 1 \leq {\bar{μ}}_{A} (x) + {\bar{γ}}_{A} (x) \leq 0$ for all x ∈ X . An intuitionistic N-fuzzy set $A = {〈 x; {\bar{μ}}_{A}, {\bar{γ}}_{A}$ : x ∈ X〉} in X can be identified to an ordered pair $({\bar{μ}}_{A}, {\bar{γ}}_{A})$ in F (X, [-1, 0]) × F (X, [-1, 0]) , where F (X, [-1, 0]) denotes the set of all functions from X to [-1, 0] . For the sake of simplicity, we shall use the notation $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A})$ instead of $A = {〈 x; {\bar{μ}}_{A}, {\bar{γ}}_{A}$ : x ∈ X〉} .

Definition 10. Let $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A})$ be an INFS in X. Then the set $N {({\bar{μ}}_{A}, {\bar{γ}}_{A}); (t, s)} = {x \in X | {\bar{μ}}_{A} (x) \leq t, {\bar{γ}}_{A} (x) \geq s}$ where t, s ∈ [-1, 0] with t + s ≥ -1 is called an N (t, s)-level set of A . An N (t, t)-level set of $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A})$ is called an N-level set of A . For simplicity, we shall use the notation N_A (t, s) instead of $N {({\bar{μ}}_{A}, {\bar{γ}}_{A}); (t, s)}$ for N (t, s)-level set of $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A}), i . e .$ $N_{A} (t, s) = {x \in X | {\bar{μ}}_{A} (x) \leq t, {\bar{γ}}_{A} (x) \geq s} .$

Definition 11. Let $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A})$ and $B = ({\bar{μ}}_{B}, {\bar{γ}}_{B})$ be two INFSs in X . If for all x ∈ X, ${\bar{μ}}_{A} (x) \geq {\bar{μ}}_{B} (x)$ and ${\bar{γ}}_{A} (x) \leq {\bar{γ}}_{B} (x),$ then A is called an intuitionistic N-fuzzy subset (INFSS) of B and is written as $A \bar{\subseteq} B .$ We say A = B if and only if $A \bar{\subseteq} B$ and $B \bar{\subseteq} A .$

Definition 12. Let $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A})$ and $B = ({\bar{μ}}_{B}, {\bar{γ}}_{B})$ be two INFSs in X . Then their union and intersection is also an intuitionistic N-fuzzy set in X, defined as, for all x ∈ X $\begin{matrix} (A \bar{\cup} B) \\ = {x, min {{\bar{μ}}_{A} (x), {\bar{μ}}_{B} (x)}, max {{\bar{γ}}_{A} (x), {\bar{γ}}_{B} (x)}}, \\ = ({\bar{μ}}_{A} (x) \land {\bar{μ}}_{B} (x), {\bar{γ}}_{A} (x) \lor {\bar{γ}}_{B} (x)), \\ and (A \bar{\cap} B) \\ = {x, max {{\bar{μ}}_{A} (x), {\bar{μ}}_{B} (x)}, min {{\bar{γ}}_{A} (x), {\bar{γ}}_{B} (x)}}, \\ = ({\bar{μ}}_{A} (x) \lor {\bar{μ}}_{B} (x), {\bar{γ}}_{A} (x) \land {\bar{γ}}_{B} (x)) . \end{matrix}$

Example 5. Let X = {a, b, c, d} be a nonempty set. Define ${\bar{μ}}_{A} : X \to [- 1, 0]$ and ${\bar{γ}}_{A} : X \to [- 1, 0]$ as, ${\bar{μ}}_{A} (a) = - 0.7,$ ${\bar{μ}}_{A} (b) = - 0.4,$ ${\bar{μ}}_{A} (c) = - 0.4,$ ${\bar{μ}}_{A} (d) = - 0.2$ and ${\bar{γ}}_{A} (a) = - 0.2,$ ${\bar{γ}}_{A} (b) = - 0.3,$ ${\bar{γ}}_{A} (c) = - 0.4,$ ${\bar{γ}}_{A} (d) = - 0.5,$ then A = {< a, - 0.7,-0.2 > , < b, - 0.4, - 0.3 > , < c, - 0.3, - 0.4 > , < d,-0.2, - 0.5 >} . It is easy to verify that $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A})$ is an intuitionistic N-fuzzy set in T .

Example 6. Let X = {a, b, c, d} and

A = {< a, - 1, 0 > , < b, - 0.7, - 0.2 > , < c, - 0.5, - 0.5 > , < d, - 0.3, - 0.6 >},

B = {< a, - 1, 0 > , < b, - 0.8, - 0.1 > , < c, - 0.6, - 0.3 > , < d, - 0.5, - 0.4 >} .

Then $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A})$ and $B = ({\bar{μ}}_{B}, {\bar{γ}}_{B})$ are intuitionistic N-fuzzy sets in X . Easily we can verify that $A \bar{\subseteq} B .$

Example 7. Let X = {a, b, c, d} and

A = {< a, - 0.8, - 0.1 > , < b, - 0.7, - 0.2 > , < c, - 0.5, - 0.5 > , < d, - 0.3, - 0.4 >}

B = {< a, - 0.7, - 0.2 > , < b, - 0.8, - 0.1 > , < c, - 0.6, - 0.3 > , < d, - 0.5, - 0.5 >} .

Then $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A})$ and $B = ({\bar{μ}}_{B}, {\bar{γ}}_{B})$ are intuitionistic N-fuzzy sets in X and

$A \bar{\cup} B = {< a, - 0.8, - 0.1 >, < b, - 0.8, - 0.1 >, < c, - 0.6, - 0.3 >, < d, - 0.5, - 0.4 >}$

$A \bar{\cap} B = {< a, - 0.7, - 0.2 >, < b, - 0.7, - 0.2 >, < c, - 0.5, - 0.5 >, < d, - 0.3, - 0.5 >} .$

Obviously, $A \bar{\cup} B$ and $A \bar{\cap} B$ are intuitionistic N-fuzzy sets in X .

Definition 13. Let S be a non-empty subset of X . Then the intuitionistic N-fuzzy characteristic function of S is a function ${\bar{χ}}_{S} = ({\bar{μ}}_{{\bar{χ}}_{S}}, {\bar{γ}}_{{\bar{χ}}_{S}})$ defined as, for any x ∈ X, $\begin{array}{l} {\bar{μ}}_{{\bar{χ}}_{S}} (x) = [\begin{matrix} - 1, if x \in S \\ 0, if x \notin S % \end{matrix} %] and \\ {\bar{γ}}_{{\bar{χ}}_{S}} (x) = [\begin{matrix} 0, if x \in S \\ - 1, if x \notin S % \end{matrix} %] . \end{array}$

We denotes the intuitionistic N-fuzzy characteristic function of X by ${\bar{χ}}_{X} = ({\bar{μ}}_{X}, {\bar{γ}}_{X}) .$

4 Institutionistic N-fuzzy sets in biΓ-ternary semigroup

In this section will apply the concept of intuitionistic N-fuzzy set to the ideals of biΓ-ternary semigroup and we will characterize these ideals in terms of intuitionistic N-fuzzy sets.

From here let T denotes a biΓ-ternary semigroup unless otherwise specified.

Definition 14. Let $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A}), B = ({\bar{μ}}_{B}, {\bar{γ}}_{B})$ and $C = ({\bar{μ}}_{C}, {\bar{γ}}_{C})$ be the three INFSs in T . Then their product is defined as, $A \circ_{Γ} B \circ_{Γ} C = ({\bar{μ}}_{A} \circ_{Γ} {\bar{μ}}_{B} \circ_{Γ} {\bar{μ}}_{C}, {\bar{γ}}_{A} \circ_{Γ} {\bar{γ}}_{B} \circ_{Γ} {\bar{γ}}_{C}) = ({\bar{μ}}_{A \circ_{Γ} B \circ_{Γ} C}, {\bar{γ}}_{A \circ_{Γ} B \circ_{Γ} C}),$ where for any x ∈ T, $\begin{matrix} ({\bar{μ}}_{A \circ_{Γ} B \circ_{Γ} C}) (x) \\ = [\begin{matrix} \underset{x = a α b β c}{\land} {{\bar{μ}}_{A} (a) \lor {\bar{μ}}_{B} (b) \lor {\bar{μ}}_{B} (c)}, if x = a α b β c \\ 0 otherwise \end{matrix}] \end{matrix}$ and $\begin{matrix} ({\bar{γ}}_{A \circ_{Γ} B \circ_{Γ} C}) (x) \\ = [\begin{matrix} \underset{x = a α b β c}{\lor} {{\bar{γ}}_{A} (a) \land {\bar{γ}}_{B} (b) \land {\bar{γ}}_{B} (c)}, if x = a α b β c \\ - 1 otherwise, \end{matrix}] \end{matrix}$

where a, b, c ∈ T, α, β ∈ Γ . Note that ${\bar{μ}}_{A \circ_{Γ} B \circ_{Γ} C} = {\bar{μ}}_{A} \circ_{Γ} {\bar{μ}}_{B} \circ_{Γ} {\bar{μ}}_{C}$ and ${\bar{γ}}_{A \circ_{Γ} B \circ_{Γ} C} = {\bar{γ}}_{A} \circ_{Γ} {\bar{γ}}_{B} \circ_{Γ} {\bar{γ}}_{C}$

4.1 Institutionistic N-fuzzy biΓ-ideals

Definition 15. Let $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A})$ be an INFS in T . Then A is called an intuitionistic N-fuzzy biΓ-ternary subsemigroup of T if for all x, y, z ∈ T, α, β ∈ Γ, $\begin{matrix} {\bar{μ}}_{A} (x α y β z) & \leq & max {{\bar{μ}}_{A} (x), {\bar{μ}}_{A} (y), {\bar{μ}}_{A} (z)} and \\ {\bar{γ}}_{A} (x α y β z) & \geq & min {{\bar{γ}}_{A} (x), {\bar{γ}}_{A} (y), {\bar{γ}}_{A} (z)} . \end{matrix}$

Definition 16. Let $L = ({\bar{μ}}_{L}, {\bar{γ}}_{L})$ be an INFS in T . Then L is called an intuitionistic N-fuzzy biΓ-left ideal of T if for all x, y, z ∈ T, α, β ∈ Γ . ${\bar{μ}}_{L} (x α y β z) \leq {\bar{μ}}_{L} (z) and {\bar{γ}}_{L} (x α y β z) \geq {\bar{γ}}_{L} (z) .$

Definition 17. Let $R = ({\bar{μ}}_{R}, {\bar{γ}}_{R})$ be an INFS in T . Then R is called an intuitionistic N-fuzzy biΓ-right ideal of T if for all x, y, z ∈ T, α, β ∈ Γ, ${\bar{μ}}_{R} (x α y β z) \leq {\bar{μ}}_{R} (x), and {\bar{γ}}_{R} (x α y β z) \geq {\bar{γ}}_{R} (x) .$

Definition 18. Let $M = ({\bar{μ}}_{M}, {\bar{γ}}_{M})$ be an INFS in T . Then M is called an intuitionistic N-fuzzy biΓ-lateral ideal of T if for all x, y, z ∈ T, α, β ∈ Γ, ${\bar{μ}}_{M} (x α y β z) \leq {\bar{μ}}_{M} (y) and {\bar{γ}}_{M} (x α y β z) \geq {\bar{γ}}_{M} (y) .$

Definition 19. Let $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A})$ be an INFS in T . Then A is called an intuitionistic N-fuzzy biΓ-ideal of T if it is a biΓ-left, a biΓ-right and a biΓ-lateral ideal of T .

Example 8. Let T be the biΓ-ternary semigroup given in Example 1. Define, ${\bar{μ}}_{A} : T \to [- 1, 0]$ and ${\bar{γ}}_{A} : T \to [- 1, 0]$ as, for x ∈ T, $\begin{matrix} {\bar{μ}}_{A} (x) & = & [\begin{matrix} - 0.5, if x is even \\ - 0.1, otherwise \end{matrix}] and \\ {\bar{γ}}_{A} (x) & = & [\begin{matrix} - 0.3, if x is even \\ - 0.7, otherwise \end{matrix}] . \end{matrix}$ Then $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A})$ is an intuitionistic N-fuzzy set in T . By simple calculations we can verify that A is an intuitionistic N-fuzzy biΓ-ternary subsemigroup of T .

Example 9. Let T = {a, b, c} and Γ = {α} . Then T is a biΓ-ternary semigroup under the operation defined in the following table,

α	a	b	c
a	a	a	a
b	a	b	b
c	a	c	c

Define, ${\bar{μ}}_{A} : T \to [- 1, 0]$ and ${\bar{γ}}_{A} : T \to [- 1, 0]$ such that $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A}) = {< a, - 0.3, - 0.1 >, < b, - 0.7, - 0.2 >, < c, - 0.5, - 0.5 >} .$ Then A is an intuitionistic N-fuzzy set in T which is an intuitionistic N-fuzzy biΓ-ternary subsemigroup of T . Further we can verify that A is not an intuitionistic N-fuzzy biΓ-left (biΓ-right, biΓ-lateral) ideal of T .If we take $B = ({\bar{μ}}_{B}, {\bar{γ}}_{B}) = {< a, - 0.7, - 0.2 >, < b, - 0.5, - 0.4 >, < c, - 0.5, - 0.4 >},$ then B is an intuitionistic N-fuzzy biΓ-left, a biΓ-right and a biΓ-lateral ideal of T, hence an intuitionistic N-fuzzy biΓ-ideal of T . Obviously it is an intuitionistic N-fuzzy biΓ-ternary subsemigroup of T .

Example 10. Let T = {2n, n ∈ N} and Γ = {α, β, γ, . . .} . Define (xαyβz) =2x + 2y + z, for x, y, z ∈ T, α, β ∈ Γ, then T is a biΓ-ternary semigroup. Now define, ${\bar{μ}}_{A} : T \to [- 1, 0]$ and ${\bar{γ}}_{A} : T \to [- 1, 0]$ as

$\begin{array}{l} {\bar{μ}}_{A} (x) = [\begin{matrix} - 0.2, if x = 4 n for some n \in N \\ - 0.5, otherwise \end{matrix}] and \\ {\bar{γ}}_{A} (x) = [\begin{matrix} - 0.7, if x = 4 n for some n \in N \\ - 0.3, otherwise \end{matrix}] . \end{array}$

$A = ({\bar{μ}}_{A}, {\bar{γ}}_{A}) is an intuitionistic N - fuzzy bi Γ - l e f t i d e a l o f T . N o w, f o r 2, 4, 6 T, α, β \in Γ, (2 α 6) β 4 = 2 (2) + 2 (6) + 4 = 20 = 4 (5) \Rightarrow {\bar{μ}}_{A} (2 α 6 β 4) = {\bar{μ}}_{A} (4 (5))$ $= - 0.2 a n d {\bar{μ}}_{A} (2) = - 0.5 i m p l i e s t h a t {\bar{μ}}_{A} (2 α 6 β 4) ≰ {\bar{μ}}_{A} (2) a l s o {\bar{γ}}_{A} (2 α 6 β 4) ≱ {\bar{γ}}_{A} (2) . H e n c e A = ({\bar{μ}}_{A}, {\bar{γ}}_{A}) i s n o t a n i n t u i t i o n i s t i c N - f u z z y b i Γ - r i g h t i d e a l o f T .$ $S i m i l a r l y, w e c a n v e r i f y t h a t A = ({\bar{μ}}_{A}, {\bar{γ}}_{A}) i s n o t a n i n t u i t i o n i s t i c N - f u z z y b i Γ - l a t e r a l i d e a l o f T . I f w e d e f i n e (x α y β z) = x + 2 y + 2 z o r (x α y β z) = 2 x + y + 2 z,$ $t h e n A = ({\bar{μ}}_{A}, {\bar{γ}}_{A}) i s a n i n t u i t i o n i s t i c N - f u z z y b i Γ - r i g h t i d e a l o f T a n d i n t u i t i o n i s t i c N - f u z z y b i Γ - l a t e r a l i d e a l o f T, r e s p e c t i v e l y b u t n o t i n t u i t i o n i s t i c$ $N - f u z z y b i Γ - l e f t i d e a l o f T . F u r t h e r m o r e i f (x α y β z) = 2 x + 2 y + 2 z, t h e n A = ({\bar{μ}}_{A}, {\bar{γ}}_{A}) i s a n i n t u i t i o n i s t i c N - f u z z y b i Γ - i d e a l o f T$

From Example 9 & 10, we can write the following remark.

Remark 1. In a biΓ-ternary semigroup T,

An intuitionistic N-fuzzy biΓ-left (right, lateral) ideal of T is an intuitionistic N-fuzzy biΓ-ternary subsemigroup of T but the converse is not true.

An intuitionistic N-fuzzy biΓ-left ideal of T may not be an intuitionistic N-fuzzy biΓ-right (lateral) ideal of T and vice versa.

Lemma 1. Let T be a biΓ-ternary semigroup then,

The intersection of any collection of intuitionistic N-fuzzy biΓ-ternary subsemigroups of T is an intuitionistic N-fuzzy biΓ-ternary subsemigroup of T .

The intersection of any collection of intuitionistic N-fuzzy biΓ-left (right, lateral) ideals of T is an intuitionistic N-fuzzy biΓ-left (right, lateral) ideal of T .

Proof. Straightforward. □

$\begin{array}{l} Proposition 2 . L e t A = ({\bar{μ}}_{A}, {\bar{γ}}_{A}) b e a n I N F S i n T t h e n \\ (i) A i s a n i n t u i t i o n i s t i c N - f u z z y b i Γ - t e r n a r y s u b s e m i g r o u p o f T i f a n d o n l y i f \\ {\bar{μ}}_{A} °_{Γ} {\bar{μ}}_{A} °_{Γ} % {\bar{μ}}_{A} \geq {\bar{μ}}_{A} and \bar{γ} %_{A} °_{Γ} {\bar{γ}}_{A} °_{Γ} {\bar{% γ}}_{A} \leq {\bar{γ}}_{A} . \\ (ii) A is an intuitionistic N -fuzzy bi Γ -left ideal of T if and only if \\ {\bar{μ}}_{T} °_{Γ} {\bar{μ}}_{T} °_{Γ} % {\bar{μ}}_{A} \geq {\bar{μ}}_{A} and \bar{γ} %_{T} °_{Γ} {\bar{γ}}_{T} °_{Γ} {\bar{γ}}_{A} \leq {\bar{γ}}_{A} . \end{array}$

(iii) A is an intuitionistic N-fuzzy biΓ-lateral ideal of T if and only if ${\bar{μ}}_{T} °_{Γ} {\bar{μ}}_{A} °_{Γ} {\bar{μ}}_{T} \geq {\bar{μ}}_{A} and {\bar{γ}}_{T} °_{Γ} {\bar{γ}}_{A} °_{Γ} {\bar{γ}}_{T} \leq {\bar{γ}}_{A} .$

(iv) A is an intuitionistic N-fuzzy biΓ-right ideal of T if and only if ${\bar{μ}}_{A} °_{Γ} {\bar{μ}}_{T} °_{Γ} {\bar{μ}}_{T} \geq {\bar{μ}}_{A} and {\bar{γ}}_{A} °_{Γ} {\bar{γ}}_{T} °_{Γ} {\bar{γ}}_{T} \leq {\bar{γ}}_{A} .$

Proof. (i) Let A is an intuitionistic N-fuzzy biΓ-ternary subsemigroup of T and x ∈ T .

Case 1. If x ≠ aαbβc, for α, β ∈ Γ, a, b, c ∈ T, then $\begin{matrix} ({\bar{μ}}_{A \circ_{Γ} A \circ_{Γ} A}) (x) & = & 0 \geq ({\bar{μ}}_{A}) (x) and \\ ({\bar{γ}}_{A \circ_{Γ} A \circ_{Γ} A}) (x) & = & - 1 \leq ({\bar{γ}}_{A}) (x) . \end{matrix}$ Case 2. If x = aαbβc, for α, β ∈ Γ and a, b, c ∈ T, then

$\begin{array}{l} ({\bar{μ}}_{A °_{Γ} A °_{Γ} A}) (x) \\ = \min_{x = a α b β c} {\max (({\bar{μ}}_{A}) (a), ({\bar{μ}}_{A}) (b), ({\bar{μ}}_{A}) (c))} \\ \geq \min_{x = a α b β c} ({\bar{μ}}_{A}) (a α b β c) \geq ({\bar{μ}}_{A}) (x), \forall x \in T . \end{array}$

Also $\begin{array}{l} ({\bar{γ}}_{A °_{Γ} A °_{Γ} A}) (x) \\ = \max_{x = a α b β c} {\min (({\bar{γ}}_{A}) (a), ({\bar{γ}}_{A}) (b), ({\bar{γ}}_{A}) (c))} \\ \leq \max_{x = a α b β c} ({\bar{γ}}_{A}) (a α b β c) \leq ({\bar{γ}}_{A}) (x), \forall x \in T . \end{array}$

This implies that ${\bar{μ}}_{A} \circ_{Γ} {\bar{μ}}_{A} \circ_{Γ} {\bar{μ}}_{A} \geq {\bar{μ}}_{A}$ and ${\bar{γ}}_{A} \circ_{Γ} {\bar{γ}}_{A} \circ_{Γ} {\bar{γ}}_{A} \leq {\bar{γ}}_{A} .$

Conversely, we suppose that ${\bar{μ}}_{A} \circ_{Γ} {\bar{μ}}_{A} \circ_{Γ} {\bar{μ}}_{A} \geq {\bar{μ}}_{A}$ and ${\bar{γ}}_{A} \circ_{Γ} {\bar{γ}}_{A} \circ_{Γ} {\bar{γ}}_{A} \leq {\bar{γ}}_{A} .$ Let, α, β ∈ Γ, a, b, c ∈ T and x = aαbβc then $\begin{matrix} ({\bar{μ}}_{A}) (a α b β c) \\ = ({\bar{μ}}_{A}) (x) \leq ({\bar{μ}}_{A \circ_{Γ} A \circ_{Γ} A}) (x) \\ = min_{x = u δ v θ w} {max (({\bar{μ}}_{A}) (u), ({\bar{μ}}_{A}) (v), ({\bar{μ}}_{A}) (w))} \\ \leq max (({\bar{μ}}_{A}) (a), ({\bar{μ}}_{A}) (b), ({\bar{μ}}_{A}) (c)) . \end{matrix}$

Also, $\begin{matrix} ({\bar{γ}}_{A}) (a α b β c) \\ = ({\bar{γ}}_{A}) (x) \leq ({\bar{γ}}_{A \circ_{Γ} A \circ_{Γ} A}) (x) \\ = max_{x = u δ v θ w} {min (({\bar{γ}}_{A}) (u), ({\bar{γ}}_{A}) (v), ({\bar{γ}}_{A}) (w))} \\ \geq min (({\bar{γ}}_{A}) (a), ({\bar{γ}}_{A}) (b), ({\bar{γ}}_{A}) (c)) . \end{matrix}$ Hence A is an intuitionistic N-fuzzy biΓ-ternary subsemigroup of T . Similarly, we can prove (ii), (iii) and (iv).□

Lemma 2. Let $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A})$ be an INFS in T then

T ∘ _ΓT ∘ _ΓA is an intuitionistic N-fuzzy biΓ-left ideal of T .

A ∘ _ΓT ∘ _ΓT is an intuitionistic N-fuzzy biΓ-right ideal of T .

T ∘ _ΓA ∘ _ΓT is an intuitionistic N-fuzzy biΓ-lateral ideal of T .

Proof. (i) Let L = T ∘ _ΓT ∘ _ΓA, then $L = ({\bar{μ}}_{L}, {\bar{γ}}_{L}) = ({\bar{μ}}_{T \circ_{Γ} T \circ_{Γ} A}, {\bar{γ}}_{T \circ_{Γ} T \circ_{Γ} A}) .$ Hence $\begin{matrix} {\bar{μ}}_{T} \circ_{Γ} {\bar{μ}}_{T} \circ_{Γ} {\bar{μ}}_{L} & = & {\bar{μ}}_{T \circ_{Γ} T \circ_{Γ} L} = {\bar{μ}}_{T \circ_{Γ} T \circ_{Γ} T \circ_{Γ} T \circ_{Γ} A} \\ \geq & {\bar{μ}}_{T \circ_{Γ} T \circ_{Γ} A} = {\bar{μ}}_{L} . \end{matrix}$ This implies that ${\bar{μ}}_{T} \circ_{Γ} {\bar{μ}}_{T} \circ_{Γ} {\bar{μ}}_{L} \geq {\bar{μ}}_{L} .$ Similarly, ${\bar{γ}}_{T} \circ_{Γ} γ_{T} \circ_{Γ} γ_{L} \leq {\bar{γ}}_{L} .$ Hence L = T ∘ _ΓT ∘ _ΓA is an intuitionistic N-fuzzy biΓ-left ideal of T . (ii) and (iii) can be proved in the same way.□

Theorem 1. Let $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A})$ be an INFS in T . Then A is an intuitionistic N-fuzzy biΓ-ternary subsemigroup of T if and only if N_A (t, s) is a biΓ-ternary subsemigroup of T, for all t, s ∈ [-1, 0] with t + s ≥ -1 .

Proof. Let $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A})$ be an intuitionistic N-fuzzy biΓ-ternary subsemigroup of T. Let x, y, z ∈ N_A (t, s) , where t, s ∈ [-1, 0] with t + s ≥ -1 then ${\bar{μ}}_{A} (x) \leq t, {\bar{γ}}_{A} (x) \geq s, {\bar{μ}}_{A} (y) \leq t, {\bar{γ}}_{A} (y) \geq s$ and ${\bar{μ}}_{A} (z) \leq t, {\bar{γ}}_{A} (z) \geq s .$ Now for α, β ∈ Γ, $\begin{matrix} {\bar{μ}}_{A} (x α y β z) & \leq & max {{\bar{μ}}_{A} (x), {\bar{μ}}_{A} (y), {\bar{μ}}_{A} (z)} \leq t and \\ {\bar{γ}}_{A} (x α y β z) & \geq & min {{\bar{γ}}_{A} (x), {\bar{γ}}_{A} (y), {\bar{γ}}_{A} (z)} \geq s . \end{matrix}$

This implies that xαyβz ∈ N_A (t, s), for all x, y, z ∈ N_A (t, s) and α, β ∈ Γ . Hence N_A (t, s) is a biΓ-ternary subsemigroup of T .

Conversely, we suppose that N_A (t, s) is a biΓ-ternary subsemigroup of T, for all t, s ∈ [-1, 0] with t + s ≥ -1 . Let x, y, z ∈ T such that ${\bar{μ}}_{A} (x) = t_{x}, {\bar{γ}}_{A} (x) = s_{x},$ ${\bar{μ}}_{A} (y) = t_{y}, {\bar{γ}}_{A} (y) = s_{y}$ and ${\bar{μ}}_{A} (z) = t_{z}, {\bar{γ}}_{A} (z) = s_{z}$ with -1 ≤ t_x + s_x ≤ 0, -1 ≤ t_y + s_y ≤ 0 and -1 ≤ t_z + s_z ≤ 0 then x ∈ N_A (t_x, s_x) , y ∈ N_A (t_y, s_y) and z ∈ N_A (t_z, s_z) . We may assume that t_x ≤ t_y ≤ t_z and s_x ≥ s_y ≥ s_z then N_A (t_x, s_x) ⊆ N_A (t_y, s_y) ⊆ N_A (t_z, s_z) , which implies that x, y, z ∈ N_A (t_z, s_z) . Since, N_A (t_z, s_z) is a biΓ-ternary subsemigroup of T implies that xαyβz ∈ N_A (t_z, s_z) , for α, β ∈ Γ . Then $\begin{matrix} {\bar{μ}}_{A} (x α y β z) & \leq & t_{z} = max (t_{x}, t_{y}, t_{z}) \\ = & max ({\bar{μ}}_{A} (x), {\bar{μ}}_{A} (y), {\bar{μ}}_{A} (z)), and \\ {\bar{γ}}_{A} (x α y β z) & \geq & s_{z} = min (s_{x}, s_{y}, s_{z}) \\ = & min ({\bar{γ}}_{A} (x), {\bar{γ}}_{A} (y), {\bar{γ}}_{A} (z)), \end{matrix}$

for all x, y, z ∈ T and α, β ∈ Γ . Hence $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A})$ is an intuitionistic N-fuzzy biΓ-ternary subsemigroup of T.□

Theorem 2. Let $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A})$ be an INFS in T . Then A is an intuitionistic N-fuzzy biΓ-left (right, lateral) ideal of T if and only if N_A (t, s) is a biΓ-left (right, lateral) ideal of T, for all t, s ∈ [-1, 0] with t + s ≥ -1 .

Proof. Straightforward.□

Theorem 3. A nonempty subset S of T is a biΓ-ternary subsemigroup of T if and only if ${\bar{χ}}_{S} = ({\bar{μ}}_{{\bar{χ}}_{S}}, {\bar{γ}}_{{\bar{χ}}_{S}})$ is an intuitionistic N-fuzzy biΓ-ternary subsemigroupof T .

Proof. Let S be a biΓ-ternary subsemigroup of T then SΓSΓS ⊆ S . Let x, y, z ∈ T, α, β ∈ Γ then we have following cases.

Case 1. If x, y, z ∈ S then xαyβz ∈ S andhence ${\bar{μ}}_{{\bar{χ}}_{S}} (x) = {\bar{μ}}_{{\bar{χ}}_{S}} (y) = {\bar{μ}}_{{\bar{χ}}_{S}} (z) = {\bar{μ}}_{{\bar{χ}}_{S}} (x α y β z) = - 1$ implies that ${\bar{μ}}_{{\bar{χ}}_{S}} (x α y β z) = max {{\bar{μ}}_{{\bar{χ}}_{S}} (x), {\bar{μ}}_{{\bar{χ}}_{S}} (y), {\bar{μ}}_{{\bar{χ}}_{S}}$ (z)} . Also, ${\bar{γ}}_{{\bar{χ}}_{S}} (x) = {\bar{γ}}_{{\bar{χ}}_{S}} (y) = {\bar{γ}}_{{\bar{χ}}_{S}} (z) = {\bar{γ}}_{{\bar{χ}}_{S}} (x α y β z)$ = 0 implies that ${\bar{γ}}_{{\bar{χ}}_{S}} (x α y β z) = min {{\bar{γ}}_{{\bar{χ}}_{S}} (x), {\bar{γ}}_{{\bar{χ}}_{S}} (y), {\bar{γ}}_{{\bar{χ}}_{S}} (z)} .$

Case 2. If either x ∉ A or y ∉ A or z ∉ A then either ${\bar{μ}}_{{\bar{χ}}_{S}} (x) = 0, {\bar{γ}}_{{\bar{χ}}_{S}} (x) = - 1$ or ${\bar{μ}}_{{\bar{χ}}_{S}} (y) = 0, {\bar{γ}}_{{\bar{χ}}_{S}} (y) = - 1$ or ${\bar{μ}}_{{\bar{χ}}_{S}} (z) = 0, {\bar{γ}}_{{\bar{χ}}_{S}} (z) = - 1 .$ This implies that, $max {{\bar{μ}}_{{\bar{χ}}_{S}} (x), {\bar{μ}}_{{\bar{χ}}_{S}} (y), {\bar{μ}}_{{\bar{χ}}_{S}} (z)} = 0$ but ${\bar{μ}}_{{\bar{χ}}_{S}} (x α y β z) \leq 0$ implies that ${\bar{μ}}_{{\bar{χ}}_{S}} (x α y β z) \leq max {{\bar{μ}}_{{\bar{χ}}_{S}} (x), {\bar{μ}}_{{\bar{χ}}_{S}} (y)$ , ${\bar{μ}}_{{\bar{χ}}_{S}} (z)}$ . Also, either ${\bar{γ}}_{{\bar{χ}}_{S}} (x) = - 1$ or ${\bar{γ}}_{{\bar{χ}}_{S}} (y) = - 1$ or ${\bar{γ}}_{{\bar{χ}}_{S}} (z) = - 1,$ which implies that, $min {{\bar{γ}}_{{\bar{χ}}_{S}} (x)$ , ${\bar{γ}}_{{\bar{χ}}_{S}} (y), {\bar{γ}}_{{\bar{χ}}_{S}} (z)} = - 1$ but ${\bar{γ}}_{{\bar{χ}}_{S}} (x α y β z) \geq - 1$ implies that ${\bar{γ}}_{{\bar{χ}}_{S}} (x α y β z) \geq min {{\bar{γ}}_{{\bar{χ}}_{S}} (x), {\bar{γ}}_{{\bar{χ}}_{S}} (y), {\bar{γ}}_{{\bar{χ}}_{S}} (z)}$ . Thisimplies that ${\bar{μ}}_{{\bar{χ}}_{S}} (x α y β z) \leq max {{\bar{μ}}_{{\bar{χ}}_{S}} (x), {\bar{μ}}_{{\bar{χ}}_{S}} (y), {\bar{μ}}_{{\bar{χ}}_{S}}$ (z)} and ${\bar{γ}}_{{\bar{χ}}_{S}} (x α y β z) \geq min {{\bar{γ}}_{{\bar{χ}}_{S}} (x), {\bar{γ}}_{{\bar{χ}}_{S}} (y),$ ${\bar{γ}}_{{\bar{χ}}_{S}} (z)},$ for all x, y, z ∈ T, α, β ∈ Γ .

Case 3. When any two of x, y, z are not in S.

Case 4. When all x, y, z are not in S .

Above both cases gives the same results as in Case 2. Hence ${\bar{χ}}_{S} = ({\bar{μ}}_{{\bar{χ}}_{S}}, {\bar{γ}}_{{\bar{χ}}_{S}})$ is an intuitionistic N-fuzzy biΓ-ternary subsemigroup of T .

Conversely, we suppose that ${\bar{χ}}_{S} = ({\bar{μ}}_{{\bar{χ}}_{S}}, {\bar{γ}}_{{\bar{χ}}_{S}})$ is an intuitionistic N-fuzzy biΓ-ternary subsemigroup of T . Let x, y, z ∈ S and α, β ∈ Γ then xαyβz ∈ SΓSΓS . By definition of ${\bar{χ}}_{S},$ ${\bar{μ}}_{{\bar{χ}}_{S}} (x) = {\bar{μ}}_{{\bar{χ}}_{S}} (y) = {\bar{μ}}_{{\bar{χ}}_{S}} (z) = - 1$ implies that $max {{\bar{μ}}_{{\bar{χ}}_{S}} (x), {\bar{μ}}_{{\bar{χ}}_{S}} (y), {\bar{μ}}_{{\bar{χ}}_{S}} (z)}$ = -1 . Since ${\bar{χ}}_{S}$ is an intuitionistic N-fuzzy biΓ-ternary subsemigroup of T then ${\bar{μ}}_{{\bar{χ}}_{S}} (x α y β z) \leq max {{\bar{μ}}_{{\bar{χ}}_{S}} (x), {\bar{μ}}_{{\bar{χ}}_{S}} (y), {\bar{μ}}_{{\bar{χ}}_{S}} (z)} = - 1$ implies that ${\bar{μ}}_{{\bar{χ}}_{S}} (x α y β z) \leq - 1$ but by definition ${\bar{μ}}_{{\bar{χ}}_{S}} (x α y β z) \geq - 1,$ which implies that ${\bar{μ}}_{{\bar{χ}}_{S}} (x α y β z) = - 1 .$ Similarly, we can show that ${\bar{γ}}_{{\bar{χ}}_{S}} (x α y β z) = 0 .$ This gives that xαyβz ∈ S implies that SΓSΓS ⊆ S . Hence S is a biΓ-ternary subsemigroup of T . □

Theorem 4. A non-empty subset S of T is a biΓ-left (right, lateral) ideal of T if and only if ${\bar{χ}}_{S} = ({\bar{μ}}_{{\bar{χ}}_{S}}, {\bar{γ}}_{{\bar{χ}}_{S}})$ is an intuitionistic N-fuzzy biΓ-left (right, lateral) ideal of T .

Proof. Straightforward.□

Definition 20. Let S be a nonempty subset of T and a, b ∈ [-1, 0] with a ≤ b . Define an intuitionistic N-fuzzy set in T as $S_{a}^{b} = ({\bar{μ}}_{S_{a}^{b}}, {\bar{γ}}_{S_{a}^{b}}),$ where, ${\bar{μ}}_{S_{a}^{b}} (x) = [\begin{matrix} b if x \notin S \\ a if x \in S \end{matrix}] and {\bar{γ}}_{S_{a}^{b}} (x) = [\begin{matrix} a if x \notin S \\ b if x \in S \end{matrix}] .$

Lemma 3. A nonempty subset S of T is a biΓ-ternary subsemigroup (left ideal, right ideal, lateral ideal) of T if and only if $S_{a}^{b}$ is an intuitionistic N-fuzzy biΓ-ternary subsemigroup (left ideal, right ideal, lateral ideal) of T .

Proof. We prove this result for biΓ-right ideals. Let S be a biΓ- right ideal of T and x, y, z ∈ T . If x ∈ S then xαyβz ∈ S implies that ${\bar{μ}}_{S_{a}^{b}} (x) = a = {\bar{μ}}_{S_{a}^{b}} (x α y β z)$ and ${\bar{γ}}_{S_{a}^{b}} (x) = b = {\bar{γ}}_{S_{a}^{b}} (x α y β z) .$ If x ∉ S then ${\bar{μ}}_{S_{a}^{b}} (x) = b \geq {\bar{μ}}_{S_{a}^{b}} (x α y β z)$ and ${\bar{γ}}_{S_{a}^{b}} (x) = a \leq {\bar{γ}}_{S_{a}^{b}} (x α y β z) .$ Hence $S_{a}^{b} = ({\bar{μ}}_{S_{a}^{b}}, {\bar{γ}}_{S_{a}^{b}})$ is an intuitionistic N-fuzzy biΓ-right ideal of T .

Conversely, we suppose that $S_{a}^{b} = ({\bar{μ}}_{S_{a}^{b}}, {\bar{γ}}_{S_{a}^{b}})$ is an intuitionistic N-fuzzy biΓ-right ideal of T . Let x ∈ S then ${\bar{μ}}_{S_{a}^{b}} (x) = a$ and ${\bar{γ}}_{S_{a}^{b}} (x) = b .$ For y, z ∈ T and $α, β \in Γ, {\bar{μ}}_{S_{a}^{b}} (x α y β z) \leq {\bar{μ}}_{S_{a}^{b}} (x) = a$ but ${\bar{μ}}_{S_{a}^{b}} (x α y β z) \geq a$ implies that ${\bar{μ}}_{S_{a}^{b}} (x α y β z) = a$ implies that xαyβz ∈ S ⇒ SΓTΓT ⊆ S . Hence S is a biΓ-right ideal of T . The result for other cases is similar.□

4.2 Intuitionistic N-fuzzy biΓ-bi-ideals

Definition 21. Let $B = ({\bar{μ}}_{B}, {\bar{γ}}_{B})$ be an INFS in T . Then B is called an intuitionistic N-fuzzy biΓ-bi-ideal ofT if,

B is an intuitionistic N-fuzzy biΓ-ternary subsemigroup of T .

For all x, y, z ∈ T, α, β, η, δ ∈ Γ, $\begin{matrix} {\bar{μ}}_{B} (x α u β y η v δ z) & \leq & max ({\bar{μ}}_{B} (x), {\bar{μ}}_{B} (y), {\bar{μ}}_{B} (z)) \\ and \\ {\bar{γ}}_{B} (x α u β y η v δ z) & \geq & min ({\bar{γ}}_{B} (x), {\bar{γ}}_{B} (y), {\bar{γ}}_{B} (z)) . \end{matrix}$

Example 11. Let T be a biΓ-ternary semigroup as given in Example 9 . Define, ${\bar{μ}}_{A} : T \to [- 1, 0]$ and ${\bar{γ}}_{A} : T \to [- 1, 0]$ such that $\begin{matrix} A & = & ({\bar{μ}}_{A}, {\bar{γ}}_{A}) = {< a, - 0.9, - 0.1 >, < b, - 0.7, \\ - 0.2 >, < c, - 0.5, - 0.5 >} . \end{matrix}$

Then $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A})$ is an intuitionistic N-fuzzy biΓ-bi-ideal of T .

Example 12. Let T be the biΓ-ternary semigroup as given in Example 1. Define, ${\bar{μ}}_{A} : T \to [- 1, 0]$ and ${\bar{γ}}_{A} : T \to [- 1, 0]$ as, for x ∈ T, $\begin{matrix} {\bar{μ}}_{A} (x) & = & [\begin{matrix} - 0.7, if x is even \\ - 0.1, otherwise \end{matrix}] and \\ {\bar{γ}}_{A} (x) & = & [\begin{matrix} - 0.3, if x is even \\ - 0.7, otherwise \end{matrix}] . \end{matrix}$ Then $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A})$ is an intuitionistic N-fuzzy set in T . By simple calculations we can verify that A is an intuitionistic N-fuzzy biΓ-bi-ideal of T .

Proposition 3. Let $B = ({\bar{μ}}_{B}, {\bar{γ}}_{B})$ be an INFS in T . Then B is an intuitionistic N-fuzzy biΓ-bi-ideal of T if and only if $(1) {\bar{μ}}_{B} °_{Γ} {\bar{μ}}_{B} °_{Γ} {\bar{μ}}_{B} \geq {\bar{μ}}_{B} and {\bar{γ}}_{B} °_{Γ} {\bar{γ}}_{B} °_{Γ} {\bar{γ}}_{B} \leq {\bar{γ}}_{B}$

$\begin{array}{l} (2) {\bar{μ}}_{B} °_{Γ} {\bar{μ}}_{T} °_{Γ} {\bar{μ}}_{B} °_{Γ} {\bar{μ}}_{T} °_{Γ} {\bar{μ}}_{B} \geq {\bar{μ}}_{B} and \\ {\bar{γ}}_{B} °_{Γ} {\bar{γ}}_{T} °_{Γ} {\bar{γ}}_{B} °_{Γ} {\bar{γ}}_{T} °_{Γ} {\bar{γ}}_{B} \leq {\bar{γ}}_{B} \end{array}$

Proof. We suppose that $B = ({\bar{μ}}_{B}, {\bar{γ}}_{B})$ is an intuitionistic N-fuzzy biΓ-bi-ideal of T then it is biΓ-ternary subsemigroup of T and by Proposition 2, (1) holds. Now for (2) , let m ∈ T . If m ≠ xαyβz, for x, y, z ∈ T and α, β ∈ Γ then $\begin{matrix} ({\bar{μ}}_{B} \circ_{Γ} {\bar{μ}}_{T} \circ_{Γ} {\bar{μ}}_{B} \circ_{Γ} {\bar{μ}}_{T} \circ_{Γ} {\bar{μ}}_{B}) (m) & = & 0 \geq {\bar{μ}}_{B} (m) and \\ ({\bar{γ}}_{B} \circ_{Γ} γ_{T} \circ_{Γ} {\bar{γ}}_{B} \circ_{Γ} γ_{T} \circ_{Γ} {\bar{γ}}_{B}) (m) & = & - 1 \leq {\bar{γ}}_{B} (m) . \end{matrix}$ If m = xαyβz and x = uδvθw, for u, v, w ∈ T and δ, θ ∈ Γ then $\begin{matrix} ({\bar{μ}}_{B} \circ_{Γ} {\bar{μ}}_{T} \circ_{Γ} {\bar{μ}}_{B} \circ_{Γ} {\bar{μ}}_{T} \circ_{Γ} {\bar{μ}}_{B}) (m) \\ = min_{m = x α y β z} {max {({\bar{μ}}_{B} \circ_{Γ} {\bar{μ}}_{T} \circ_{Γ} {\bar{μ}}_{B}) (x), \\ {\bar{μ}}_{T} (y), {\bar{μ}}_{B} (z)}} \\ = min_{m = x α y β z} {max {min_{x = u δ v θ w} {max ({\bar{μ}}_{B} (u), {\bar{μ}}_{T} (v), \\ {\bar{μ}}_{B} (w)), {\bar{μ}}_{T} (y), {\bar{μ}}_{B} (z)}} \\ = min_{m = x α y β z} {max {min_{x = u δ v θ w} {max ({\bar{μ}}_{B} (u), - 1, \\ {\bar{μ}}_{B} (w)), - 1, {\bar{μ}}_{B} (z)}} \end{matrix}$

$\begin{matrix} = min_{m = x α y β z} {min_{x = u δ v θ w} {max {max ({\bar{μ}}_{B} (u), \\ {\bar{μ}}_{B} (w)), {\bar{μ}}_{B} (z)}} \\ = min_{m = u δ v θ w α y β z} {max ({\bar{μ}}_{B} (u), {\bar{μ}}_{B} (w), {\bar{μ}}_{B} (z)} \\ \geq min_{m = u δ v θ w α y β z} {\bar{μ}}_{B} (u δ v θ w α y β z), = {\bar{μ}}_{B} (m) \\ \Rightarrow ({\bar{μ}}_{B} \circ_{Γ} {\bar{μ}}_{T} \circ_{Γ} {\bar{μ}}_{B} \circ_{Γ} {\bar{μ}}_{T} \circ_{Γ} {\bar{μ}}_{B}) (m) \geq {\bar{μ}}_{B} (m), \end{matrix}$ for allm ∈ T. This implies that

${\bar{μ}}_{B} \circ_{Γ} {\bar{μ}}_{T} \circ_{Γ} {\bar{μ}}_{B} \circ_{Γ} {\bar{μ}}_{T} \circ_{Γ} {\bar{μ}}_{B} \geq {\bar{μ}}_{B} .$ Similarly, we can prove that ${\bar{γ}}_{B} \circ_{Γ} {\bar{γ}}_{T} \circ_{Γ} {\bar{γ}}_{B} \circ_{Γ} {\bar{γ}}_{T} \circ_{Γ} {\bar{γ}}_{B} \leq {\bar{γ}}_{B} .$ Conversely, we suppose that (1) and (2) holds for any intuitionistic N-fuzzy subset $B = ({\bar{μ}}_{B}, {\bar{γ}}_{B})$ of T . Let m = xαuβyδvθz for x, y, z, u, v ∈ T, α, β, δ, θ ∈ Γ then $\begin{matrix} {\bar{μ}}_{B} (x α u β y δ v θ z) \\ = {\bar{μ}}_{B} (m) \leq ({\bar{μ}}_{B} \circ_{Γ} {\bar{μ}}_{T} \circ_{Γ} {\bar{μ}}_{B} \circ_{Γ} {\bar{μ}}_{T} \circ_{Γ} {\bar{μ}}_{B}) (m) \\ = ({\bar{μ}}_{B} \circ_{Γ} {\bar{μ}}_{T} \circ_{Γ} {\bar{μ}}_{B} \circ_{Γ} {\bar{μ}}_{T} \circ_{Γ} {\bar{μ}}_{B}) (x α u β y δ v θ z) \\ = min_{m = x α u β y δ v θ z} {max {({\bar{μ}}_{B} \circ_{Γ} {\bar{μ}}_{T} \circ_{Γ} {\bar{μ}}_{B}) (x α u β y), \\ {\bar{μ}}_{T} (v), {\bar{μ}}_{B} (z)}} \\ \leq max {{max ({\bar{μ}}_{B} (x), {\bar{μ}}_{T} (u), {\bar{μ}}_{B} (y))}, {\bar{μ}}_{T} (v), \\ {\bar{μ}}_{B} (z)} \\ = min_{n = x α u β y} {max {max ({\bar{μ}}_{B} (x), - 1, {\bar{μ}}_{B} (y)), - 1, \\ {\bar{μ}}_{B} (z)}} \\ \leq max ({\bar{μ}}_{B} (x), {\bar{μ}}_{B} (y), {\bar{μ}}_{B} (z)) . \\ {\bar{γ}}_{B} (x α u β y δ v θ z) \geq min ({\bar{γ}}_{B} (x), {\bar{γ}}_{B} (y), {\bar{γ}}_{B} (z)) . \end{matrix}$

Hence $B = ({\bar{μ}}_{B}, {\bar{γ}}_{B})$ is an intuitionistic N-fuzzy biΓ-bi-ideal of T .□

Lemma 4. Let X = (μ_X, γ_X) and Y = (μ_Y, γ_Y) be two INFSs in T then X ∘ _ΓT ∘ _ΓY is an intuitionistic N-fuzzy biΓ-bi-ideal of T .

Proof. Straightforward.□

Theorem 5. Let X = (μ_X, γ_X) , Y = (μ_Y, γ_Y) and Z = (μ_Z, γ_Z) be three INFSs in T . Then X ∘ _ΓZ ∘ _ΓY is an intuitionistic N-fuzzy biΓ-bi-ideal of T if any one of X, Y or Z is either an intuitionistic N-fuzzy biΓ-left ideal or an intuitionistic N-fuzzy biΓ-right ideal or an intuitionistic N-fuzzy biΓ-lateral ideal of T .

Proof. Straightforward.

Lemma 5. Every intuitionistic N-fuzzy biΓ-left (right, lateral)-ideal of T is an intuitionistic N-fuzzy biΓ-bi-ideal of T .

Proof. Straightforward.

Lemma 6. Let {B_i, i ∈ I} be a collection of intuitionistic N-fuzzy biΓ-bi-ideals of T then $\underset{i \in I}{\bar{\cap}} B_{i}$ is also an intuitionistic N-fuzzy biΓ-bi-ideal of T .

Proof. Straightforward.□

Theorem 6. Let $B = ({\bar{μ}}_{B}, {\bar{γ}}_{B})$ be an INFS in T . Then B is an intuitionistic N-fuzzy biΓ-bi-ideal of T if and only if N_B (t, s) is a biΓ-bi-ideal of T, for all t, s ∈ [-1, 0] with t + s ≥ -1 .

Proof. We suppose that $B = ({\bar{μ}}_{B}, {\bar{γ}}_{B})$ is an intuitionistic N-fuzzy biΓ-bi-ideal of T and m ∈ N_B (t, s) ΓTΓN_B (t, s) ΓTΓN_B (t, s) .

Then m = nαxβoδyθq for n, o, q ∈ N_B (t, s) , x, y ∈ T and α, β, δ, θ ∈ Γ . Since n, o, q ∈ N_B (t, s) implies that ${\bar{μ}}_{B} (n), {\bar{μ}}_{B} (o), {\bar{μ}}_{B} (q) \leq t$ and ${\bar{γ}}_{B} (n), {\bar{γ}}_{B} (o), {\bar{γ}}_{B} (q) \geq s .$ Now, since B is intuitionistic N-fuzzy biΓ-bi-ideal of T then $\begin{matrix} {\bar{μ}}_{B} (m) & = & {\bar{μ}}_{B} (n α x β o δ y θ q) \\ \leq & max ({\bar{μ}}_{B} (n), {\bar{μ}}_{B} (o), {\bar{μ}}_{B} (q))} \leq max (t, t, t) \\ \leq & t, and \\ {\bar{γ}}_{B} (m) & = & {\bar{γ}}_{B} (n α x β o δ y θ q) \\ \geq & min ({\bar{γ}}_{B} (n), {\bar{γ}}_{B} (o), {\bar{γ}}_{B} (q))} \geq min (s, s, s) \\ \geq & s . \end{matrix}$

This implies that m ∈ N_B (t, s) and hence, N_B (t, s) ΓTΓN_B (t, s) ΓTΓN_B (t, s) ⊆ N_B (t, s) , which showsthat, N_B (t, s) is a biΓ-bi-ideal of T .

Conversely, we suppose that N_B (t, s) is a biΓ-bi-ideal of T, for all t, s ∈ [-1, 0] with t + s ≥ -1 . Let x, y, z ∈ T such that ${\bar{μ}}_{B} (x) = t_{x}, {\bar{γ}}_{B} (x) = s_{x}$ with -1 ≤ t_x + s_x ≤ 0, ${\bar{μ}}_{B} (y) = t_{y}, {\bar{γ}}_{B} (y) = s_{y}$ with -1 ≤ t_y + s_y ≤ 0 and ${\bar{μ}}_{B} (z) = t_{z}, {\bar{γ}}_{B} (z) = s_{z}$ with -1 ≤ t_z + s_z ≤ 0 . Then x ∈ N_B (t_x, s_x) , y ∈ N_B (t_y, s_y) and z ∈ N_B (t_z, s_z) . We may assume that t_x ≤ t_y ≤ t_z and s_x ≥ s_y ≥ s_z then N_B (t_x, s_x) ⊆ N_B (t_y, s_y) ⊆N_B (t_z, s_z) . This implies that x, y, z ∈ N_B (t_z, s_z) . Since N_B (t_z, s_z) is a biΓ-bi-ideal of T then for u, v ∈ T, α, β, η, θ ∈ Γ, xαuβyδvθz ∈ N_B (t_z, s_z), we have

$\begin{matrix} {\bar{μ}}_{B} (x α u β y δ v θ z) \leq t_{z} & = & max (t_{x}, t_{y}, t_{z}) \\ = & max ({\bar{μ}}_{B} (x), {\bar{μ}}_{B} (y), {\bar{μ}}_{B} (z)), and \\ {\bar{γ}}_{B} (x α u β y δ v θ z) \geq s_{z} & = & min (s_{x}, s_{y}, s_{z}) \\ = & min ({\bar{γ}}_{B} (x), {\bar{γ}}_{B} (y), {\bar{γ}}_{B} (z)) . \end{matrix}$

Above holds for all x, y, z, u, v ∈ T and α, β, η, θ ∈ Γ . Hence $B = ({\bar{μ}}_{B}, {\bar{γ}}_{B})$ is an intuitionistic N-fuzzy biΓ-bi-ideal of T .□

Theorem 7. A nonempty subset S of T is a biΓ-bi-ideal of T if and only if ${\bar{χ}}_{S} = ({\bar{μ}}_{{\bar{χ}}_{S}}, {\bar{γ}}_{{\bar{χ}}_{S}})$ is an intuitionistic N-fuzzy biΓ-bi-ideal of T .

Proof. We suppose that S is a biΓ-bi-ideal of T then it is a biΓ-ternary subsemigroup of T and by Theorem 3, ${\bar{χ}}_{S} = ({\bar{μ}}_{{\bar{χ}}_{S}}, {\bar{γ}}_{{\bar{χ}}_{S}})$ is an intuitionistic N-fuzzy biΓ-ternary subsemigroup of T . Also S ΓTΓSΓTΓS ⊆ S . Now for any x, y, z, u, v ∈ T, α, β, η, θ ∈ Γ, xαuβyηvθz ∈ T. We have following cases,

(i) If x, y, z ∈ S then, xαuβyηvθz ∈ SΓTΓSΓTΓS ⊆ S implies that ${\bar{μ}}_{{\bar{χ}}_{S}} (x) = {\bar{μ}}_{{\bar{χ}}_{S}} (y) = {\bar{μ}}_{{\bar{χ}}_{S}} (z) = - 1 = {\bar{μ}}_{{\bar{χ}}_{S}} (x α u β y η v θ z)$ and ${\bar{γ}}_{{\bar{χ}}_{S}} (x) = {\bar{γ}}_{{\bar{χ}}_{S}} (y) = {\bar{γ}}_{{\bar{χ}}_{S}} (z) = 0 = {\bar{γ}}_{{\bar{χ}}_{S}} (x α u β y η v θ z) .$ Hence

${\bar{μ}}_{{\bar{χ}}_{S}} (x α u β y η v θ z) = max ({\bar{μ}}_{{\bar{χ}}_{S}} (x), {\bar{μ}}_{{\bar{χ}}_{S}} (y), {\bar{μ}}_{{\bar{χ}}_{S}} (z)) .$ Also

${\bar{γ}}_{{\bar{χ}}_{S}} (x α u β y η v θ z) = min ({\bar{γ}}_{{\bar{χ}}_{S}} (x), {\bar{γ}}_{{\bar{χ}}_{S}} (y), {\bar{γ}}_{{\bar{χ}}_{S}} (z)) .$

(ii) If either x ∉ S or y ∉ S or z ∉ S then either ${\bar{μ}}_{{\bar{χ}}_{S}} (x) = 0, {\bar{γ}}_{{\bar{χ}}_{S}} (x) = - 1$ or ${\bar{μ}}_{{\bar{χ}}_{S}} (y) = 0, {\bar{γ}}_{{\bar{χ}}_{S}} (y) = - 1$ or ${\bar{μ}}_{{\bar{χ}}_{S}} (z) = 0, {\bar{γ}}_{{\bar{χ}}_{S}} (z) = - 1$ implies that $\begin{matrix} max ({\bar{μ}}_{{\bar{χ}}_{S}} (x), {\bar{μ}}_{{\bar{χ}}_{S}} (y), {\bar{μ}}_{{\bar{χ}}_{S}} (z)) = 0 and \\ min ({\bar{γ}}_{{\bar{χ}}_{S}} (x), {\bar{γ}}_{{\bar{χ}}_{S}} (y), {\bar{γ}}_{{\bar{χ}}_{S}} (z)) = - 1 . \end{matrix}$ But

${\bar{μ}}_{{\bar{χ}}_{S}} (x α u β y η v θ z) \leq 0$ and ${\bar{γ}}_{{\bar{χ}}_{S}} (x α u β y η v θ z) \geq - 1$ implies that

${\bar{μ}}_{{\bar{χ}}_{S}} (x α u β y η v θ z) \leq max ({\bar{μ}}_{{\bar{χ}}_{S}} (x), {\bar{μ}}_{{\bar{χ}}_{S}} (y), {\bar{μ}}_{{\bar{χ}}_{S}} (z))$ and

${\bar{γ}}_{{\bar{χ}}_{S}} (x α u β y η v θ z) \geq min ({\bar{γ}}_{{\bar{χ}}_{S}} (x), {\bar{γ}}_{{\bar{χ}}_{S}} (y), {\bar{γ}}_{{\bar{χ}}_{S}} (z)) .$

(iii) If any two of x, y, z are not in S .

(iv) If x ∉ S and y ∉ S and z ∉ S .

Both (iii) and (iv) are same like (ii).

Hence, ${\bar{χ}}_{S} = ({\bar{μ}}_{{\bar{χ}}_{S}}, {\bar{γ}}_{{\bar{χ}}_{S}})$ is an intuitionistic N-fuzzy biΓ-bi-ideal of T .

Conversely, we suppose that ${\bar{χ}}_{S} = ({\bar{μ}}_{{\bar{χ}}_{S}}, {\bar{γ}}_{{\bar{χ}}_{S}})$ is an intuitionistic N-fuzzy biΓ-bi-ideal of T . For any t ∈ SΓTΓSΓTΓS there exists x, y, z ∈ S, u, v ∈ T and α, β, η, θ ∈ Γ such that t = xαuβyηvθz . Then ${\bar{μ}}_{{\bar{χ}}_{S}} (x) = {\bar{μ}}_{{\bar{χ}}_{S}} (y) = {\bar{μ}}_{{\bar{χ}}_{S}} (z) = - 1$ implies that $max ({\bar{μ}}_{{\bar{χ}}_{S}} (x), {\bar{μ}}_{{\bar{χ}}_{S}} (y), {\bar{μ}}_{{\bar{χ}}_{S}} (z)) = - 1$ and ${\bar{γ}}_{{\bar{χ}}_{S}} (x) = {\bar{γ}}_{{\bar{χ}}_{S}}$ $(y) = {\bar{γ}}_{{\bar{χ}}_{S}} (z) = 0$ implies that $min ({\bar{γ}}_{{\bar{χ}}_{S}} (x), {\bar{γ}}_{{\bar{χ}}_{S}} (y)$ , ${\bar{γ}}_{{\bar{χ}}_{S}} (z)) = 0 .$

Since ${\bar{χ}}_{S} = ({\bar{μ}}_{{\bar{χ}}_{S}}, {\bar{γ}}_{{\bar{χ}}_{S}})$ is an intuitionistic N-fuzzy biΓ-bi-ideal of T implies that $\begin{matrix} {\bar{μ}}_{{\bar{χ}}_{S}} (x α u β y η v θ z) & \leq & max ({\bar{μ}}_{{\bar{χ}}_{S}} (x), {\bar{μ}}_{{\bar{χ}}_{S}} (y), {\bar{μ}}_{{\bar{χ}}_{S}} (z)) \\ = & - 1 and \\ {\bar{γ}}_{{\bar{χ}}_{S}} (x α u β y η v θ z) & \geq & min ({\bar{γ}}_{{\bar{χ}}_{S}} (x), {\bar{γ}}_{{\bar{χ}}_{S}} (y), {\bar{γ}}_{{\bar{χ}}_{S}} (z)) \\ = & 0 . \end{matrix}$

But by definition ${\bar{μ}}_{{\bar{χ}}_{S}} (x α u β y η v θ z) \geq - 1$ and ${\bar{γ}}_{{\bar{χ}}_{S}} (x α u β y η v θ z) \leq 0 .$ This gives that ${\bar{μ}}_{{\bar{χ}}_{S}}$ (xαuβyηvθz) = -1 and ${\bar{γ}}_{{\bar{χ}}_{S}} (x α u β y η v θ z) = 0$ impliesthat t = xαuβyηvθz ∈ S . This gives that SΓTΓSΓTΓS ⊆ S . Hence S is a biΓ-bi-ideal of T . □

Lemma 7. A non-empty subset S of T is a biΓ-bi-ideal of T if and only if $S_{a}^{b}$ is an intuitionistic N-fuzzy biΓ-bi-ideal of T .

Proof. Straightforward. □

The following examples shows the use of level sets to characterize biΓ-ternary semigroups.

Example 13. Let T = {a, b, c} and Γ = {α} . Then T is a biΓ-ternary semigroup with the operation as given in Example 9 . Then {b, c} is a biΓ-ternary subsemigroup of T but not a biΓ-bi-ideal of T . Now, define ${\bar{μ}}_{A} : T \to [- 1, 0]$ and ${\bar{γ}}_{A} : T \to [- 1, 0]$ as ${\bar{μ}}_{A} (a) = {\bar{μ}}_{A} (c) = - 0.5,$ ${\bar{μ}}_{A} (b) = - 0.8$ and ${\bar{γ}}_{A} (a) = {\bar{γ}}_{A} (c) = - 0.2,$ ${\bar{γ}}_{Q} (b) = - 0.1 .$ Then $N_{A} (t, s) = [\begin{matrix} T if t, s \in [-0 . 2, 0] \\ {a, c} if t, s \in [-0 . 5, -0 . 2) \\ {b, c} if t, s \in [-0 . 8, -0 . 5) \\ Φ if t, s \in [-1, -0 . 8) \end{matrix}] .$

Obviously, N_A (t, s) is a biΓ-ternary subsemigroup of T but not a biΓ-bi-ideal of T, for all t, s ∈ [-1, 0] with t + s ≥ -1 . Then by Theorem 1 and Theorem 6, $A = ({\bar{μ}}_{A}, {\bar{γ}}_{A})$ is an intuitionistic N-fuzzy biΓ-ternary subsemigroup of T but not an intuitionistic N-fuzzy biΓ-bi-ideal of T .

Example 14. In above example {a, c} is a biΓ-bi-ideal of T . Further more A is a biΓ-left ideal of T but neither a biΓ-right ideal nor a biΓ-lateral ideal of T . Now, define ${\bar{μ}}_{B} : T \to [- 1, 0]$ and ${\bar{γ}}_{B} : T \to [- 1, 0]$ as ${\bar{μ}}_{B} (a) = {\bar{μ}}_{B} (c) = - 0.8,$ ${\bar{μ}}_{B} (b) = - 0.5$ and ${\bar{γ}}_{B} (a) = - 0.2,$ ${\bar{γ}}_{B} (b) = 0 = {\bar{γ}}_{B} (c) .$ Then $N_{B} (t, s) = [\begin{matrix} T if t, s \in [-0 . 5, 0] \\ {a, c} if t, s \in [-0 . 8, -0 . 5) \\ Φ if t, s \in [-1, -0 . 8) \end{matrix}] .$

Obviously, N_B (t, s) is a biΓ-quasi ideal of T, for all t, s ∈ [-1, 0] with t + s ≥ -1 but neither a biΓ-right ideal nor a biΓ-lateral ideal of T for t, s ∈ [-0.8, - 0.5). Hence by Theorem 6, $B = ({\bar{μ}}_{B}, {\bar{γ}}_{B})$ is an intuitionistic N-fuzzy biΓ-bi-ideal of T and by Theorem 2, $B = ({\bar{μ}}_{B}, {\bar{γ}}_{B})$ is neither an intuitionistic N-fuzzy biΓ-right ideal nor an intuitionistic N-fuzzy biΓ-lateral ideal of T . Similarly, we can construct examples of intuitionistic N-fuzzy biΓ-bi-ideal of T which are not intuitionistic N-fuzzy biΓ-left ideal of T .

5 Conclusion

In this research, we introduced the notion of intuitionistic N-fuzzy set as a new generalization of fuzzy and intuitionistic fuzzy set. As an application of this set, we applied the concept to a newly defined structure biΓ-ternary semigroup [2] . Intuitionistic N-fuzzy set being a new tool containing the negative information may be used to explain and solve the real life problems more easily like the fuzzy set and intuitionistic fuzzy sets. After the introduction of this set, researchers may find some better options to deal with the problems of uncertainty with regard to intuitionistic fuzzy set. In our future work, we are planning to build some further theory on this set. We will apply this notion to characterize some algebraic structures by their quasi and bi-ideals. Furthermore, for the applications of intuitionistic N-fuzzy set we will apply this tool to the real life problems and we will try to explain these problems more specifically.

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