Redefined intuitionistic fuzzy bi-ideals of ordered semigroups

Abstract

In this article, we try to obtain a more general form than $(\in ,\in \vee \mathrm{q})$-intuitionistic fuzzy bi-ideals in ordered semigroups. The notion of an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal is introduced, and several properties are investigated. Characterizations of an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal are established. A condition for an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal to be an intuitionistic fuzzy bi-ideal is provided. It is shown that every $(\in ,\in )$-intuitionistic fuzzy bi-ideal is an $(\in ,\in \vee \mathrm{q})$-intuitionistic fuzzy bi-ideal, and every $(\in ,\in \vee \mathrm{q})$-intuitionistic fuzzy bi-ideal is an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal but the converse is not true. The important achievement of the study with an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal is that the notion of an $(\in ,\in \vee \mathrm{q})$-intuitionistic fuzzy bi-ideal is a special case of an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal, and, thus, several results in the paper (Jun et al. in Bi-ideals of ordered semigroups based on the intuitionistic fuzzy points (submitted)) are the corollaries of our results obtained in this paper.

1 Introduction

In mathematics, an ordered semigroup is a semigroup together with a partial order that is compatible with the semigroup operation. Ordered semigroups have many applications in the theory of sequential machines, formal languages, computer arithmetics, design of fast adders and error-correcting codes. The concept of a fuzzy filter in ordered semigroups was first introduced by Kehayopulu and Tsingelis in [1], where some basic properties of fuzzy filters and prime fuzzy ideals were discussed. A theory of fuzzy generalized sets on ordered semigroups can be developed. Mordeson et al. in [2] presented an up-to-date account of fuzzy sub-semigroups and fuzzy ideals of a semigroup. Murali [3] proposed the definition of a fuzzy point belonging to a fuzzy subset under a natural equivalence on fuzzy subset. The idea of quasi-coincidence of a fuzzy point with a fuzzy set played a vital role in generating different types of fuzzy subgroups. Bhakat and Das [4, 5] gave the concepts of $(\alpha ,\beta )$-fuzzy subgroups by using the ‘belong to’ $(\in )$ relation and ‘quasi-coincident with’ (q) relation between a fuzzy point and a fuzzy subgroup, and introduced the concept of $(\in ,\in \vee \mathrm{q})$-fuzzy subgroup. In [6], Davvaz started the generalized fuzzification in algebra. In [7], Jun et al. initiated the study of $(\alpha ,\beta )$-fuzzy bi-ideals of an ordered semigroup. In [8], Davvaz and Khan studied $(\in ,\in \vee \mathrm{q})$-fuzzy generalized bi-ideals of an ordered semigroup. Shabir et al. [9] studied characterization of regular semigroups by $(\alpha ,\beta )$-fuzzy ideals. Jun et al. [10] discussed a generalization of $(\in ,\in \vee \mathrm{q})$-fuzzy ideals of a $BCK/BCI$-algebra. Using the idea of a quasi-coincidence of a fuzzy point with a fuzzy set, Jun et al. [11] introduced the concept of $(\alpha ,\beta )$-intuitionistic fuzzy bi-ideals in an ordered semigroup. They introduced a new sort of intuitionistic fuzzy bi-ideals, called $(\alpha ,\beta )$-intuitionistic fuzzy bi-ideals, and studied $(\in ,\in \vee \mathrm{q})$-intuitionistic fuzzy bi-ideals.

In this paper, we try to have more general form of an $(\in ,\in \vee \mathrm{q})$-intuitionistic bi-ideal of an ordered semigroup. We introduce the notion of an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic bi-ideal of an ordered semigroup, and give examples which are $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideals but not $(\in ,\in \vee \mathrm{q})$-intuitionistic fuzzy bi-ideals. We discuss characterizations of $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideals in ordered semigroups. We provide a condition for an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal to be an intuitionistic fuzzy bi-ideal. The important achievement of the study with an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal is that the notion of an $(\in ,\in \vee \mathrm{q})$-intuitionistic fuzzy bi-ideal is a special case of an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal, and, thus, several results in the paper [11] are the corollaries of our results obtained in this paper.

2 Basic definitions and preliminary results

By an ordered semigroup (or po-semigroup) we mean a structure $(S,\cdot ,\le )$, in which the following are satisfied:

(OS1) $(S,\cdot )$ is a semigroup,

(OS2) $(S,\le )$ is a poset,

(OS3) ($\mathrm{\forall }x,a,b\in S$) ($a\le b\Rightarrow a\cdot x\le b\cdot x$, $x\cdot a\le x\cdot b$).

In what follows, $x\cdot y$ is simply denoted by xy for all $x,y\in S$.

A nonempty subset A of an ordered semigroup S is called a subsemigroup of S if $A^2\subseteq A$. A non-empty subset A of an ordered semigroup S is called a bi-ideal of S if it satisfies

(b1) ($\mathrm{\forall }b\in S$) ($\mathrm{\forall }b\in A$) ($a\le b\Rightarrow b\in A$),

(b2) ($\mathrm{\forall }a,b\in S$) ($a,b\in A\Rightarrow ab\in A$),

(b3) $ASA\subseteq A$.

An intuitionistic fuzzy set (briefly IFS) A in a non-empty set X is an object having the form $A=\{〈x,{\mu }_A(x),{\gamma }_A(x)〉|x\in X\}$, where the function ${\mu }_A:X⟶[0,1]$ and ${\gamma }_A:X⟶[0,1]$ denote the degree of membership (namely, ${\mu }_A(x)$) and the degree of non-membership (namely, ${\gamma }_A(x)$) for each element $x\in X$ to the set A, respectively, and $0\le {\mu }_A(x)+{\gamma }_A(x)\le 1$ for all $x\in X$. For the sake of simplicity, we shall use the symbol $A=〈x,{\mu }_A,{\gamma }_A〉$ for the intuitionistic fuzzy set $A=\{〈x,{\mu }_A(x),{\gamma }_A(x)〉|x\in X\}$.

Let $(S,\cdot ,\le )$ be an ordered semigroup and $A=〈x,{\mu }_A,{\gamma }_A〉$ be an IFS of S. Then $A=〈x,{\mu }_A,{\gamma }_A〉$ is called an intuitionistic fuzzy subsemigroup of S [11] if

$$(\mathrm{\forall }x,y\in S)({\mu }_A(xy)\ge min\{{\mu }_A(x),{\mu }_A(y)\}\text{ and }{\gamma }_A(xy)\le max\{{\gamma }_A(x),{\gamma }_A(y)\}).$$

Let $(S,\cdot ,\le )$ be an ordered semigroup and $A=〈x,{\mu }_A,{\gamma }_A〉$ be an intuitionistic fuzzy subsemigroup of S. Then $A=〈x,{\mu }_A,{\gamma }_A〉$ is called an intuitionistic fuzzy bi-ideal of S [11] if

(b4) ($\mathrm{\forall }x,y\in S$) ($x\le y⟹{\mu }_A(x)\ge {\mu }_A(y)$ and ${\gamma }_A(x)\le {\gamma }_A(y)$),

(b5) ($\mathrm{\forall }x,y\in S$) (${\mu }_A(xy)\ge min\{{\mu }_A(x),{\mu }_A(y)\}$ and ${\gamma }_A(xy)\le max\{{\gamma }_A(x),{\gamma }_A(y)\}$),

(b6) ($\mathrm{\forall }x,y,z\in S$) (${\mu }_A(xyz)\ge min\{{\mu }_A(x),{\mu }_A(z)\}$ and ${\gamma }_A(xyz)\le max\{{\gamma }_A(x),{\gamma }_A(z)\}$).

Let $A=〈x,{\mu }_A,{\gamma }_A〉$ be an IFS of S and $\alpha \in (0,1]$ and $\beta \in [0,1)$. Then the sets

$$U({\mu }_A;\alpha )=\{x\in S|{\mu }_A(x)\ge \alpha \}\text{and}L({\gamma }_A;\beta )=\{x\in S|{\gamma }_A(x)\le \beta \}$$

are called ${\mu }_A$-level and ${\gamma }_A$-level cuts of the intuitionistic fuzzy set $A=〈x,{\mu }_A,{\gamma }_A〉$, respectively. For an IFS $A=〈x,{\mu }_A,{\gamma }_A〉$ and $\alpha \in (0,1]$, $\beta \in [0,1)$, we define the $({\mu }_A,{\gamma }_A)$-level cut as follows

$$C_{(\alpha ,\beta )}(A)=\{x\in S|{\mu }_A(x)\ge \alpha \text{ and }{\gamma }_A(x)\le \beta \}.$$

Clearly, $C_{(\alpha ,\beta )}(A)=U({\mu }_A;\alpha )\cap L({\gamma }_A;\beta )$.

Let x be a point of a non-empty set X. If $\alpha \in (0,1]$ and $\beta \in [0,1)$ are two real numbers such that $0\le \alpha +\beta \le 1$, then the IFS of the form

$$〈x;(\alpha ,\beta )〉=〈x;x_{\alpha },1-x_{1-\beta }〉$$

is called an intuitionistic fuzzy point (IFP for short) in X, where α (resp. β) is the degree of membership (resp. non-membership) of $〈x;(\alpha ,\beta )〉$ and $x\in X$ is the support of $〈x;(\alpha ,\beta )〉$.

Consider an IFP $〈x;(\alpha ,\beta )〉$ in S, an IFS $A=〈x,{\mu }_A,{\gamma }_A〉$ and $\alpha \in \{\in ,\mathrm{q},\in \vee \mathrm{q}\}$, we define $〈x;(\alpha ,\beta )〉\alpha A$ as follows

(b7) $〈x;(\alpha ,\beta )〉\in A$ (resp. $〈x;(\alpha ,\beta )〉\mathrm{q}A$) means that ${\mu }_A(x)\ge \alpha$ and ${\gamma }_A(x)\le \beta$ (resp. ${\mu }_A(x)+\alpha >1$ and ${\gamma }_A(x)+\beta <1$), and in this case, we say that $〈x;(\alpha ,\beta )〉$ belongs to (resp. quasi-coincident with) an IFS $A=〈x,{\mu }_A,{\gamma }_A〉$.

(b8) $〈x;(\alpha ,\beta )〉\in \vee \mathrm{q}A$ (resp. $〈x;(\alpha ,\beta )〉\in \wedge \mathrm{q}A$) means that $〈x;(\alpha ,\beta )〉\in A$ or $〈x;(\alpha ,\beta )〉\mathrm{q}A$ (resp. $〈x;(\alpha ,\beta )〉\in A$ and $〈x;(\alpha ,\beta )〉\mathrm{q}A$).

By $〈x;(\alpha ,\beta )〉\overline{\alpha }A$, we mean that $〈x;(\alpha ,\beta )〉\alpha A$ does not hold.

3 $(\in ,\in \vee {\mathrm{q}}_k)$-Intuitionistic fuzzy bi-ideals

Let k denote an arbitrary element of $[0,1)$ unless specified otherwise. For an IFP $〈x;(\alpha ,\beta )〉$ and an IFS $A=〈x,{\mu }_A,{\gamma }_A〉$ of X, we say that

(c1) $〈x;(\alpha ,\beta )〉{\mathrm{q}}_{k}A$ if ${\mu }_A(x)+k+\alpha >1$ and ${\gamma }_A(x)+k+\beta <1$.

(c2) $〈x;(\alpha ,\beta )〉\in \vee {\mathrm{q}}_{k}A$ if $〈x;(\alpha ,\beta )〉\in A$ or $〈x;(\alpha ,\beta )〉{\mathrm{q}}_{k}A$.

(c3) $〈x;(\alpha ,\beta )〉\overline{\alpha }A$ if $〈x;(\alpha ,\beta )〉\alpha A$ does not hold for $\alpha \in \{{\mathrm{q}}_k,\in \vee {\mathrm{q}}_k\}$.

Theorem 3.1 Let $A=〈x,{\mu }_A,{\gamma }_A〉$ be an IFS of an ordered semigroup S. Then the following are equivalent

• (1) ($\mathrm{\forall }\alpha \in (\frac{1-k}{2},1]$) ($\mathrm{\forall }\beta \in [0,\frac{1-k}{2})$) ($C_{(\alpha ,\beta )}(A)\ne \mathrm{\varnothing }⟹C_{(\alpha ,\beta )}(A)$ is a bi-ideal of S).

• (2) $A=〈x,{\mu }_A,{\gamma }_A〉$ satisfies the following assertions

  $$\begin{array}{c}\text{(2.1)}\left(\begin{array}{l}x\le y⟹{\mu }_A(y)\le max\{{\mu }_A(x),\frac{1-k}{2}\}\\ \mathit{\text{and }}{\gamma }_A(y)\ge min\{{\gamma }_A(x),\frac{1-k}{2}\}\end{array}\right),\hfill \\ \text{(2.2)}\left(\begin{array}{l}min\{{\mu }_A(x),{\mu }_A(y)\}\le max\{{\mu }_A(xy),\frac{1-k}{2}\}\\ \mathit{\text{and }}max\{{\gamma }_A(x),{\gamma }_A(y)\}\ge min\{{\gamma }_A(xy),\frac{1-k}{2}\}\end{array}\right),\hfill \\ \text{(2.3)}\left(\begin{array}{l}min\{{\mu }_A(x),{\mu }_A(z)\}\le max\{{\mu }_A(xyz),\frac{1-k}{2}\}\mathit{\text{ and}}\\ max\{{\gamma }_A(x),{\gamma }_A(z)\}\ge min\{{\gamma }_A(xyz),\frac{1-k}{2}\}\end{array}\right),\hfill \end{array}$$

for all $x,y,z\in S$.

Proof Assume that $C_{(\alpha ,\beta )}(A)$ is a bi-ideal of S for all $\alpha \in (\frac{1-k}{2},1]$ and $\beta \in [0,\frac{1-k}{2})$ with $C_{(\alpha ,\beta )}(A)\ne \mathrm{\varnothing }$. If there exist $a,b\in S$ such that condition (2.1) is not valid, that is, there exist $a,b\in S$ with $a\le b$ and ${\mu }_A(b)>max\{{\mu }_A(a),\frac{1-k}{2}\}$, ${\gamma }_A(b)<min\{{\gamma }_A(a),\frac{1-k}{2}\}$. Then ${\mu }_A(b)\in (\frac{1-k}{2},1]$, ${\gamma }_A(b)\in [0,\frac{1-k}{2})$ and $b\in C_{({\mu }_A(b),{\gamma }_A(b))}(A)$. But ${\mu }_A(a)<{\mu }_A(b)$ and ${\gamma }_A(a)>{\gamma }_A(b)$ imply that $a\notin C_{({\mu }_A(b),{\gamma }_A(b))}(A)$. This is not possible. Hence (2.1) is valid. Suppose that (2.2) is false, that is,

$$s:=min\{{\mu }_A(a),{\mu }_A(c)\}>max\{{\mu }_A(ac),\frac{1-k}{2}\}$$

and

$$t:=max\{{\gamma }_A(a),{\gamma }_A(c)\}<min\{{\gamma }_A(ac),\frac{1-k}{2}\}$$

for some $a,c\in S$. Then $s\in (\frac{1-k}{2},1]$, $t\in [0,\frac{1-k}{2})$ and $a,c\in C_{(s,t)}(A)$. But $ac\notin C_{(s,t)}(A)$, since ${\mu }_A(ac)<s$ and ${\gamma }_A(ac)>t$. This is not possible, and so (2.2) is valid. If there exist $a,b,c\in S$ such that (2.3) is not valid, that is,

$$s_0:=min\{{\mu }_A(a),{\mu }_A(b)\}>max\{{\mu }_A(acb),\frac{1-k}{2}\}$$

and

$$t_0:=max\{{\gamma }_A(a),{\gamma }_A(b)\}<min\{{\gamma }_A(acb),\frac{1-k}{2}\}.$$

Then $s_0\in (\frac{1-k}{2},1]$, $t_0\in [0,\frac{1-k}{2})$ and $a,b\in C_{(s_0,t_0)}(A)$. But $acb\notin C_{(s_0,t_0)}(A)$, since ${\mu }_A(acb)<s_0$ and ${\gamma }_A(acb)>t_0$. This is a contradiction, and hence (2.3) is valid.

Conversely, assume that $A=〈x,{\mu }_A,{\gamma }_A〉$ satisfies the three conditions (2.1), (2.2) and (2.3). Suppose that $C_{(\alpha ,\beta )}(A)\ne \mathrm{\varnothing }$ for all $\alpha \in (\frac{1-k}{2},1]$, and $\beta \in [0,\frac{1-k}{2})$. Let $x,y\in S$ be such that $x\le y$ and $y\in C_{(\alpha ,\beta )}(A)$. Then ${\mu }_A(y)\ge \alpha$ and ${\gamma }_A(y)\le \beta$. Using (2.1), we have $max\{{\mu }_A(x),\frac{1-k}{2}\}\ge {\mu }_A(y)\ge \alpha >\frac{1-k}{2}$ and $min\{{\gamma }_A(x),\frac{1-k}{2}\}\le {\gamma }_A(y)\le \beta <\frac{1-k}{2}$. Hence ${\mu }_A(x)\ge \alpha$ and ${\gamma }_A(x)\le \beta$, i.e., $x\in C_{(\alpha ,\beta )}(A)$. If $x,y\in C_{(\alpha ,\beta )}(A)$, then ${\mu }_A(x)\ge \alpha$, ${\gamma }_A(x)\le \beta$ and ${\mu }_A(y)\ge \alpha$, ${\gamma }_A(y)\le \beta$. By using (2.2), we have

$$max\{{\mu }_A(xy),\frac{1-k}{2}\}\ge min\{{\mu }_A(x),{\mu }_A(y)\}\ge \alpha >\frac{1-k}{2}$$

and

$$min\{{\gamma }_A(xy),\frac{1-k}{2}\}\le max\{{\gamma }_A(x),{\gamma }_A(y)\}\le \beta <\frac{1-k}{2},$$

so that ${\mu }_A(xy)\ge \alpha$ and ${\gamma }_A(xy)\le \beta$, i.e., $xy\in C_{(\alpha ,\beta )}(A)$. Finally, if $x,z\in C_{(\alpha ,\beta )}(A)$ and $y\in S$, then ${\mu }_A(x)\ge \alpha$, ${\gamma }_A(x)\le \beta$ and ${\mu }_A(z)\ge \alpha$, ${\gamma }_A(z)\le \beta$. By using (2.3), we have

$$max\{{\mu }_A(xyz),\frac{1-k}{2}\}\ge min\{{\mu }_A(x),{\mu }_A(z)\}\ge \alpha >\frac{1-k}{2}$$

and

$$min\{{\gamma }_A(xyz),\frac{1-k}{2}\}\le max\{{\gamma }_A(x),{\gamma }_A(z)\}\le \beta <\frac{1-k}{2},$$

so that ${\mu }_A(xyz)\ge \alpha$ and ${\gamma }_A(xyz)\le \beta$, i.e., $xyz\in C_{(\alpha ,\beta )}(A)$. Therefore, $C_{(\alpha ,\beta )}(A)$ is a bi-ideal of S. □

If we take $k=0$ in Theorem 3.1, then we have the following corollary.

Corollary 3.2 [[11], Theorem 3.1]

Let $A=〈x,{\mu }_A,{\gamma }_A〉$ be an IFS of S. Then the following assertions are equivalent

• (1) ($\mathrm{\forall }\alpha \in (0.5,1]$) ($\mathrm{\forall }\beta \in [0,0.5)$) ($C_{(\alpha ,\beta )}(A)\ne \mathrm{\varnothing }⟹C_{(\alpha ,\beta )}(A)$ is a bi-ideal of S).

• (2) $A=〈x,{\mu }_A,{\gamma }_A〉$ satisfies the following conditions

  $$\begin{array}{c}\text{(2.1)}\left(\begin{array}{l}x\le y⟹{\mu }_A(y)\le max\{{\mu }_A(x),0.5\}\\ \mathit{\text{and }}{\gamma }_A(y)\ge min\{{\gamma }_A(x),0.5\}\end{array}\right),\hfill \\ \text{(2.2)}\left(\begin{array}{l}min\{{\mu }_A(x),{\mu }_A(y)\}\le max\{{\mu }_A(xy),0.5\}\\ \mathit{\text{and }}max\{{\gamma }_A(x),{\gamma }_A(y)\}\ge min\{{\gamma }_A(xy),0.5\}\end{array}\right),\hfill \\ \text{(2.3)}\left(\begin{array}{l}min\{{\mu }_A(x),{\mu }_A(z)\}\le max\{{\mu }_A(xyz),0.5\}\\ \mathit{\text{and }}max\{{\gamma }_A(x),{\gamma }_A(z)\}\ge min\{{\gamma }_A(xyz),0.5\}\end{array}\right),\hfill \end{array}$$

for all $x,y,z\in S$.

Definition 3.3 An IFS $A=〈x,{\mu }_A,{\gamma }_A〉$ in S is called an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S if for all $x,y,z\in S$, $t,t_1,t_2\in (0,1]$ and $s,s_1,s_2\in [0,1)$ it satisfies the following conditions

(q1) ($x\le y$, $〈y;(t,s)〉\in A⟹〈x;(t,s)〉\in \vee {\mathrm{q}}_{k}A$),

(q2) ($〈x;(t_1,s_1)〉\in A$ and $〈y;(t_2,s_2)〉\in A⟹〈xy;min\{t_1,t_2\},max\{s_1,s_2\}〉\in \vee {\mathrm{q}}_{k}A$),

(q3) ($〈x;(t_1,s_1)〉\in A$ and $〈z;(t_2,s_2)〉\in A⟹〈xyz;min\{t_1,t_2\},max\{s_1,s_2\}〉\in \vee {\mathrm{q}}_{k}A$).

An $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S with $k=0$ is an $(\in ,\in \vee \mathrm{q})$-intuitionistic fuzzy bi-ideal of S.

Example 3.4 Consider the set $S=\{a,b,c,d,e\}$ with the order relation $a\le c\le e$, $a\le d\le e$, $b\le d$ and $b\le e$ and ∗-multiplication table (see Table 1 above).

• (1) Define an IFS $A=〈x,{\mu }_A,{\gamma }_A〉$ by

  $${\mu }_A:S\to [0,1]\mid {\mu }_A(x)=\{\begin{array}{cc}0.40\hfill & \text{if }x=a,\hfill \\ 0.35\hfill & \text{if }x=b,\hfill \\ 0.30\hfill & \text{if }x=c,\hfill \\ 0.50\hfill & \text{if }x=d,\hfill \\ 0.20\hfill & \text{if }x=e\hfill \end{array}$$

and

$${\gamma }_A:S\to [0,1]\mid {\gamma }_A(x)=\{\begin{array}{cc}0.40\hfill & \text{if }x=a,\hfill \\ 0.30\hfill & \text{if }x=b,\hfill \\ 0.50\hfill & \text{if }x=c,\hfill \\ 0.40\hfill & \text{if }x=d,\hfill \\ 0.80\hfill & \text{if }x=e.\hfill \end{array}$$

Then $A=〈x,{\mu }_A,{\gamma }_A〉$ is an $(\in ,\in \vee {\mathrm{q}}_{0.4})$-intuitionistic fuzzy bi-ideal of S.

• (2) Let $A=〈x,{\mu }_A,{\gamma }_A〉$ be an intuitionistic fuzzy set given by

  $${\mu }_A:S\to [0,1]\mid {\mu }_A(x)=\{\begin{array}{cc}0.80\hfill & \text{if }x=a,\hfill \\ 0.60\hfill & \text{if }x=b,e,\hfill \\ 0.30\hfill & \text{if }x=c,\hfill \\ 0.50\hfill & \text{if }x=d\hfill \end{array}$$

and

$${\gamma }_A:S\to [0,1]\mid {\gamma }_A(x)=\{\begin{array}{cc}0.20\hfill & \text{if }x=a,\hfill \\ 0.30\hfill & \text{if }x=b,e,\hfill \\ 0.60\hfill & \text{if }x=c,\hfill \\ 0.50\hfill & \text{if }x=d.\hfill \end{array}$$

Then $A=〈x,{\mu }_A,{\gamma }_A〉$ is an $(\in ,\in \vee {\mathrm{q}}_{0.04})$-intuitionistic fuzzy bi-ideal of S.

Table 1 ∗-multiplication table for S

Theorem 3.5 An IFS $A=〈x,{\mu }_A,{\gamma }_A〉$ of S is an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S if and only if it satisfies the following conditions

$$\begin{array}{c}\text{(1)}\left(\begin{array}{l}x\le y⟹{\mu }_A(x)\ge min\{{\mu }_A(y),\frac{1-k}{2}\}\\ \mathit{\text{and }}{\gamma }_A(x)\le max\{{\gamma }_A(y),\frac{1-k}{2}\}\end{array}\right),\hfill \\ \text{(2)}\left(\begin{array}{l}{\mu }_A(xy)\ge min\{{\mu }_A(x),{\mu }_A(y),\frac{1-k}{2}\}\\ \mathit{\text{and }}{\gamma }_A(xy)\le max\{{\gamma }_A(x),{\gamma }_A(y),\frac{1-k}{2}\}\end{array}\right),\hfill \\ \text{(3)}\left(\begin{array}{l}{\mu }_A(xyz)\ge min\{{\mu }_A(x),{\mu }_A(z),\frac{1-k}{2}\}\\ \mathit{\text{and }}{\gamma }_A(xyz)\le max\{{\gamma }_A(x),{\gamma }_A(z),\frac{1-k}{2}\}\end{array}\right).\hfill \end{array}$$

Proof Suppose that $A=〈x,{\mu }_A,{\gamma }_A〉$ is an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S. Let $x,y\in S$ be such that $x\le y$. Assume that ${\mu }_A(y)\le \frac{1-k}{2}$ and ${\gamma }_A(y)\ge \frac{1-k}{2}$. If ${\mu }_A(x)<{\mu }_A(y)$ and ${\gamma }_A(x)>{\gamma }_A(y)$, then ${\mu }_A(x)<t\le {\mu }_A(y)$ and ${\gamma }_A(x)>s\ge {\gamma }_A(y)$ for some $t\in (0,\frac{1-k}{2})$ and $s\in (\frac{1-k}{2},1)$. It follows $〈y;(t,s)〉\in A$, but $〈x;(t,s)〉\overline{\in }A$. Since ${\mu }_A(x)+t<2t<1-k$ and ${\gamma }_A(x)+s>2s>1-k$, we get $〈x;(t,s)〉\overline{{\mathrm{q}}_k}A$. Therefore, $〈x;(t,s)〉\overline{\in \vee {\mathrm{q}}_k}A$, which is a contradiction. Hence ${\mu }_A(x)\ge {\mu }_A(y)$ and ${\gamma }_A(x)\le {\gamma }_A(y)$. Now, if ${\mu }_A(y)\ge \frac{1-k}{2}$ and ${\gamma }_A(x)\le \frac{1-k}{2}$, then $〈y;(\frac{1-k}{2},\frac{1-k}{2})〉\in A$, and so, $〈x;(\frac{1-k}{2},\frac{1-k}{2})〉\in \vee {\mathrm{q}}_{k}A$, which implies that $〈x;(\frac{1-k}{2},\frac{1-k}{2})〉\in A$ or $〈x;(\frac{1-k}{2},\frac{1-k}{2})〉{\mathrm{q}}_{k}A$, that is, ${\mu }_A(x)\ge \frac{1-k}{2}$ and ${\gamma }_A(x)\le \frac{1-k}{2}$ or ${\mu }_A(x)+\frac{1-k}{2}>1$ and ${\gamma }_A(x)+\frac{1-k}{2}<1$. Hence ${\mu }_A(x)\ge \frac{1-k}{2}$ and ${\gamma }_A(x)\le \frac{1-k}{2}$. Otherwise, ${\mu }_A(x)+\frac{1-k}{2}<\frac{1-k}{2}+\frac{1-k}{2}=1$ and ${\gamma }_A(x)+\frac{1-k}{2}>\frac{1-k}{2}+\frac{1-k}{2}=1$, a contradiction. Consequently,

$${\mu }_A(x)\ge min\{{\mu }_A(y),\frac{1-k}{2}\}$$

and

$${\gamma }_A(x)\le max\{{\gamma }_A(y),\frac{1-k}{2}\}$$

for all $x,y\in S$ with $x\le y$. Let $x,y\in S$ be such that $min\{{\mu }_A(x),{\mu }_A(y)\}<\frac{1-k}{2}$ and $max\{{\gamma }_A(x),{\gamma }_A(y)\}>\frac{1-k}{2}$. We claim that ${\mu }_A(xy)\ge min\{{\mu }_A(x),{\mu }_A(y)\}$ and ${\gamma }_A(xy)\le max\{{\gamma }_A(x),{\gamma }_A(y)\}$. If not, then ${\mu }_A(xy)<t\le min\{{\mu }_A(x),{\mu }_A(y)\}$ and ${\gamma }_A(xy)>s\ge max\{{\gamma }_A(x),{\gamma }_A(y)\}$ for some $t\in (0,\frac{1-k}{2})$ and $s\in (\frac{1-k}{2},1)$. It follows that $〈x;(t,s)〉\in A$ and $〈y;(t,s)〉\in A$, but $〈xy;(t,s)〉\overline{\in }A$ and ${\mu }_A(xy)+t<2t<1-k$ and ${\gamma }_A(xy)+s>2s>1-k$, i.e., $〈xy;(t,s)〉\overline{{\mathrm{q}}_k}A$. This is a contradiction. Thus, ${\mu }_A(xy)\ge min\{{\mu }_A(x),{\mu }_A(y)\}$ and ${\gamma }_A(xy)\le max\{{\gamma }_A(x),{\gamma }_A(y)\}$ for all $x,y\in S$ with $min\{{\mu }_A(x),{\mu }_A(y)\}<\frac{1-k}{2}$ and $max\{{\gamma }_A(x),{\gamma }_A(y)\}>\frac{1-k}{2}$. If $min\{{\mu }_A(x),{\mu }_A(y)\}\ge \frac{1-k}{2}$ and $max\{{\gamma }_A(x),{\gamma }_A(y)\}\le \frac{1-k}{2}$, then $〈x;(\frac{1-k}{2},\frac{1-k}{2})〉\in A$ and $〈y;(\frac{1-k}{2},\frac{1-k}{2})〉\in A$. Using (q2), we have

$$\begin{array}{c}〈xy;(\frac{1-k}{2},\frac{1-k}{2})〉\hfill \\ =〈xy;min\{\frac{1-k}{2},\frac{1-k}{2}\},max\{\frac{1-k}{2},\frac{1-k}{2}\}〉\in \vee {\mathrm{q}}_{k}A,\hfill \end{array}$$

and so, ${\mu }_A(xy)\ge \frac{1-k}{2}$ and ${\gamma }_A(xy)\le \frac{1-k}{2}$ or ${\mu }_A(xy)+\frac{1-k}{2}>1$ and ${\gamma }_A(xy)+\frac{1-k}{2}<1$. If ${\mu }_A(xy)<\frac{1-k}{2}$ and ${\gamma }_A(xy)>\frac{1-k}{2}$, then

$${\mu }_A(xy)+\frac{1-k}{2}<\frac{1-k}{2}+\frac{1-k}{2}=1$$

and

$${\gamma }_A(xy)+\frac{1-k}{2}>\frac{1-k}{2}+\frac{1-k}{2}=1,$$

which is impossible. Consequently, ${\mu }_A(xy)\ge min\{{\mu }_A(x),{\mu }_A(y),\frac{1-k}{2}\}$ and ${\gamma }_A(xy)\le max\{{\gamma }_A(x),{\gamma }_A(y),\frac{1-k}{2}\}$ for all $x,y\in S$. Let $a,b,c\in S$ be such that $min\{{\mu }_A(a),{\mu }_A(c)\}<\frac{1-k}{2}$ and $max\{{\gamma }_A(a),{\gamma }_A(c)\}>\frac{1-k}{2}$. We claim that

$${\mu }_A(abc)\ge min\{{\mu }_A(a),{\mu }_A(c)\}\text{and}{\gamma }_A(abc)\le max\{{\gamma }_A(a),{\gamma }_A(c)\}.$$

If not, then ${\mu }_A(abc)<t_0\le min\{{\mu }_A(a),{\mu }_A(c)\}$ and ${\gamma }_A(abc)>s_0\ge max\{{\gamma }_A(a),{\gamma }_A(c)\}$ for some $t_0\in (0,\frac{1-k}{2})$ and $s_0\in (\frac{1-k}{2},1)$. It follows that $〈a;(t_0,s_0)〉\in A$ and $〈c;(t_0,s_0)〉\in A$, but $〈abc;(t_0,s_0)〉\overline{\in }A$ and ${\mu }_A(abc)+t_0<2t_0<1-k$ and ${\gamma }_A(abc)+s_0>2s_0>1-k$, i.e., $〈abc;(t_0,s_0)〉\overline{{\mathrm{q}}_k}A$. This is a contradiction. Thus, ${\mu }_A(abc)\ge min\{{\mu }_A(a),{\mu }_A(c)\}$ and ${\gamma }_A(abc)\le max\{{\gamma }_A(a),{\gamma }_A(c)\}$ for all $a,b,c\in S$ with $min\{{\mu }_A(a),{\mu }_A(c)\}<\frac{1-k}{2}$ and $max\{{\gamma }_A(a),{\gamma }_A(c)\}>\frac{1-k}{2}$. If $min\{{\mu }_A(a),{\mu }_A(c)\}\ge \frac{1-k}{2}$ and $max\{{\gamma }_A(a),{\gamma }_A(c)\}\le \frac{1-k}{2}$, then $〈a;(\frac{1-k}{2},\frac{1-k}{2})〉\in A$ and $〈c;(\frac{1-k}{2},\frac{1-k}{2})〉\in A$. Using (q3), we have

$$\begin{array}{c}〈abc;(\frac{1-k}{2},\frac{1-k}{2})〉\hfill \\ =〈abc;min\{\frac{1-k}{2},\frac{1-k}{2}\},max\{\frac{1-k}{2},\frac{1-k}{2}\}〉\in \vee \mathrm{q}A,\hfill \end{array}$$

and so ${\mu }_A(abc)\ge \frac{1-k}{2}$ and ${\gamma }_A(abc)\le \frac{1-k}{2}$ or ${\mu }_A(abc)+\frac{1-k}{2}>1$ and ${\gamma }_A(abc)+\frac{1-k}{2}<1$. If ${\mu }_A(abc)<\frac{1-k}{2}$ and ${\gamma }_A(abc)>\frac{1-k}{2}$, then

$${\mu }_A(abc)+\frac{1-k}{2}<\frac{1-k}{2}+\frac{1-k}{2}=1$$

and

$${\gamma }_A(abc)+\frac{1-k}{2}>\frac{1-k}{2}+\frac{1-k}{2}=1,$$

which is impossible. Therefore, ${\mu }_A(xyz)\ge min\{{\mu }_A(x),{\mu }_A(z),\frac{1-k}{2}\}$ and ${\gamma }_A(xyz)\le max\{{\gamma }_A(x),{\gamma }_A(z),\frac{1-k}{2}\}$ for all $x,y,z\in S$.

Conversely, let $A=〈x,{\mu }_A,{\gamma }_A〉$ be an IFS of S that satisfies the three conditions (1), (2) and (3). Let $x,y\in S$, $t\in (0,1]$ and $s\in [0,1)$ be such that $x\le y$ and $[y;(t,s)]\in A$. Then ${\mu }_A(y)\ge t$ and ${\gamma }_A(y)\le s$, and so,

$$\begin{array}{rcl}{\mu }_A(x)& \ge & min\{{\mu }_A(y),\frac{1-k}{2}\}\ge min\{t,\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}t\hfill & \text{if }t\le \frac{1-k}{2},\hfill \\ \frac{1-k}{2}\hfill & \text{if }t>\frac{1-k}{2}\hfill \end{array}\end{array}$$

and

$$\begin{array}{rcl}{\gamma }_A(x)& \le & max\{{\gamma }_A(y),\frac{1-k}{2}\}\le max\{s,\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}s\hfill & \text{if }s\ge \frac{1-k}{2},\hfill \\ \frac{1-k}{2}\hfill & \text{if }s<\frac{1-k}{2}.\hfill \end{array}\end{array}$$

It follows that ${\mu }_A(x)\ge t$ and ${\gamma }_A(x)\le s$ or ${\mu }_A(x)+t\ge \frac{1-k}{2}+t>1-k$ and ${\gamma }_A(x)+s\le \frac{1-k}{2}+s<1-k$, i.e., $〈x;(t,s)〉\in A$ or $〈x;(t,s)〉{\mathrm{q}}_{k}A$. Hence, $〈x;(t,s)〉\in \vee {\mathrm{q}}_{k}A$. Let $x,y\in S$, $t_1,t_2\in (0,1]$ and $s_1,s_2\in [0,1)$ be such that $〈x;(t_1,s_1)〉\in A$ and $〈y;(t_2,s_2)〉\in A$. Then ${\mu }_A(x)\ge t_1$, ${\gamma }_A(x)\le s_1$ and ${\mu }_A(y)\ge t_2$, ${\gamma }_A(y)\le s_2$. It follows from (2) that

$$\begin{array}{rcl}{\mu }_A(xy)& \ge & min\{{\mu }_A(x),{\mu }_A(y),\frac{1-k}{2}\}\ge min\{t_1,t_2,\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}min\{t_1,t_2\}\hfill & \text{if }min\{t_1,t_2\}\le \frac{1-k}{2},\hfill \\ \frac{1-k}{2}\hfill & \text{if }min\{t_1,t_2\}\le \frac{1-k}{2}\hfill \end{array}\end{array}$$

and

$$\begin{array}{rcl}{\gamma }_A(xy)& \le & max\{{\gamma }_A(x),{\gamma }_A(y),\frac{1-k}{2}\}\le max\{s_1,s_2,\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}max\{s_1,s_2\}\hfill & \text{if }max\{s_1,s_2\}\ge \frac{1-k}{2},\hfill \\ \frac{1-k}{2}\hfill & \text{if }max\{s_1,s_2\}<\frac{1-k}{2}.\hfill \end{array}\end{array}$$

It follows that $〈xy;min\{t_1,t_2\},max\{s_1,s_2\}〉\in A$ or ${\mu }_A(xy)+min\{t_1,t_2\}\ge \frac{1-k}{2}+min\{t_1,t_2\}>\frac{1-k}{2}+\frac{1-k}{2}=1-k$ and ${\gamma }_A(xy)+max\{s_1,s_2\}\le \frac{1-k}{2}+max\{s_1,s_2\}<\frac{1-k}{2}+\frac{1-k}{2}=1-k$, i.e., $〈xy;min\{t_1,t_2\},max\{s_1,s_2\}〉{\mathrm{q}}_{k}A$. Therefore, $〈xy;min\{t_1,t_2\},max\{s_1,s_2\}〉\in \vee {\mathrm{q}}_{k}A$. Let $x,y,z\in S$, $t_1,t_2\in (0,1]$ and $s_1,s_2\in [0,1)$ be such that $〈x;(t_1,s_1)〉\in A$ and $〈z;(t_2,s_2)〉\in A$. Then ${\mu }_A(x)\ge t_1$, ${\gamma }_A(x)\le s_1$ and ${\mu }_A(z)\ge t_2$, ${\gamma }_A(z)\le s_2$. It follows from (3) that

$$\begin{array}{rcl}{\mu }_A(xyz)& \ge & min\{{\mu }_A(x),{\mu }_A(z),\frac{1-k}{2}\}\ge min\{t_1,t_2,\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}min\{t_1,t_2\}\hfill & \text{if }min\{t_1,t_2\}\le \frac{1-k}{2},\hfill \\ \frac{1-k}{2}\hfill & \text{if }min\{t_1,t_2\}\le \frac{1-k}{2}\hfill \end{array}\end{array}$$

and

$$\begin{array}{rcl}{\gamma }_A(xyz)& \le & max\{{\gamma }_A(x),{\gamma }_A(z),\frac{1-k}{2}\}\le max\{s_1,s_2,\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}max\{s_1,s_2\}\hfill & \text{if }max\{s_1,s_2\}\ge \frac{1-k}{2},\hfill \\ \frac{1-k}{2}\hfill & \text{if }max\{s_1,s_2\}<\frac{1-k}{2}.\hfill \end{array}\end{array}$$

Thus, we have $〈xyz;min\{t_1,t_2\},max\{s_1,s_2\}〉\in A$ or ${\mu }_A(xyz)+min\{t_1,t_2\}\ge \frac{1-k}{2}+min\{t_1,t_2\}>\frac{1-k}{2}+\frac{1-k}{2}=1-k$ and ${\gamma }_A(xyz)+max\{s_1,s_2\}\le \frac{1-k}{2}+max\{s_1,s_2\}<\frac{1-k}{2}+\frac{1-k}{2}=1-k$, i.e., $〈xyz;min\{t_1,t_2\},max\{s_1,s_2\}〉{\mathrm{q}}_{k}A$. Therefore, $〈xyz;min\{t_1,t_2\},max\{s_1,s_2\}〉\in \vee {\mathrm{q}}_{k}A$. Thus, $A=〈x,{\mu }_A,{\gamma }_A〉$ is an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S. □

If we take $k=0$ in Theorem 3.5, then we have the following corollary.

Corollary 3.6 [[11], Theorem 3.5]

An IFS $A=〈x,{\mu }_A,{\gamma }_A〉$ of S is an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S if and only if it satisfies the conditions

$$\begin{array}{c}\text{(1)}\left(\begin{array}{l}x\le y⟹{\mu }_A(x)\ge min\{{\mu }_A(y),0.5\}\\ \mathit{\text{and }}{\gamma }_A(x)\le max\{{\gamma }_A(y),0.5\}\end{array}\right),\hfill \\ \text{(2)}\left(\begin{array}{l}{\mu }_A(xy)\ge min\{{\mu }_A(x),{\mu }_A(y),0.5\}\\ \mathit{\text{and }}{\gamma }_A(xy)\le max\{{\gamma }_A(x),{\gamma }_A(y),0.5\}\end{array}\right),\hfill \\ \text{(3)}\left(\begin{array}{l}{\mu }_A(xyz)\ge min\{{\mu }_A(x),{\mu }_A(z),0.5\}\\ \mathit{\text{and }}{\gamma }_A(xyz)\le max\{{\gamma }_A(x),{\gamma }_A(z),0.5\}\end{array}\right).\hfill \end{array}$$

Obviously, every intuitionistic fuzzy bi-ideal is an $(\in ,\in )$-intuitionistic fuzzy bi-ideal, and we know that every $(\in ,\in )$-intuitionistic fuzzy bi-ideal of S is an $(\in ,\in \vee \mathrm{q})$-intuitionistic fuzzy bi-ideal of S, and every $(\in ,\in \vee \mathrm{q})$-intuitionistic fuzzy bi-ideal is an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S. But the converse may not be true. The following example shows that every $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S may not be an $(\in ,\in \vee \mathrm{q})$-intuitionistic fuzzy bi-ideal nor an intuitionistic fuzzy bi-ideal of S.

Example 3.7 Consider the ordered semigroup of Example 3.4 and define an IFS $A=〈x,{\mu }_A,{\gamma }_A〉$ by

$${\mu }_A:S\to [0,1]\mid {\mu }_A(x)=\{\begin{array}{cc}0.80\hfill & \text{if }x=a,\hfill \\ 0.60\hfill & \text{if }x=b,\hfill \\ 0.40\hfill & \text{if }x=c,\hfill \\ 0.30\hfill & \text{if }x=d,e\hfill \end{array}$$

and

$${\gamma }_A:S\to [0,1]\mid {\gamma }_A(x)=\{\begin{array}{cc}0.20\hfill & \text{if }x=a,\hfill \\ 0.30\hfill & \text{if }x=b,\hfill \\ 0.40\hfill & \text{if }x=c,\hfill \\ 0.70\hfill & \text{if }x=d,e.\hfill \end{array}$$

Then $A=〈x,{\mu }_A,{\gamma }_A〉$ is an $(\in ,\in \vee {\mathrm{q}}_{0.4})$-intuitionistic fuzzy bi-ideal of S. But

• (1) $A=〈x,{\mu }_A,{\gamma }_A〉$ is not an $(\in ,\in \vee \mathrm{q})$-intuitionistic fuzzy bi-ideal of S. Since $〈a;(0.8,0.2)〉\in A$ and $〈b;(0.6,0.3)〉\in A$ but $〈ab;(0.6,0.3)〉\overline{\in \vee \mathrm{q}}A$.

• (2) $A=〈x,{\mu }_A,{\gamma }_A〉$ is not an intuitionistic fuzzy bi-ideal of S. Since

  $${\mu }_A(ab)={\mu }_A(d)=0.30<m\{{\mu }_A(a)=0.80,{\mu }_A(b)=0.60\}$$

and

$${\gamma }_A(ab)={\gamma }_A(d)=0.70>M\{{\gamma }_A(a)=0.20,{\mu }_A(b)=0.30\}.$$

In the following, we give a condition for an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S to be an ordinary intuitionistic fuzzy bi-ideal of S.

Theorem 3.8 Let $A=〈x,{\mu }_A,{\gamma }_A〉$ be an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S. If ${\mu }_A(x)\ge \frac{1-k}{2}$ and ${\gamma }_A(x)\le \frac{1-k}{2}$ for all $x\in S$, then $A=〈x,{\mu }_A,{\gamma }_A〉$ is an $(\in ,\in )$-intuitionistic fuzzy bi-ideal of S.

Proof The proof is straightforward by Theorem 3.5. □

Corollary 3.9 [[11], Theorem 3.8]

Let $A=〈x,{\mu }_A,{\gamma }_A〉$ be an $(\in ,\in \vee \mathrm{q})$-intuitionistic fuzzy bi-ideal of S. If ${\mu }_A(x)\ge 0.5$ and ${\gamma }_A(x)\le 0.5$ for all $x\in S$, then $A=〈x,{\mu }_A,{\gamma }_A〉$ is an $(\in ,\in )$-intuitionistic fuzzy bi-ideal of S.

Proof The proof follows from Theorem 3.8, by taking $k=0$. □

Theorem 3.10 For an IFS $A=〈x,{\mu }_A,{\gamma }_A〉$ of S, the following are equivalent:

• (1) $A=〈x,{\mu }_A,{\gamma }_A〉$ is an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S.

• (2) ($\mathrm{\forall }t\in (0,\frac{1-k}{2}]$) ($\mathrm{\forall }s\in [\frac{1-k}{2},1)$) ($C_{(\alpha ,\beta )}(A)\ne \mathrm{\varnothing }⟹C_{(\alpha ,\beta )}(A)\mathit{\text{is a bi-ideal of}}S$) .

Proof Assume that $A=〈x,{\mu }_A,{\gamma }_A〉$ is an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S, let $t\in (0,\frac{1-k}{2}]$ and $s\in [\frac{1-k}{2},1)$ be such that $C_{(\alpha ,\beta )}(A)\ne \mathrm{\varnothing }$. Using Theorem 3.5(1), we have

$${\mu }_A(x)\ge min\{{\mu }_A(y),\frac{1-k}{2}\}\text{and}{\gamma }_A(x)\le max\{{\gamma }_A(y),\frac{1-k}{2}\}$$

for any $x,y\in S$ with $x\le y$ and $x\in C_{(t,s)}(A)$. It follows that ${\mu }_A(x)\ge min\{t,0.5\}=t$ and ${\gamma }_A(x)\le max\{s,0.5\}=s$, so that $y\in C_{(t,s)}(A)$. Let $x,y\in C_{(\alpha ,\beta )}(A)$. Then ${\mu }_A(x)\ge t$, ${\gamma }_A(x)\le s$ and ${\mu }_A(y)\ge t$, ${\gamma }_A(y)\le s$. Theorem 3.5(2) implies that

$${\mu }_A(xy)\ge min\{{\mu }_A(x),{\mu }_A(y),\frac{1-k}{2}\}\ge min\{t,\frac{1-k}{2}\}=t$$

and

$${\gamma }_A(xy)\le max\{{\gamma }_A(x),{\gamma }_A(y),\frac{1-k}{2}\}\le max\{s,\frac{1-k}{2}\}=s.$$

Thus, $xy\in C_{(\alpha ,\beta )}(A)$. Now let $x,z\in C_{(\alpha ,\beta )}(A)$. Then ${\mu }_A(x)\ge t$, ${\gamma }_A(x)\le s$ and ${\mu }_A(z)\ge t$, ${\gamma }_A(z)\le s$. Theorem 3.5(3) induces that

$${\mu }_A(xyz)\ge min\{{\mu }_A(x),{\mu }_A(z),\frac{1-k}{2}\}\ge min\{t,\frac{1-k}{2}\}=t$$

and

$${\gamma }_A(xyz)\le max\{{\gamma }_A(x),{\gamma }_A(z),\frac{1-k}{2}\}\le max\{s,\frac{1-k}{2}\}=s.$$

Thus, $xyz\in C_{(\alpha ,\beta )}(A)$, therefore, $C_{(\alpha ,\beta )}(A)$ is a bi-ideal of S.

Conversely, let $A=〈x,{\mu }_A,{\gamma }_A〉$ be an IFS of S such that $C_{(\alpha ,\beta )}(A)$ is non-empty and is a bi-ideal of S for all $t\in (0,\frac{1-k}{2}]$ and $s\in [\frac{1-k}{2},1)$. If there exist $a,b\in S$ with $a\le b$ and $b\in C_{(\alpha ,\beta )}(A)$ such that ${\mu }_A(a)<min\{{\mu }_A(b),\frac{1-k}{2}\}$ and ${\gamma }_A(a)>max\{{\gamma }_A(b),\frac{1-k}{2}\}$, then ${\mu }_A(a)<t_a\le min\{{\mu }_A(b),\frac{1-k}{2}\}$ and ${\gamma }_A(a)>s_a\ge max\{{\gamma }_A(b),\frac{1-k}{2}\}$ for some $t_a\in (0,\frac{1-k}{2}]$ and $s_a\in [\frac{1-k}{2},1)$. Then $a\notin C_{(t_a,s_a)}(A)$, a contradiction. Therefore, ${\mu }_A(x)\ge min\{{\mu }_A(y),\frac{1-k}{2}\}$ and ${\gamma }_A(x)>max\{{\gamma }_A(y),\frac{1-k}{2}\}$ for all $x,y\in S$ with $x\le y$. Assume that there exist $a,b\in S$ such that

$${\mu }_A(ab)<min\{{\mu }_A(a),{\mu }_A(b),\frac{1-k}{2}\}$$

and ${\gamma }_A(ab)>max\{{\gamma }_A(a),{\gamma }_A(b),\frac{1-k}{2}\}$. Then

$${\mu }_A(ab)<t_0\le min\{{\mu }_A(a),{\mu }_A(b),\frac{1-k}{2}\}$$

and

$${\gamma }_A(ab)>s_0\ge max\{{\gamma }_A(a),{\gamma }_A(b),\frac{1-k}{2}\}$$

for some $t_0\in (0,\frac{1-k}{2}]$ and $s_0\in [\frac{1-k}{2},1)$. It follows that $a\in C_{(t_0,s_0)}(A)$ and $b\in C_{(t_0,s_0)}(A)$, but $ab\notin C_{(t_0,s_0)}(A)$. This is a contradiction. Hence

$${\mu }_A(xy)\ge min\{{\mu }_A(x),{\mu }_A(y),\frac{1-k}{2}\}$$

and

$${\gamma }_A(xy)\le max\{{\gamma }_A(x),{\gamma }_A(y),\frac{1-k}{2}\}$$

for all $x,y\in S$. Suppose that

$${\mu }_A(abc)<min\{{\mu }_A(a),{\mu }_A(c),\frac{1-k}{2}\}$$

and

$${\gamma }_A(abc)>max\{{\gamma }_A(a),{\gamma }_A(c),\frac{1-k}{2}\}$$

for some $a,b,c\in S$. Then there exist $t_1\in (0,\frac{1-k}{2}]$ and $s_1\in [\frac{1-k}{2},1)$ such that

$${\mu }_A(abc)<t_1\le min\{{\mu }_A(a),{\mu }_A(c),\frac{1-k}{2}\}$$

and

$${\gamma }_A(abc)>s_1\ge max\{{\gamma }_A(a),{\gamma }_A(c),\frac{1-k}{2}\}.$$

Then $a\in C_{(t_1,s_1)}(A)$ and $c\in C_{(t_1,s_1)}(A)$, but $abc\notin C_{(t_1,s_1)}(A)$. This is impossible, and hence ${\mu }_A(xyz)<min\{{\mu }_A(x),{\mu }_A(z),\frac{1-k}{2}\}$ and

$${\gamma }_A(xyz)>max\{{\gamma }_A(x),{\gamma }_A(z),\frac{1-k}{2}\}$$

for all $x,y,z\in S$. Therefore, $A=〈x,{\mu }_A,{\gamma }_A〉$ is an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S. □

By taking $k=0$ in Theorem 3.10, we get the following corollary.

Corollary 3.11 [[11], Theorem 3.10]

For an IFS $A=〈x,{\mu }_A,{\gamma }_A〉$ of an ordered semigroup $(S,\cdot ,\le )$, the following are equivalent

• (1) $A=〈x,{\mu }_A,{\gamma }_A〉$ is an $(\in ,\in \vee \mathrm{q})$-intuitionistic fuzzy bi-ideal of S.

• (2) ($\mathrm{\forall }t\in (0,0.5]$) ($\mathrm{\forall }s\in [0.5,1)$) ($C_{(\alpha ,\beta )}(A)\ne \mathrm{\varnothing }⟹C_{(\alpha ,\beta )}(A)$ is a bi-ideal of S) .

For an IFP $〈x;(\alpha ,\beta )〉$ of S and an IFS $A=〈x,{\mu }_A,{\gamma }_A〉$ of S, we say that

(c4) $〈x;(\alpha ,\beta )〉\underline{\mathrm{q}}A$ if ${\mu }_A(x)+\alpha \ge 1$ and ${\gamma }_A(x)+\beta \le 1$,

(c5) $〈x;(\alpha ,\beta )〉\underline{{\mathrm{q}}_k}A$ if ${\mu }_A(x)+\alpha +k\ge 1$ and ${\gamma }_A(x)+\beta +k\le 1$.

We denote by $Q_{(\alpha ,\beta )}^k(A)$ (resp. ${\underline{Q^k}}_{(\alpha ,\beta )}(A)$) the set $\{x\in S|〈x;(\alpha ,\beta )〉{\mathrm{q}}_{k}A\}$ (resp. $\{x\in S|〈x;(\alpha ,\beta )〉\underline{{\mathrm{q}}_k}A\}$), and ${[A]}_{(t,s)}^k:=\{x\in S|[x;(t,s)]\in \vee {\mathrm{q}}_{k}A\}$. It is obvious that ${[A]}_{(t,s)}^k=C_{(t,s)}(A)\cup Q_{(t,s)}^k(A)$.

Proposition 3.12 If $A=〈x,{\mu }_A,{\gamma }_A〉$ is an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S, then

$$(\mathrm{\forall }t\in (\frac{1-k}{2},1])(\mathrm{\forall }s\in [0,\frac{1-k}{2}))(Q_{(t,s)}^k(A)\ne \mathrm{\varnothing }⟹Q_{(t,s)}^k(A)\mathit{\text{is a bi-ideal of}}S).$$

Proof Assume that $A=〈x,{\mu }_A,{\gamma }_A〉$ is an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S. Let $t\in [\frac{1-k}{2},1]$ and $s\in [0,\frac{1-k}{2}]$ be such that $Q_{(t,s)}^k(A)\ne \mathrm{\varnothing }$. Let $y\in Q_{(t,s)}^k(A)$ and $x\in S$ be such that $x\le y$ . Then ${\mu }_A(y)+t+k>1$ and ${\gamma }_A(y)+s+k<1$. By means of Theorem 3.5(1), we have

$$\begin{array}{rcl}{\mu }_A(x)& \ge & min\{{\mu }_A(y),\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}\frac{1-k}{2}\hfill & \text{if }{\mu }_A(y)\ge \frac{1-k}{2},\hfill \\ {\mu }_A(y)\hfill & \text{if }{\mu }_A(y)<\frac{1-k}{2}\hfill \end{array}\\ >& 1-t-k\end{array}$$

and

$$\begin{array}{rcl}{\gamma }_A(x)& \le & max\{{\gamma }_A(y),\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}\frac{1-k}{2}\hfill & \text{if }{\gamma }_A(y)<\frac{1-k}{2},\hfill \\ {\gamma }_A(y)\hfill & \text{if }{\gamma }_A(y)\ge \frac{1-k}{2}\hfill \end{array}\\ <& 1-s-k.\end{array}$$

It follows that $x\in Q_{(t,s)}(A)$. Let $x,y\in Q_{(t,s)}(A)$. Then ${\mu }_A(x)+t>1-k$ and ${\gamma }_A(x)+s<1-k$, ${\mu }_A(y)+t>1-k$ and ${\gamma }_A(y)+s<1-k$. Using (2) of Theorem 3.5, we have that

$$\begin{array}{rcl}{\mu }_A(xy)& \ge & min\{{\mu }_A(x),{\mu }_A(y),\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}\frac{1-k}{2}\hfill & \text{if }min\{{\mu }_A(x),{\mu }_A(y)\}\ge \frac{1-k}{2},\hfill \\ min\{{\mu }_A(x),{\mu }_A(y)\}\hfill & \text{if }min\{{\mu }_A(x),{\mu }_A(y)\}<\frac{1-k}{2}\hfill \end{array}\\ >& 1-t-k,\end{array}$$

and

$$\begin{array}{rcl}{\gamma }_A(xy)& \le & max\{{\gamma }_A(x),{\gamma }_A(y),\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}\frac{1-k}{2}\hfill & \text{if }max\{{\gamma }_A(x),{\gamma }_A(y)\}<\frac{1-k}{2},\hfill \\ max\{{\gamma }_A(x),{\gamma }_A(y)\}\hfill & \text{if }max\{{\gamma }_A(x),{\gamma }_A(y)\}\ge \frac{1-k}{2}\hfill \end{array}\\ <& 1-s-k.\end{array}$$

Thus, $xy\in Q_{(t,s)}(A)$. Let $x,z\in Q_{(t,s)}(A)$ and $y\in S$. Then ${\mu }_A(x)+t>1-k$ and ${\gamma }_A(x)+s<1-k$, ${\mu }_A(z)+t>1-k$ and ${\gamma }_A(z)+s<1-k$. Using (3) of Theorem 3.5, we have that

$$\begin{array}{rcl}{\mu }_A(xyz)& \ge & min\{{\mu }_A(x),{\mu }_A(z),\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}\frac{1-k}{2}\hfill & \text{if }min\{{\mu }_A(x),{\mu }_A(z)\}\ge \frac{1-k}{2},\hfill \\ min\{{\mu }_A(x),{\mu }_A(z)\}\hfill & \text{if }min\{{\mu }_A(x),{\mu }_A(z)\}<\frac{1-k}{2}\hfill \end{array}\\ >& 1-t-k\end{array}$$

and

$$\begin{array}{rcl}{\gamma }_A(xyz)& \le & max\{{\gamma }_A(x),{\gamma }_A(z),\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}\frac{1-k}{2}\hfill & \text{if }max\{{\gamma }_A(x),{\gamma }_A(z)\}<\frac{1-k}{2},\hfill \\ max\{{\gamma }_A(x),{\gamma }_A(z)\}\hfill & \text{if }max\{{\gamma }_A(x),{\gamma }_A(z)\}\ge \frac{1-k}{2}\hfill \end{array}\\ <& 1-s.\end{array}$$

Hence $xyz\in Q_{(t,s)}^k(A)$. Therefore, $Q_{(t,s)}^k(A)$ is a bi-ideal of S. □

Theorem 3.13 For any IFS $A=〈x,{\mu }_A,{\gamma }_A〉$ of S, the following are equivalent

• (1) $A=〈x,{\mu }_A,{\gamma }_A〉$ is an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S.

• (2) ($\mathrm{\forall }t\in (0,1]$) ($\mathrm{\forall }s\in [0,1)$) (${[A]}_{(t,s)}^k\ne \mathrm{\varnothing }⟹{[A]}_{(t,s)}^k$ is a bi-ideal of S).

We call ${[A]}_{(t,s)}^k$ an $(\in \vee \mathrm{q})$-level bi-ideal of $A=〈x,{\mu }_A,{\gamma }_A〉$.

Proof Assume that $A=〈x,{\mu }_A,{\gamma }_A〉$ is an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S, and let $t\in (0,1]$ and $s\in [0,1)$ be such that ${[A]}_{(t,s)}^k\ne \mathrm{\varnothing }$. Let $y\in {[A]}_{(t,s)}^k$ and $x\in S$ be such that $x\le y$. Then $y\in C_{(t,s)}(A)$ or $y\in Q_{(t,s)}^k(A)$, i.e., ${\mu }_A(y)\ge t$ and ${\gamma }_A(y)\le s$ or ${\mu }_A(y)+t>1-k$ and ${\gamma }_A(y)+s<1-k$. Using Theorem 3.5(1), we get

$${\mu }_A(x)\ge min\{{\mu }_A(y),\frac{1-k}{2}\}\text{and}{\gamma }_A(x)\le max\{{\gamma }_A(y),\frac{1-k}{2}\}.$$

(3.1)

We consider two cases ${\mu }_A(y)\le \frac{1-k}{2}$, ${\gamma }_A(y)\ge \frac{1-k}{2}$ and ${\mu }_A(y)>\frac{1-k}{2}$, ${\gamma }_A(y)<\frac{1-k}{2}$. The first case implies from (3.1) that ${\mu }_A(x)\ge {\mu }_A(y)$ and ${\gamma }_A(x)\le {\gamma }_A(y)$. Thus, if ${\mu }_A(y)\ge t$ and ${\gamma }_A(y)\le s$, then ${\mu }_A(x)\ge t$ and ${\gamma }_A(x)\le s$, and so, $x\in C_{(t,s)}(A)\subseteq {[A]}_{(t,s)}^k$. If ${\mu }_A(y)+t>1-k$ and ${\gamma }_A(y)+s<1-k$, then ${\mu }_A(x)+t\ge {\mu }_A(y)+t>1-k$ and ${\gamma }_A(x)+s\le {\gamma }_A(y)+s<1-k$, which implies that $[x;(t,s)]{\mathrm{q}}_{k}A$, i.e., $x\in Q_{(t,s)}^k(A)\subseteq {[A]}_{(t,s)}^k$. Combining the second case and (3.1), we have ${\mu }_A(x)\ge \frac{1-k}{2}$ and ${\gamma }_A(x)\le \frac{1-k}{2}$. If $t\le \frac{1-k}{2}$ and $s\ge \frac{1-k}{2}$, then ${\mu }_A(x)\ge t$ and ${\gamma }_A(x)\le s$, and hence $x\in C_{(t,s)}(A)\subseteq {[A]}_{(t,s)}^k$. If $t>\frac{1-k}{2}$ and $s<\frac{1-k}{2}$, then ${\mu }_A(x)+t>\frac{1-k}{2}+\frac{1-k}{2}=1-k$ and ${\gamma }_A(x)+s<\frac{1-k}{2}+\frac{1-k}{2}=1-k$, which implies that $x\in Q_{(t,s)}^k(A)\subseteq {[A]}_{(t,s)}^k$. Therefore, ${[A]}_{(t,s)}^k$ satisfies the condition (b1). Let $x,y\in {[A]}_{(t,s)}^k$. Then $x\in C_{(t,s)}(A)$ or $〈x;(t,s)〉{\mathrm{q}}_{k}A$ and $y\in C_{(t,s)}(A)$ or $〈y;(t,s)〉{\mathrm{q}}_{k}A$, that is, ${\mu }_A(x)\ge t$, ${\gamma }_A(x)\le s$ or ${\mu }_A(x)+t+k>1$, ${\gamma }_A(x)+s+k<1$ and ${\mu }_A(y)\ge t$, ${\gamma }_A(y)\le s$ or ${\mu }_A(y)+t+k>1$, ${\gamma }_A(y)+s+k<1$. We consider the following four cases

• (i) ${\mu }_A(x)\ge t$, ${\gamma }_A(x)\le s$ and ${\mu }_A(y)\ge t$, ${\gamma }_A(y)\le s$,

• (ii) ${\mu }_A(x)\ge t$, ${\gamma }_A(x)\le s$ and ${\mu }_A(y)+t+k>1$, ${\gamma }_A(y)+s+k<1$,

• (iii) ${\mu }_A(x)+t+k>1$, ${\gamma }_A(x)+s+k<1$ and ${\mu }_A(y)\ge t$, ${\gamma }_A(y)\le s$,

• (iv) ${\mu }_A(x)+t+k>1$, ${\gamma }_A(x)+s+k<1$ and ${\mu }_A(y)+t+k>1$, ${\gamma }_A(y)+s+k<1$.

For the case (i), Theorem 3.5(2) implies that

$$\begin{array}{rcl}{\mu }_A(xy)& \ge & min\{{\mu }_A(x),{\mu }_A(y),\frac{1-k}{2}\}\ge min\{t,\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}\frac{1-k}{2}\hfill & \text{if }t>\frac{1-k}{2},\hfill \\ t\hfill & \text{if }t\le \frac{1-k}{2}\hfill \end{array}\end{array}$$

and

$$\begin{array}{rcl}{\gamma }_A(xy)& \le & max\{{\gamma }_A(x),{\gamma }_A(y),\frac{1-k}{2}\}\le max\{s,\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}\frac{1-k}{2}\hfill & \text{if }s<\frac{1-k}{2},\hfill \\ s\hfill & \text{if }s\ge \frac{1-k}{2}.\hfill \end{array}\end{array}$$

Then $xy\in C_{(t,s)}(A)$ or ${\mu }_A(xy)+t+k>\frac{1-k}{2}+\frac{1-k}{2}+k=1$ and ${\gamma }_A(xy)+s+k<\frac{1-k}{2}+\frac{1-k}{2}+k=1$, that is, $xy\in Q_{(t,s)}^k(A)$. Hence $xy\in C_{(t,s)}(A)\cup Q_{(t,s)}^k(A)={[A]}_{(t,s)}^k$. For the second case, assume that $t>\frac{1-k}{2}$ and $s<\frac{1-k}{2}$, then $1-t-k\le 1-t<\frac{1-k}{2}$ and $1-s-k\ge 1-s\ge \frac{1-k}{2}$. Hence

$$\begin{array}{rcl}{\mu }_A(xy)& \ge & min\{{\mu }_A(x),{\mu }_A(y),\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}min\{{\mu }_A(y),\frac{1-k}{2}\}>1-t-k\hfill & \text{if }min\{{\mu }_A(y),\frac{1-k}{2}\}\le {\mu }_A(x),\hfill \\ {\mu }_A(x)\ge t\hfill & \text{if }min\{{\mu }_A(y),\frac{1-k}{2}\}>{\mu }_A(x)\hfill \end{array}\end{array}$$

and

$$\begin{array}{rcl}{\gamma }_A(xy)& \le & max\{{\gamma }_A(x),{\gamma }_A(y),\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}max\{{\gamma }_A(y),\frac{1-k}{2}\}<1-s-k\hfill & \text{if }max\{{\gamma }_A(y),\frac{1-k}{2}\}\ge {\gamma }_A(x),\hfill \\ {\gamma }_A(x)\le s\hfill & \text{if }max\{{\gamma }_A(y),\frac{1-k}{2}\}<{\gamma }_A(x).\hfill \end{array}\end{array}$$

Thus $xy\in C_{(t,s)}(A)\cup Q_{(t,s)}^k(A)={[A]}_{(t,s)}^k$. Suppose that $t\le \frac{1-k}{2}$ and $s\ge \frac{1-k}{2}$. Then

$$\begin{array}{rcl}{\mu }_A(xy)& \ge & min\{{\mu }_A(x),{\mu }_A(y),\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}min\{{\mu }_A(x),\frac{1-k}{2}\}\ge t\hfill & \text{if }min\{{\mu }_A(x),\frac{1-k}{2}\}\le {\mu }_A(y),\hfill \\ {\mu }_A(y)>1-t-k\hfill & \text{if }min\{{\mu }_A(x),\frac{1-k}{2}\}>{\mu }_A(y),\hfill \end{array}\end{array}$$

and

$$\begin{array}{rcl}{\gamma }_A(xy)& \le & max\{{\gamma }_A(x),{\gamma }_A(y),\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}max\{{\gamma }_A(x),\frac{1-k}{2}\}\le s\hfill & \text{if }max\{{\gamma }_A(x),\frac{1-k}{2}\}\ge {\gamma }_A(y),\hfill \\ {\gamma }_A(y)<1-s-k\hfill & \text{if }max\{{\gamma }_A(x),\frac{1-k}{2}\}<{\gamma }_A(y).\hfill \end{array}\end{array}$$

Thus $xy\in C_{(t,s)}(A)\cup Q_{(t,s)}^k(A)={[A]}_{(t,s)}^k$. We have a similar result for the case (iii). For the final case, if $t>\frac{1-k}{2}$ and $s<\frac{1-k}{2}$, then $1-t-k<\frac{1-k}{2}$ and $1-s-k>\frac{1-k}{2}$. Hence

$$\begin{array}{rcl}{\mu }_A(xy)& \ge & min\{{\mu }_A(x),{\mu }_A(y),\frac{1-k}{2}\}\\ =& \frac{1-k}{2}>1-t-k\text{whenever }min\{{\mu }_A(x),{\mu }_A(y)\}\ge \frac{1-k}{2}\end{array}$$

and

$$\begin{array}{rcl}{\mu }_A(xy)& \le & min\{{\mu }_A(x),{\mu }_A(y),\frac{1-k}{2}\}\\ =& min\{{\mu }_A(x),{\mu }_A(y)\}\\ >& 1-s-k\text{whenever }min\{{\mu }_A(x),{\mu }_A(y)\}\ge \frac{1-k}{2}\end{array}$$

and

$$\begin{array}{rcl}{\gamma }_A(xy)& \le & max\{{\gamma }_A(x),{\gamma }_A(y),\frac{1-k}{2}\}\\ =& \frac{1-k}{2}<1-s-k\text{whenever }max\{{\gamma }_A(x),{\gamma }_A(y)\}\le \frac{1-k}{2}\end{array}$$

and

$$\begin{array}{rcl}{\gamma }_A(xy)& \le & max\{{\gamma }_A(x),{\gamma }_A(y),\frac{1-k}{2}\}\\ =& max\{{\gamma }_A(x),{\gamma }_A(y)\}\\ <& 1-s-k,\text{whenever }max\{{\gamma }_A(x),{\gamma }_A(y)\}\le \frac{1-k}{2}.\end{array}$$

Thus, $xy\in Q_{(t,s)}^k(A)\subseteq {[A]}_{(t,s)}^k$. If $t\le \frac{1-k}{2}$ and $s\ge \frac{1-k}{2}$, then

$$\begin{array}{rcl}{\mu }_A(xy)& \ge & min\{{\mu }_A(x),{\mu }_A(y),\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}\frac{1-k}{2}\ge t\hfill & \text{if }min\{{\mu }_A(x),{\mu }_A(y)\}\ge \frac{1-k}{2},\hfill \\ min\{{\mu }_A(x),{\mu }_A(y)\}>1-t-k\hfill & \text{if }min\{{\mu }_A(x),{\mu }_A(y)\}<\frac{1-k}{2},\hfill \end{array}\end{array}$$

and

$$\begin{array}{rcl}{\gamma }_A(xy)& \le & max\{{\gamma }_A(x),{\gamma }_A(y),\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}\frac{1-k}{2}\le s\hfill & \text{if }max\{{\gamma }_A(x),{\gamma }_A(y)\}\le \frac{1-k}{2},\hfill \\ max\{{\gamma }_A(x),{\gamma }_A(y)\}<1-s-k\hfill & \text{if }max\{{\gamma }_A(x),{\gamma }_A(y)\}>\frac{1-k}{2},\hfill \end{array}\end{array}$$

which implies that $xy\in Q_{(t,s)}^k(A)\subseteq {[A]}_{(t,s)}^k$. Let $x,z\in {[A]}_{(t,s)}^k$. Then $x\in C_{(t,s)}(A)$ or $〈x;(t,s)〉{\mathrm{q}}_{k}A$ and $z\in C_{(t,s)}(A)$ or $〈z;(t,s)〉{\mathrm{q}}_{k}A$, that is, ${\mu }_A(x)\ge t$, ${\gamma }_A(x)\le s$ or ${\mu }_A(x)+t+k>1$, ${\gamma }_A(x)+s+k<1$ and ${\mu }_A(z)\ge t$, ${\gamma }_A(z)\le s$ or ${\mu }_A(z)+t+k>1$, ${\gamma }_A(z)+s+k<1$. We consider the following four cases

• (i) ${\mu }_A(x)\ge t$, ${\gamma }_A(x)\le s$ and ${\mu }_A(z)\ge t$, ${\gamma }_A(z)\le s$,

• (ii) ${\mu }_A(x)\ge t$, ${\gamma }_A(x)\le s$ and ${\mu }_A(z)+t>1-k$, ${\gamma }_A(z)+s<1-k$,

• (iii) ${\mu }_A(x)+t>1-k$, ${\gamma }_A(x)+s<1-k$ and ${\mu }_A(z)\ge t$, ${\gamma }_A(z)\le s$,

• (iv) ${\mu }_A(x)+t>1-k$, ${\gamma }_A(x)+s<1-k$ and ${\mu }_A(z)+t>1-k$, ${\gamma }_A(z)+s<1-k$.

For the case (i), Theorem 3.5(3) implies that

$${\mu }_A(xyz)\ge min\{{\mu }_A(x),{\mu }_A(z),\frac{1-k}{2}\}\ge min\{t,\frac{1-k}{2}\}=\{\begin{array}{cc}\frac{1-k}{2}\hfill & \text{if }t>\frac{1-k}{2},\hfill \\ t\hfill & \text{if }t\le \frac{1-k}{2}\hfill \end{array}$$

and

$${\gamma }_A(xyz)\le max\{{\gamma }_A(x),{\gamma }_A(z),\frac{1-k}{2}\}\le max\{s,\frac{1-k}{2}\}=\{\begin{array}{cc}\frac{1-k}{2}\hfill & \text{if }s<\frac{1-k}{2},\hfill \\ s\hfill & \text{if }s\ge \frac{1-k}{2}.\hfill \end{array}$$

Then $xyz\in C_{(t,s)}(A)$ or ${\mu }_A(xyz)+t+k>\frac{1-k}{2}+\frac{1-k}{2}+k=1$ and ${\gamma }_A(xyz)+s+k<\frac{1-k}{2}+\frac{1-k}{2}+k=1$, that is, $xyz\in Q_{(t,s)}^k(A)$. Hence $xyz\in C_{(t,s)}(A)\cup Q_{(t,s)}^k(A)={[A]}_{(t,s)}^k$. For the second case, assume that $t>\frac{1-k}{2}$ and $s<\frac{1-k}{2}$, then $1-t-k\le 1-t<\frac{1-k}{2}$ and $1-s-k\ge 1-s\ge \frac{1-k}{2}$. Hence

$$\begin{array}{rcl}{\mu }_A(xyz)& \ge & min\{{\mu }_A(x),{\mu }_A(z),\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}min\{{\mu }_A(x),\frac{1-k}{2}\}>1-t-k\hfill & \text{if }min\{{\mu }_A(z),\frac{1-k}{2}\}\le {\mu }_A(x),\hfill \\ {\mu }_A(x)\ge t\hfill & \text{if }min\{{\mu }_A(z),\frac{1-k}{2}\}>{\mu }_A(x)\hfill \end{array}\end{array}$$

and

$$\begin{array}{rcl}{\gamma }_A(xyz)& \le & max\{{\gamma }_A(x),{\gamma }_A(z),\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}max\{{\gamma }_A(z),\frac{1-k}{2}\}<1-s-k\hfill & \text{if }max\{{\gamma }_A(z),\frac{1-k}{2}\}\ge {\gamma }_A(x),\hfill \\ {\gamma }_A(x)\le s\hfill & \text{if }max\{{\gamma }_A(z),\frac{1-k}{2}\}<{\gamma }_A(x).\hfill \end{array}\end{array}$$

Thus, $xyz\in C_{(t,s)}(A)\cup Q_{(t,s)}^k(A)={[A]}_{(t,s)}^k$. Suppose that $t\le \frac{1-k}{2}$ and $s\ge \frac{1-k}{2}$. Then

$$\begin{array}{rcl}{\mu }_A(xyz)& \ge & min\{{\mu }_A(x),{\mu }_A(z),\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}min\{{\mu }_A(x),\frac{1-k}{2}\}\ge t\hfill & \text{if }min\{{\mu }_A(x),\frac{1-k}{2}\}\le {\mu }_A(z),\hfill \\ {\mu }_A(z)>1-t-k\hfill & \text{if }min\{{\mu }_A(x),\frac{1-k}{2}\}>{\mu }_A(z),\hfill \end{array}\end{array}$$

and

$$\begin{array}{rcl}{\gamma }_A(xyz)& \le & max\{{\gamma }_A(x),{\gamma }_A(z),\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}max\{{\gamma }_A(x),\frac{1-k}{2}\}\le s\hfill & \text{if }max\{{\gamma }_A(x),\frac{1-k}{2}\}\ge {\gamma }_A(z),\hfill \\ {\gamma }_A(y)<1-s-k\hfill & \text{if }max\{{\gamma }_A(x),\frac{1-k}{2}\}<{\gamma }_A(z).\hfill \end{array}\end{array}$$

Thus, $xyz\in C_{(t,s)}(A)\cup Q_{(t,s)}^k(A)={[A]}_{(t,s)}^k$. We have a similar result for the case (iii). For the final case, if $t>\frac{1-k}{2}$ and $s<\frac{1-k}{2}$, then $1-t-k<\frac{1-k}{2}$ and $1-s-k>\frac{1-k}{2}$. Hence

$$\begin{array}{rcl}{\mu }_A(xyz)& \ge & min\{{\mu }_A(x),{\mu }_A(z),\frac{1-k}{2}\}\\ =& \frac{1-k}{2}>1-t-k\text{whenever }min\{{\mu }_A(x),{\mu }_A(z)\}\ge \frac{1-k}{2}\end{array}$$

and

$$\begin{array}{rcl}{\mu }_A(xyz)& \le & min\{{\mu }_A(x),{\mu }_A(z),\frac{1-k}{2}\}\\ =& min\{{\mu }_A(x),{\mu }_A(z)\}\\ >& 1-s-k\text{whenever }min\{{\mu }_A(x),{\mu }_A(z)\}\ge \frac{1-k}{2}\end{array}$$

and

$$\begin{array}{rcl}{\gamma }_A(xyz)& \le & max\{{\gamma }_A(x),{\gamma }_A(z),\frac{1-k}{2}\}\\ =& \frac{1-k}{2}<1-s-k\text{whenever }max\{{\gamma }_A(x),{\gamma }_A(z)\}\le \frac{1-k}{2}\end{array}$$

and

$$\begin{array}{rcl}{\gamma }_A(xyz)& \le & max\{{\gamma }_A(x),{\gamma }_A(z),\frac{1-k}{2}\}\\ =& max\{{\gamma }_A(x),{\gamma }_A(z)\}\\ <& 1-s-k\text{whenever }max\{{\gamma }_A(x),{\gamma }_A(z)\}\le \frac{1-k}{2}.\end{array}$$

Thus, $xyz\in Q_{(t,s)}^k(A)\subseteq {[A]}_{(t,s)}^k$. If $t\le \frac{1-k}{2}$ and $s\ge \frac{1-k}{2}$, then

$$\begin{array}{rcl}{\mu }_A(xyz)& \ge & min\{{\mu }_A(x),{\mu }_A(z),\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}\frac{1-k}{2}\ge t\hfill & \text{if }min\{{\mu }_A(x),{\mu }_A(z)\}\ge \frac{1-k}{2},\hfill \\ min\{{\mu }_A(x),{\mu }_A(z)\}>1-t-k\hfill & \text{if }min\{{\mu }_A(x),{\mu }_A(z)\}<\frac{1-k}{2},\hfill \end{array}\end{array}$$

and

$$\begin{array}{rcl}{\gamma }_A(xyz)& \le & max\{{\gamma }_A(x),{\gamma }_A(z),\frac{1-k}{2}\}\\ =& \{\begin{array}{cc}\frac{1-k}{2}\le s\hfill & \text{if }max\{{\gamma }_A(x),{\gamma }_A(z)\}\le \frac{1-k}{2},\hfill \\ max\{{\gamma }_A(x),{\gamma }_A(z)\}<1-s-k\hfill & \text{if }max\{{\gamma }_A(x),{\gamma }_A(z)\}>\frac{1-k}{2},\hfill \end{array}\end{array}$$

which implies that $xyz\in Q_{(t,s)}^k(A)\subseteq {[A]}_{(t,s)}^k$. Therefore, ${[A]}_{(t,s)}^k$ is a bi-ideal of S.

Conversely, suppose that (2) is valid. If there exist $a,b\in S$ such that $a\le b$ and

$${\mu }_A(a)<min\{{\mu }_A(b),\frac{1-k}{2}\}\text{and}{\gamma }_A(a)>max\{{\gamma }_A(b),\frac{1-k}{2}\}.$$

Then ${\mu }_A(a)<t_a\le min\{{\mu }_A(b),\frac{1-k}{2}\}$ and ${\gamma }_A(a)>s_a\ge max\{{\gamma }_A(b),\frac{1-k}{2}\}$ for some $t_a\in (0,1]$ and $s_a\in [0,1)$. It follows that $b\in C_{(t_a,s_a)}(A)\subseteq {[A]}_{(t_a,s_a)}^k$ but $a\notin C_{(t_a,s_a)}(A)$. Also we have ${\mu }_A(a)+t_a<2t_a\le 1-k$ and ${\gamma }_A(a)+s_a>2s_a\ge 1-k$ and so $〈a;(t_a,s_a)〉\overline{{\mathrm{q}}_k}A$, i.e., $b\notin Q_{(t_a,s_a)}^k(A)$. Therefore, $a\notin {[A]}_{(t_a,s_a)}$, a contradiction. Hence ${\mu }_A(x)\ge min\{{\mu }_A(y),\frac{1-k}{2}\}$ and ${\gamma }_A(x)\le max\{{\gamma }_A(y),\frac{1-k}{2}\}$ for all $x,y\in S$ with $x\le y$. Suppose that there exist $a,b\in S$ such that

$${\mu }_A(ab)<min\{{\mu }_A(a),{\mu }_A(b),\frac{1-k}{2}\}$$

and

$${\gamma }_A(ab)>max\{{\gamma }_A(a),{\gamma }_A(b),\frac{1-k}{2}\}.$$

Then

$${\mu }_A(ab)<t\le min\{{\mu }_A(a),{\mu }_A(b),\frac{1-k}{2}\}$$

and

$${\gamma }_A(ab)>s\ge max\{{\gamma }_A(a),{\gamma }_A(b),\frac{1-k}{2}\}$$

for $t\in (0,1]$ and $s\in [0,1)$. It follows that $a\in C_{(t,s)}(A)\subseteq {[A]}_{(t,s)}^k$ and $b\in C_{(t,s)}(A)\subseteq {[A]}_{(t,s)}^k$, so from (b2) $ab\in {[A]}_{(t,s)}^k$. Thus, ${\mu }_A(ab)\ge t$, ${\gamma }_A(ab)\le s$ or ${\mu }_A(ab)+t+k>1$, ${\gamma }_A(ab)+s+k<1$, a contradiction. Therefore, ${\mu }_A(xy)\ge min\{{\mu }_A(x),{\mu }_A(y),\frac{1-k}{2}\}$ and ${\gamma }_A(xy)\le max\{{\gamma }_A(x),{\gamma }_A(y),\frac{1-k}{2}\}$ for all $x,y\in S$. Assume that there exist $a,b,c\in S$ such that

$${\mu }_A(abc)<min\{{\mu }_A(a),{\mu }_A(c),\frac{1-k}{2}\}$$

and

$${\gamma }_A(abc)>max\{{\gamma }_A(a),{\gamma }_A(c),\frac{1-k}{2}\}.$$

Then ${\mu }_A(abc)<t_0\le min\{{\mu }_A(a),{\mu }_A(c),\frac{1-k}{2}\}$ and ${\gamma }_A(abc)>s_0\ge max\{{\gamma }_A(a),{\gamma }_A(c),\frac{1-k}{2}\}$ for $t_0\in (0,1]$ and $s_0\in [0,1)$. It follows that $a\in C_{(t_0,s_0)}(A)\subseteq {[A]}_{(t_0,s_0)}^k$ and $c\in C_{(t_0,s_0)}(A)\subseteq {[A]}_{(t_0,s_0)}^k$ so from (b2) $abc\in {[A]}_{(t_0,s_0)}$. Thus, ${\mu }_A(ab)\ge t_0$, ${\gamma }_A(ab)\le s_0$ or ${\mu }_A(ab)+t_0+k>1$, ${\gamma }_A(ab)+s_0+k<1$, a contradiction. Therefore, ${\mu }_A(xyz)\ge min\{{\mu }_A(x),{\mu }_A(z),\frac{1-k}{2}\}$ and ${\gamma }_A(xyz)\le max\{{\gamma }_A(x),{\gamma }_A(z),\frac{1-k}{2}\}$ for all $x,y,z\in S$. Thus, $A=〈x,{\mu }_A,{\gamma }_A〉$ is an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S. □

Theorem 3.14 Let $\{A_i|i\in \mathrm{\Lambda }\}$ be a family of $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideals of S. Then $A={\bigcap }_{i\in \mathrm{\Lambda }}{A}_i$ is an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S, where ${\bigcap }_{i\in \mathrm{\Lambda }}{A}_i=〈x,{\bigwedge }_{i\in \mathrm{\Lambda }}{\mu }_{A_i},{\bigvee }_{i\in \mathrm{\Lambda }}{\gamma }_{Ai}〉$ and

$$\begin{array}{rcl}\underset{i\in \mathrm{\Lambda }}{\bigwedge }{\mu }_{A_i}(x)& =& inf\{{\mu }_{A_i}(x)|i\in \mathrm{\Lambda }\mathit{\text{and}}x\in S\},\\ \underset{i\in \mathrm{\Lambda }}{\bigvee }{\gamma }_{A_i}(x)& =& sup\{{\gamma }_{A_i}(x)|i\in \mathrm{\Lambda }\mathit{\text{and}}x\in S\}.\end{array}$$

Proof Let $x,y\in S$ with $x\le y$, $t\in (0,1]$ and $s\in [0,1)$ be such that $〈y;(t,s)〉\in A$. Assume that $〈x;(t,s)〉\overline{\in \vee {\mathrm{q}}_k}A$. Then ${\mu }_A(x)<t$, ${\gamma }_A(x)>s$ and ${\mu }_A(x)+t+k\le 1$, ${\gamma }_A(x)+s+k\ge 1$, which imply that

$${\mu }_A(x)<\frac{1-k}{2}\text{and}{\gamma }_A(x)>\frac{1-k}{2}.$$

Let ${\mathrm{\Omega }}_1:=\{i\in \mathrm{\Lambda }|{\mu }_{A_i}(x)\ge t,{\gamma }_{A_i}(x)\le s\}$ and ${\mathrm{\Omega }}_2:=\{i\in \mathrm{\Lambda }|〈x;(t,s)〉{\mathrm{q}}_k{A}_i\text{ and }{\mu }_{A_i}(x)<t,{\gamma }_{A_i}(x)>s\}$.

Then $\mathrm{\Lambda }={\mathrm{\Omega }}_1\cup {\mathrm{\Omega }}_2$ and ${\mathrm{\Omega }}_1\cap {\mathrm{\Omega }}_2=\mathrm{\varnothing }$. If ${\mathrm{\Omega }}_2=\mathrm{\varnothing }$, then ${\mu }_{A_i}(x)\ge t$, ${\gamma }_{A_i}(x)\le s$ for all $i\in \mathrm{\Lambda }$, and so, ${\mu }_A(x)\ge t$, ${\gamma }_A(x)\le s$, which is a contradiction. Hence ${\mathrm{\Omega }}_2\ne \mathrm{\varnothing }$, and so, ${\mu }_{A_i}(x)+t+k>1$, ${\gamma }_{A_i}(x)+s+k<1$ and ${\mu }_{A_i}(x)<t$, ${\gamma }_{A_i}(x)>s$ for every $i\in \mathrm{\Lambda }$. It follows that $t>\frac{1-k}{2}$ and $s<\frac{1-k}{2}$, so that ${\mu }_{A_i}(x)\ge {\mu }_A(x)\ge t>\frac{1-k}{2}$ and ${\gamma }_{A_i}(x)\le {\gamma }_A(x)\le s<\frac{1-k}{2}$ for all $i\in \mathrm{\Lambda }$. Now, suppose that $t_x:={\mu }_{A_i}(x)<\frac{1-k}{2}$ and $s_x:={\gamma }_{A_i}(x)>\frac{1-k}{2}$ for some $i\in \mathrm{\Lambda }$. Let $t_x^{\prime }\in (0,\frac{1-k}{2})$ and $s_x^{\prime }\in (\frac{1-k}{2},1)$ be such that $t_x<t_x^{\prime }$ and $s_x^{\prime }<s_x$. Then ${\mu }_{A_i}(y)>\frac{1-k}{2}>t_x^{\prime }$ and ${\gamma }_{A_i}(y)<\frac{1-k}{2}<s_x^{\prime }$, i.e., $〈y;(t_x^{\prime },s_x^{\prime })〉\in A_i$. But ${\mu }_{A_i}(x)=t_x<t_x^{\prime }$, ${\gamma }_{A_i}(x)=s_x>s_x^{\prime }$ and ${\mu }_{A_i}(x)+t_x^{\prime }+k<1$, ${\gamma }_{A_i}(x)+s_x^{\prime }+k>1$, that is, $〈x;(t_x^{\prime },s_x^{\prime })〉\overline{\in \vee {\mathrm{q}}_k}{A}_i$. This is a contradiction, and so, ${\mu }_{A_i}(x)\ge \frac{1-k}{2}$ and ${\gamma }_{A_i}(x)\le \frac{1-k}{2}$ for all $i\in \mathrm{\Lambda }$. Thus, ${\mu }_A(x)\ge \frac{1-k}{2}$ and ${\gamma }_A(x)\le \frac{1-k}{2}$, which is impossible. Therefore, $〈y;(t,s)〉\in \vee {\mathrm{q}}_{k}A$.

Let $x,y\in S$, $t_1,t_2\in (0,1]$ and $s_1,s_2\in [0,1)$ be such that $〈x;(t_1,s_1)〉\in A$ and $〈y;(t_2,s_2)〉\in A$. Assume that $〈xy;min\{t_1,t_2\},max\{s_1,s_2\}〉\overline{\in \vee {\mathrm{q}}_k}A$. Then

$${\mu }_A(xy)<min\{t_1,t_2\},{\gamma }_A(xy)>max\{s_1,s_2\}$$

and

$${\mu }_A(xy)+min\{t_1,t_2\}<1-k,{\gamma }_A(xy)+max\{s_1,s_2\}>1-k.$$

It follows that ${\mu }_A(xy)<\frac{1-k}{2}$ and ${\gamma }_A(xy)>\frac{1-k}{2}$. Let ${\mathrm{\Omega }}_3:=\{i\in \mathrm{\Lambda }|{\mu }_{A_i}(xy)\ge min\{t_1,t_2\}\text{ and}{\gamma }_{A_i}(xy)\le max\{s_1,s_2\}\}$ and ${\mathrm{\Omega }}_4:=\{i\in \mathrm{\Lambda }|〈xy;min\{t_1,t_2\},max\{s_1,s_2\}〉{\mathrm{q}}_k{A}_i\text{ and }{\mu }_{A_i}(xy)<min\{t_1,t_2\}\text{ and }{\gamma }_{A_i}(xy)>max\{s_1,s_2\}\}$. Then ${\mathrm{\Omega }}_3\cup {\mathrm{\Omega }}_4=\mathrm{\Lambda }$ and ${\mathrm{\Omega }}_3\cap {\mathrm{\Omega }}_4=\mathrm{\varnothing }$. If ${\mathrm{\Omega }}_4=\mathrm{\varnothing }$, then ${\mu }_{A_i}(xy)\ge min\{t_1,t_2\}$ and ${\gamma }_{A_i}(xy)\le max\{s_1,s_2\}$ for all $i\in \mathrm{\Lambda }$, and so, ${\mu }_A(xy)\ge min\{t_1,t_2\}$ and ${\gamma }_A(xy)\le max\{s_1,s_2\}$, which is a contradiction. Hence ${\mathrm{\Omega }}_4\ne \mathrm{\varnothing }$ and $〈xy;min\{t_1,t_2\},max\{s_1,s_2\}〉{\mathrm{q}}_k{A}_i$, i.e., ${\mu }_{A_i}(xy)+min\{t_1,t_2\}>1-k$, ${\gamma }_{A_i}(xy)+max\{s_1,s_2\}<1-k$. It follows that $min\{t_1,t_2\}>\frac{1-k}{2}$ and $max\{s_1,s_2\}<\frac{1-k}{2}$, so that ${\mu }_{A_i}(x)\ge {\mu }_A(x)\ge t_1\ge min\{t_1,t_2\}>\frac{1-k}{2}$ and ${\gamma }_{A_i}(x)\le {\gamma }_A(x)\le s_1\le max\{s_1,s_2\}$ for all $i\in \mathrm{\Lambda }$. By a similar way, we have ${\mu }_{A_i}(y)\ge {\mu }_A(y)\ge t_1\ge min\{t_1,t_2\}>\frac{1-k}{2}$ and ${\gamma }_{A_i}(y)\le {\gamma }_A(y)\le s_1\le max\{s_1,s_2\}$ for all $i\in \mathrm{\Lambda }$. Now, suppose that $t:={\mu }_{A_i}(xy)<\frac{1-k}{2}$ and $s:={\gamma }_{A_i}(xy)>\frac{1-k}{2}$ for some $i\in \mathrm{\Lambda }$. Let $t^{\prime }\in (0,\frac{1-k}{2})$ and $s^{\prime }\in (\frac{1-k}{2},1)$ be such that $t<t^{\prime }$ and $s>s^{\prime }$. Then ${\mu }_{A_i}(x)>\frac{1-k}{2}>t^{\prime }$, ${\gamma }_{A_i}(x)<\frac{1-k}{2}<s^{\prime }$ and ${\mu }_{A_i}(y)>\frac{1-k}{2}>t^{\prime }$, ${\gamma }_{A_i}(y)<\frac{1-k}{2}<s^{\prime }$, i.e., $〈x;(t^{\prime },s^{\prime })〉\in A$ and $〈y;(t^{\prime },s^{\prime })〉\in A$. But ${\mu }_{A_i}(xy)=t<t^{\prime }$, ${\gamma }_{A_i}(xy)=s>s^{\prime }$ and ${\mu }_{A_i}(xy)+t^{\prime }+k<1$, ${\gamma }_{A_i}(xy)+s^{\prime }+k>1$, that is, $〈xy;(t^{\prime },s^{\prime })〉\overline{\in }\vee \overline{{\mathrm{q}}_k}{A}_i$. This is a contradiction. Thus, ${\mu }_{A_i}(xy)\ge \frac{1-k}{2}$ and ${\gamma }_{A_i}(xy)\le \frac{1-k}{2}$ for all $i\in \mathrm{\Lambda }$. Therefore, ${\mu }_A(xy)\ge \frac{1-k}{2}$ and ${\gamma }_A(xy)\le \frac{1-k}{2}$, which is invalid. Consequently, $〈xy;min\{t_1,t_2\},max\{s_1,s_2\}〉\in \vee {\mathrm{q}}_{k}A$. Finally, suppose that $x,y,z\in S$, $t_1,t_2\in (0,1]$ and $s_1,s_2\in [0,1)$ be such that $〈x;(t_1,s_1)〉\in A$ and $〈z;(t_2,s_2)〉\in A$. Assume that $〈xyz;min\{t_1,t_2\},max\{s_1,s_2\}〉\overline{\in \vee {\mathrm{q}}_k}A$. Then

$${\mu }_A(xyz)<min\{t_1,t_2\},{\gamma }_A(xyz)>max\{s_1,s_2\}$$

and

$${\mu }_A(xyz)+min\{t_1,t_2\}<1-k,{\gamma }_A(xyz)+max\{s_1,s_2\}>1-k.$$

It follows that ${\mu }_A(xyz)<\frac{1-k}{2}$ and ${\gamma }_A(xyz)>\frac{1-k}{2}$. Let

$${\mathrm{\Omega }}_5:=\{i\in \mathrm{\Lambda }|{\mu }_{A_i}(xyz)\ge min\{t_1,t_2\}\text{ and }{\gamma }_{A_i}(xyz)\le max\{s_1,s_2\}\}$$

and

$$\begin{array}{rcl}{\mathrm{\Omega }}_6& :=& \{i\in \mathrm{\Lambda }|〈xyz;min\{t_1,t_2\},max\{s_1,s_2\}〉{\mathrm{q}}_k{A}_i\\ \text{and }{\mu }_{A_i}(xyz)<min\{t_1,t_2\}\text{ and }{\gamma }_{A_i}(xyz)>max\{s_1,s_2\}\}.\end{array}$$

Then ${\mathrm{\Omega }}_5\cup {\mathrm{\Omega }}_6=\mathrm{\Lambda }$ and ${\mathrm{\Omega }}_5\cap {\mathrm{\Omega }}_6=\mathrm{\varnothing }$. If ${\mathrm{\Omega }}_6=\mathrm{\varnothing }$, then ${\mu }_{A_i}(xyz)\ge min\{t_1,t_2\}$ and ${\gamma }_{A_i}(xyz)\le max\{s_1,s_2\}$ for all $i\in \mathrm{\Lambda }$, and so ${\mu }_A(xyz)\ge min\{t_1,t_2\}$ and ${\gamma }_A(xyz)\le max\{s_1,s_2\}$ which is a contradiction. Hence ${\mathrm{\Omega }}_6\ne \mathrm{\varnothing }$ and

$$〈xyz;min\{t_1,t_2\},max\{s_1,s_2\}〉{\mathrm{q}}_k{A}_i,$$

i.e.,

$${\mu }_{A_i}(xyz)+min\{t_1,t_2\}>1-k,{\gamma }_{A_i}(xyz)+max\{s_1,s_2\}<1-k.$$

It follows that $min\{t_1,t_2\}>\frac{1-k}{2}$ and $max\{s_1,s_2\}<\frac{1-k}{2}$, so that

$${\mu }_{A_i}(x)\ge {\mu }_A(x)\ge t_1\ge min\{t_1,t_2\}>\frac{1-k}{2}$$

and

$${\gamma }_{A_i}(x)\le {\gamma }_A(x)\le s_1\le max\{s_1,s_2\}$$

for all $i\in \mathrm{\Lambda }$. Similarly, we have

$${\mu }_{A_i}(z)\ge {\mu }_A(z)\ge t_1\ge min\{t_1,t_2\}>\frac{1-k}{2}$$

and

$${\gamma }_{A_i}(z)\le {\gamma }_A(z)\le s_1\le max\{s_1,s_2\}$$

for all $i\in \mathrm{\Lambda }$. Now, suppose that $t:={\mu }_{A_i}(xyz)<\frac{1-k}{2}$ and $s:={\gamma }_{A_i}(xyz)>\frac{1-k}{2}$ for some $i\in \mathrm{\Lambda }$. Let $t^{\prime }\in (0,\frac{1-k}{2})$ and $s^{\prime }\in (\frac{1-k}{2},1)$ be such that $t<t^{\prime }$ and $s>s^{\prime }$. Then ${\mu }_{A_i}(x)>\frac{1-k}{2}>t^{\prime }$, ${\gamma }_{A_i}(x)<\frac{1-k}{2}<s^{\prime }$ and ${\mu }_{A_i}(y)>\frac{1-k}{2}>t^{\prime }$, ${\gamma }_{A_i}(y)<\frac{1-k}{2}<s^{\prime }$, i.e., $〈x;(t^{\prime },s^{\prime })〉\in A$ and $〈y;(t^{\prime },s^{\prime })〉\in A$. But

$${\mu }_{A_i}(xyz)=t<t^{\prime },{\gamma }_{A_i}(xyz)=s>s^{\prime }$$

and

$${\mu }_{A_i}(xyz)+t^{\prime }<1,{\gamma }_{A_i}(xyz)+s^{\prime }>1,$$

that is, $〈xyz;(t^{\prime },s^{\prime })〉\overline{\in }\vee \overline{{\mathrm{q}}_k}{A}_i$. This is a contradiction. Thus, ${\mu }_{A_i}(xyz)\ge \frac{1-k}{2}$ and ${\gamma }_{A_i}(xyz)\le \frac{1-k}{2}$ for all $i\in \mathrm{\Lambda }$. Therefore, ${\mu }_A(xyz)\ge \frac{1-k}{2}$ and ${\gamma }_A(xyz)\le \frac{1-k}{2}$, which is invalid. Thus, $〈xyz;min\{t_1,t_2\},max\{s_1,s_2\}〉\in \vee {\mathrm{q}}_{k}A$. Therefore, ${\bigcap }_{i\in \mathrm{\Lambda }}{A}_i$ is an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S. □

The following example shows that the union of two $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideals of S may not be an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S.

Example 3.15 Consider the ordered semigroup of Example 3.4 with the ∗-multiplication Table 1 and the IFS of Examples 3.4 and 3.7, then $〈a;(0.7,0.3)〉\in A\cup B$ and $〈b;(0.5,0.4)〉\in A\cup B$, but $〈abmin\{0.7,0.5\},max\{0.3,0.4\}〉=〈d;(0.5,0.4)〉\overline{\in \vee {\mathrm{q}}_k}A\cup B$.

Definition 3.16 An IFS $A=〈x,{\mu }_A,{\gamma }_A〉$ of S is called an $(\overline{\in },\overline{\in }\vee \overline{{\mathrm{q}}_k})$-intuitionistic fuzzy bi-ideal of S if for all $x,y,z\in S$, $t,t_1,t_2\in (0,1]$ and $s,s_1,s_2\in [0,1)$, it satisfies the following conditions

(q4) ($〈x;(t,s)〉\overline{\in }A⟹〈y;(t,s)〉\overline{\in }\vee \overline{{\mathrm{q}}_k}A$ with $x\le y$),

(q5) ($〈xy;min\{t_1,t_2\},max\{s_1,s_2\}〉\overline{\in }A⟹〈x;(t_1,s_1)〉\overline{\in }\vee \overline{{\mathrm{q}}_k}A$ or $〈y;(t_2,s_2)〉\overline{\in }\vee \overline{{\mathrm{q}}_k}A$),

(q6) ($〈xyz;min\{t_1,t_2\},max\{s_1,s_2\}〉\overline{\in }A⟹〈x;(t_1,s_1)〉\overline{\in }\vee \overline{{\mathrm{q}}_k}A$ or $〈z;(t_2,s_2)〉\overline{\in }\vee \overline{{\mathrm{q}}_k}A$).

Let $A=〈x,{\mu }_A,{\gamma }_A〉$ be an $(\overline{\in },\overline{\in }\vee \overline{{\mathrm{q}}_k})$-intuitionistic fuzzy bi-ideal of an ordered semigroup S. Suppose that there exist $a,b\in S$ with $a\le b$ such that

$${\mu }_A(b)>max\{{\mu }_A(a),\frac{1-k}{2}\}\text{and}{\gamma }_A(b)<min\{{\gamma }_A(a),\frac{1-k}{2}\}.$$

Then ${\mu }_A(b)\ge t>max\{{\mu }_A(a),\frac{1-k}{2}\}$ and ${\gamma }_A(b)\le s<min\{{\gamma }_A(a),\frac{1-k}{2}\}$ for some $t\in (\frac{1-k}{2},1]$ and $s\in [0,\frac{1-k}{2})$. It follows that $〈a;(t,s)〉\overline{\in }A$, $〈b;(t,s)〉\in A$ and ${\mu }_A(b)+t\ge 2t>1-k$, ${\gamma }_A(b)+s\le 2s<1-k$, i.e., $〈b;(t,s)〉{\mathrm{q}}_{k}A$. This is a contradiction, and so the following inequalities hold.

$$\text{(e1)}(x\le y⟹{\mu }_A(b)\le max\{{\mu }_A(a),\frac{1-k}{2}\}\text{ and }{\gamma }_A(b)\ge min\{{\gamma }_A(a),\frac{1-k}{2}\}).$$

Suppose that $max\{{\mu }_A(ab),\frac{1-k}{2}\}<min\{{\mu }_A(a),{\mu }_A(b)\}$ and $min\{{\gamma }_A(ab),\frac{1-k}{2}\}>max\{{\gamma }_A(a),{\gamma }_A(b)\}$ for some $a,b\in S$. Then $max\{{\mu }_A(ab),\frac{1-k}{2}\}<t\le min\{{\mu }_A(a),{\mu }_A(b)\}$ and $min\{{\gamma }_A(ab),\frac{1-k}{2}\}>s\ge max\{{\gamma }_A(a),{\gamma }_A(b)\}$. Thus, $〈a;(t,s)〉\overline{\in }A$, $〈b;(t,s)〉\overline{\in }A$, ${\mu }_A(a)+t\ge 2t>1-k$, ${\gamma }_A(a)+s\le 2s<1-k$, i.e., $〈a;(t,s)〉{\mathrm{q}}_{k}A$ and ${\mu }_A(b)+t\ge 2t>1-k$, ${\gamma }_A(b)+s\le 2s<1-k$, i.e., $〈b;(t,s)〉{\mathrm{q}}_{k}A$. This is impossible, and hence $A=〈x,{\mu }_A,{\gamma }_A〉$ satisfies the following assertion

(e2) $max\{{\mu }_A(xy),\frac{1-k}{2}\}<min\{{\mu }_A(x),{\mu }_A(y)\}$ and $min\{{\gamma }_A(xy),\frac{1-k}{2}\}>max\{{\gamma }_A(x),{\gamma }_A(y)\}$ for all $x,y\in S$.

Now, assume that

$$max\{{\mu }_A(abc),\frac{1-k}{2}\}<min\{{\mu }_A(a),{\mu }_A(c)\}$$

and

$$min\{{\gamma }_A(abc),\frac{1-k}{2}\}>max\{{\gamma }_A(a),{\gamma }_A(c)\}$$

for some $a,b,c\in S$. Then

$$max\{{\mu }_A(abc),\frac{1-k}{2}\}<t\le min\{{\mu }_A(a),{\mu }_A(c)\}$$

and

$$min\{{\gamma }_A(abc),\frac{1-k}{2}\}>s\ge max\{{\gamma }_A(a),{\gamma }_A(c)\}.$$

Thus, $〈a;(t,s)〉\overline{\in }A,〈c;(t,s)〉\overline{\in }A$, ${\mu }_A(a)+t\ge 2t>1-k$, ${\gamma }_A(a)+s\le 2s<1-k$, i.e., $〈a;(t,s)〉{\mathrm{q}}_{k}A$ and ${\mu }_A(c)+t\ge 2t>1-k$, ${\gamma }_A(c)+s\le 2s<1-k$, i.e., $〈c;(t,s)〉{\mathrm{q}}_{k}A$. This is a contradiction, and hence we have the following assertion

(e3) $max\{{\mu }_A(xyz),\frac{1-k}{2}\}<min\{{\mu }_A(x),{\mu }_A(z)\}$ and $min\{{\gamma }_A(xyz),\frac{1-k}{2}\}>max\{{\gamma }_A(x),{\gamma }_A(z)\}$ for all $x,y,z\in S$.

Let $A=〈x,{\mu }_A,{\gamma }_A〉$ be an IFS of S satisfying the three conditions (e1), (e2) and (e3). Let $t\in (\frac{1-k}{2},1]$ and $s\in [0,\frac{1-k}{2})$ be such that $C_{(t,s)}(A)\ne \mathrm{\varnothing }$. Then there exist $b\in C_{(t,s)}(A)$ and $S\ni a\le b$, by using (e1), we get

$$\frac{1-k}{2}<t\le {\mu }_A(b)\le max\{{\mu }_A(a),\frac{1-k}{2}\}={\mu }_A(a)$$

and

$$\frac{1-k}{2}>s\ge {\gamma }_A(b)\ge min\{{\gamma }_A(a),\frac{1-k}{2}\}={\gamma }_A(a).$$

Hence $a\in C_{(t,s)}(A)$. Let $a,b\in S$ be such that $a\in C_{(t,s)}(A)$ and $b\in C_{(t,s)}(A)$. Then ${\mu }_A(a)\ge t$, ${\gamma }_A(a)\le s$ and ${\mu }_A(b)\ge t$, ${\gamma }_A(b)\le s$. Using (e2), we get

$$max\{{\mu }_A(ab),\frac{1-k}{2}\}\ge min\{{\mu }_A(a),{\mu }_A(b)\}\ge t>\frac{1-k}{2}$$

and

$$min\{{\gamma }_A(ab),\frac{1-k}{2}\}\le max\{{\gamma }_A(a),{\gamma }_A(b)\}\le s<\frac{1-k}{2},$$

which implies that ${\mu }_A(ab)=max\{{\mu }_A(ab),\frac{1-k}{2}\}\ge t$ and ${\gamma }_A(ab)=min\{{\gamma }_A(ab),\frac{1-k}{2}\}\le s$. Thus, $ab\in C_{(t,s)}(A)$. Now, suppose that $a,c\in C_{(t,s)}(A)$ and $b\in S$. Then ${\mu }_A(a)\ge t$, ${\gamma }_A(a)\le s$ and ${\mu }_A(c)\ge t$, ${\gamma }_A(c)\le s$. Using (e3), we get

$$max\{{\mu }_A(abc),\frac{1-k}{2}\}\ge min\{{\mu }_A(a),{\mu }_A(c)\}\ge t>\frac{1-k}{2}$$

and

$$min\{{\gamma }_A(abc),\frac{1-k}{2}\}\le max\{{\gamma }_A(a),{\gamma }_A(c)\}\le s<\frac{1-k}{2},$$

which implies that ${\mu }_A(abc)=max\{{\mu }_A(abc),\frac{1-k}{2}\}\ge t$ and ${\gamma }_A(abc)=min\{{\gamma }_A(abc),\frac{1-k}{2}\}\le s$. Thus, $abc\in C_{(t,s)}(A)$. Consequently, $C_{(t,s)}(A)$ is a bi-ideal of S. Therefore, we conclude that if an IFS $A=〈x,{\mu }_A,{\gamma }_A〉$ of S satisfies the three conditions (e1), (e2) and (e3), then the following assertion is valid

(e4) ($\mathrm{\forall }t\in (0.5,1]$) ($\mathrm{\forall }s\in [0,\frac{1-k}{2})$) ($C_{(t,s)}(A)\ne \mathrm{\varnothing }⟹C_{(t,s)}(A)$ is a bi-ideal of S).

Now, let $A=〈x,{\mu }_A,{\gamma }_A〉$ be an IFS of S satisfying (e4). Let $a,b\in S$ with $a\le b$ and $t\in (0,1]$ and $s\in [0,1)$ be such that $〈b;(t,s)〉\overline{\overline{\in }\vee \overline{{\mathrm{q}}_k}}A$. Then $〈b;(t,s)〉\in A$ and $〈b;(t,s)〉{\mathrm{q}}_{k}A$. Hence $b\in C_{(t,s)}(A)$ and $C_{(t,s)}(A)\ne \mathrm{\varnothing }$. Thus, by (e4), $a\in C_{(t,s)}(A)$ and so ${\mu }_A(a)\ge t$, ${\gamma }_A(a)\le s$, i.e., $〈a;(t,s)〉\in A$. This shows that (q4) is valid. Let $a,b\in S$, $t_1,t_2\in (0,1]$ and $s_1,s_2\in [0,1)$ be such that $〈a;(t_1,s_1)〉\overline{\overline{\in }\vee \overline{{\mathrm{q}}_k}}A$ and $〈b;(t_2,s_2)〉\overline{\overline{\in }\vee \overline{{\mathrm{q}}_k}}A$. Then $〈a;(t_1,s_1)〉\in A$, $〈a;(t_1,s_1)〉{\mathrm{q}}_{k}A$ and $〈b;(t_2,s_2)〉\in A$, $〈b;(t_2,s_2)〉{\mathrm{q}}_{k}A$, which implies that $a\in C_{(t,s)}(A)\subseteq C_{(min\{t_1,t_2\},max\{s_1,s_2\})}(A)$ and $b\in C_{(t,s)}(A)\subseteq C_{(min\{t_1,t_2\},max\{s_1,s_2\})}(A)$. Since $C_{(min\{t_1,t_2\},max\{s_1,s_2\})}(A)$ is bi-ideal of S by (e4), it follows by (b2) that $ab\in C_{(min\{t_1,t_2\},max\{s_1,s_2\})}(A)$, that is, ${\mu }_A(ab)\ge min\{t_1,t_2\}$, and ${\gamma }_A(ab)\le max\{s_1,s_2\}$, so that $〈ab;min\{t_1,t_2\},max\{s_1,s_2\}〉\in A$. Hence (q5) is valid. Finally, let $a,b,c\in S$, $t_1,t_2\in (0,1]$ and $s_1,s_2\in [0,1)$ be such that $〈a;(t_1,s_1)〉\overline{\overline{\in }\vee \overline{{\mathrm{q}}_k}}A$ and $〈c;(t_2,s_2)〉\overline{\overline{\in }\vee \overline{{\mathrm{q}}_k}}A$. Then $〈a;(t_1,s_1)〉\in A$, $〈a;(t_1,s_1)〉{\mathrm{q}}_{k}A$ and $〈c;(t_2,s_2)〉\in A$, $〈c;(t_2,s_2)〉{\mathrm{q}}_{k}A$, which implies that $a\in C_{(t,s)}(A)\subseteq C_{(min\{t_1,t_2\},max\{s_1,s_2\})}(A)$ and $c\in C_{(t,s)}(A)\subseteq C_{(min\{t_1,t_2\},max\{s_1,s_2\})}(A)$. Since $C_{(min\{t_1,t_2\},max\{s_1,s_2\})}(A)$ is bi-ideal of S by (e4), it follows by (b3) that $abc\in C_{(min\{t_1,t_2\},max\{s_1,s_2\})}(A)$, that is, ${\mu }_A(abc)\ge min\{t_1,t_2\}$, and ${\gamma }_A(abc)\le max\{s_1,s_2\}$, so that $〈abc;min\{t_1,t_2\},max\{s_1,s_2\}〉\in A$. Hence (q6) is valid.

Therefore, as a concluding remark, we have the following theorem.

Theorem 3.17 For an IFS $A=〈x,{\mu }_A,{\gamma }_A〉$ of S, the following are equivalent

• (1) $A=〈x,{\mu }_A,{\gamma }_A〉$ is an $(\overline{\in },\overline{\in }\vee \overline{{\mathrm{q}}_k})$-intuitionistic fuzzy bi-ideal of S.

• (2) $A=〈x,{\mu }_A,{\gamma }_A〉$ satisfies the condition (e4).

• (3) $A=〈x,{\mu }_A,{\gamma }_A〉$ satisfies the three conditions (e1), (e2) and (e3).

For an IFS $A=〈x;{\mu }_A,{\gamma }_A〉$ of S, we consider the following sets

$$\begin{array}{c}{\mathrm{\Gamma }}_1:=\{t\in (0,1]|U({\mu }_A;t)\ne \mathrm{\varnothing }⟹U({\mu }_A;t)\text{ is a bi-ideal of }S\},\hfill \\ {\mathrm{\Gamma }}_2:=\{s\in [0,1)|L({\mu }_A;t)\ne \mathrm{\varnothing }⟹L({\mu }_A;t)\text{ is a bi-ideal of }S\}.\hfill \end{array}$$

Then

• (1) If ${\mathrm{\Gamma }}_1=(0,1]$ and ${\mathrm{\Gamma }}_2=[0,1)$, then $A=〈x;{\mu }_A,{\gamma }_A〉$ is an intuitionistic fuzzy bi-ideal of S.

• (2) If ${\mathrm{\Gamma }}_1=(0,0.5]$ and ${\mathrm{\Gamma }}_2=[0.5,1)$, then $A=〈x;{\mu }_A,{\gamma }_A〉$ is an $(\in ,\in \vee \mathrm{q})$-intuitionistic fuzzy bi-ideal of S.

• (3) If ${\mathrm{\Gamma }}_1=(0,\frac{1-k}{2}]$ and ${\mathrm{\Gamma }}_2=[\frac{1-k}{2},1)$, then $A=〈x;{\mu }_A,{\gamma }_A〉$ is an $(\in ,\in \vee {\mathrm{q}}_k)$-intuitionistic fuzzy bi-ideal of S.

• (4) If ${\mathrm{\Gamma }}_1=(\frac{1-k}{2},1]$ and ${\mathrm{\Gamma }}_2=[0,\frac{1-k}{2})$, then $A=〈x;{\mu }_A,{\gamma }_A〉$ is an $(\overline{\in },\overline{\in }\vee \overline{{\mathrm{q}}_k})$-intuitionistic fuzzy bi-ideal of S.