Rough bipolar fuzzy ideals in semigroups

Abstract

Fuzzy set (FS) theory and rough sets (RSs) are constructed to accommodate the data uncertainty. In contrast, the bipolar FS (BFS) theory can tackle the uncertainty and the bipolarity of the data in different circumstances. This article aims to introduce the idea of rough bipolar fuzzy ideals in semigroup (SG), which is a generalization of the concept of rough BFSs (RBFSs) in an SG. We also investigate the roughness in the bipolar fuzzy subsemigroup (BF-SSG) with the help of congruence relation (cng-R) defined on the SG and studied some relevant structural properties. Moreover, the idea is extended to the rough bipolar fuzzy left ideal, rough bipolar fuzzy right ideal, rough bipolar fuzzy two-sided ideal, rough bipolar fuzzy interior ideal, and rough bipolar fuzzy bi-ideal in SGs. Further, it is seen that cng-Rs and complete cng-Rs play vital roles in the construction of rough approximations of bipolar fuzzy ideals. Consequently, their associated properties are explored by using cng-Rs and complete cng-Rs.

Introduction

To handle uncertainty and imprecision of the data, FS theory and RS theory are two very prominent notions. Although these theories are different, they can be combined in a very productive way.

The basic idea of FSs established by Zadeh [61] is an extension of classical sets. This idea not only brought a revolution in mathematics and logic but also in science and technology. FS theory depends on the fuzzy membership function (MF), through which we can evaluate the membership degree (MD) of an object w.r.t to a set. The larger the MD, the greater the belongingness of that object to the associated set. In line with the progress and to deal with new challenges, it has been proposed several types of extensions of FSs [6, 24, 46].

The primary and fundamental idea of RS theory [47] was initially developed by Pawlak as a valuable mathematical approach to tackle uncertainty in data analysis. In the recent, this theory has got broad attention in the research domains in both real-life applications and the theory itself [4, 22, 45]. Dubois and Prade [14] suggested the idea of rough FS (RFS) by combining FSs and RSs. Initially, Rosenfeld [49] pioneered the idea of fuzzy subgroups. The concept of fuzzy SGs was introduced and studied by Kuroki [32] in 1991 for the first time. The fuzzy bi-ideals, fuzzy radicals, and fuzzy prime ideals of ordered SGs are presented in [28, 56]. Hussain et al. [23] offered the conception of roughness for fuzzy ideals in ordered SGs. Mahmood et al. [43] offered the concept of roughness in fuzzy filters and fuzzy ideals with thresholds in ordered semigroups. Bashir et al. [10] projected the idea of rough fuzzy ideals induced by set-valued homomorphism in ternary SGs. Qurashi and Shabir [48] proposed the notion of roughness in quantales using generalized approximation space. RSs are hybridized with FSs in [8, 14, 15]. Biswas and Nanda [12] fostered the notions of rough groups and rough subgroups.

In many real-life scenarios apart from the MF, the non-MF is also required. The MF speaks about the positive aspects of the data, while non-MF reveals the negative aspects of the data. To address such conditions, the concept of bipolar FSs (BFSs) was put forward by Zhang [62] as an extension of FSs. Since its inception, several scholars have taken an eager interest in the theory of BFSs, and many applications have been reported in different domains. Han et al. [18] established a bipolar-valued rough fuzzy set model with applications to the decision information system. Luo and Hu [38] presented a bipolar three-way decision model and its application in analyzing incomplete data. Mahmood et al. [41] developed a multiple-criteria decision-making based on BFSs. For more about BFSs, we refer to References [3, 5, 44, 57].

A lot of research has been accomplished on fuzzy and bipolar fuzzy structures. Mahmood and Munir [42] initiated bipolar fuzzy subgroups. Yiarayong [60] developed some algebraic structures on SGs by applying BFS theory. Jun and Park [25], Jun et al. [26], and Lee [34] implemented BFSs to BCK/BCI-algebras. Kim et al. [30] discovered several properties of bases, neighborhoods, and continuities in bipolar fuzzy topological spaces. Gaketem et al. [16] give the concept of SGs in terms of cubic bipolar fuzzy ideals. Kang and Kang [27] applied BFS to sub-SGs with operators in SGs. In [64, 65], Zhou and Li discussed applications of BFS theory to hemirings and semiring. Hayat et al. [19] present the notions of bipolar anti-fuzzy h-ideals and bipolar anti-fuzzy interior h-ideals in hemirings. In 2019, Shabir et al. [53] worked on regular and intra-regular semirings in terms of bipolar fuzzy ideals. Recently, it has been discussed algebra and $\sigma$-algebra via soft set theory by Al-shami et al. [7] and Ameen et al. [9].

Numerous attempts have been conducted to hybridize RSs and BFS theory [18, 57, 58]. The notion of RBFSs handles the vagueness and uncertainty, as well as bipolarity in data which can not be avoided in many real-life problems. The concepts of roughness, fuzziness, and bipolarity are also correlated to the SGs in different manners by many authors to study the data having the structure of SGs. Gul and Shabir [17] put forward a novel concept of the roughness of a set based on $(\alpha , \beta )$-indiscernibility of a bipolar fuzzy relation. Yang et al. [57] fostered the idea of the transformation of bipolar fuzzy RS models. Malik and Shabir [40] developed a consensus model based on rough bipolar fuzzy approximations.

The theory of SGs is a substantial portion of algebra and this theory is frivolous without the study of ideals. Ideal theory in SGs is correlated to the FSs by many authors in different ways. Ahsan et al. [1] presented the fuzzy quasi-ideals in SGs. Ahsan et al. [2] established SGs characterized by their fuzzy bi-ideals. Fuzzy interior ideals in SGs were initiated by Hong et al. [21]. Khan and Shabir [29] pioneered the idea of ($\alpha ,\beta$)-fuzzy interior ideals in ordered SGs. In [50], Shabir and Ali studied soft ideals and generalized fuzzy ideals in SGs. Shabir and Khan [51] fostered fuzzy quasi-ideals of ordered SGs. Characterizations of regular SGs by ($\alpha ,\beta$)-fuzzy ideals are investigated by Shabir et al. [52]. Shabir et al. [54] offered the notions of SGs characterized by ($\epsilon ,\epsilon \vee qk$)-fuzzy ideals. It was applied fuzzy settings to address various practical issues as given in [55, 63].

In SGs, the idea of rough ideals was first initiated by Kuroki [33] in 1997. Bipolar fuzzy ideals in SGs were presented by Kim et al. [31]. Yiarayong [60] proposed a new approach to deal with the algebraic structure of SGs by applying BFS theory. In 2011, Hedayati [20] defined the interval-valued $(\alpha ,\beta )$-fuzzy bi-ideals of SGs, and investigated some fundamental properties. Bashir et al. [11] investigated the roughness of fuzzy ideals with three-dimensional congruence relation of ternary SGs.

Based on the research survey, we can observe that many scholars have studied the roughness of various algebraic structures. According to the best of our knowledge, in the current literature on BFSs and RS theory, there are no studies of SGs and ideals. Therefore, in this study, we attempt to fill the research gap by developing the idea of rough bipolar fuzzy ideals in SGs. In this respect, we have discussed the BFSs and bipolar fuzzy ideals (BF ideals) in an SG. Moreover, we studied the roughness in the BF-SSGs with the help of a cng-R defined on the SG and investigated some properties of the rough BF-SSG. Also, the rough BF ideal (RBF ideal), rough bipolar fuzzy interior ideal (RBFI ideal), and rough bipolar fuzzy bi-ideal (RBF bi-ideal) in the SGs are defined and discussed in this study.

The outliving part of the article is presented as follows: In section “inlinkPreliminariessec2”, some prior literature which is necessary for understanding our research work. Section “Bipolar fuzzy sets in semigroups” is dedicated to study BFSs in SGs along with their important structural properties. Section “Rough bipolar fuzzy sets in semigroups” offers a novel idea of rough BFS in SGs. In Section “Rough bipolar fuzzy ideals in semigroups”, we established the concept of RBF ideals in SGs. In Section “Comparative analysis and discussion”, a comparative study of the proposed technique is given with some already existing techniques in the literature. Finally, in Section “Conclusions”, we summarize our findings and give a few recommendations for future perspectives.

Preliminaries

In this section, we will give and deliberate several rudimentary ideas associated with RSs, FSs, BFSs, ideals, rough ideals in SGs, and BF ideals in SGs.

RS theory

One can always relate certain information to each element of the universe. The RS theory [47] uses the lower and upper approximations of a cluster of elements to describe how close the elements are to the data associated with them. Pawlak constructed these approximations from an equivalence relation (ER) $\sigma$ on a universe U of objects. The equivalence class in $U/\sigma$ consisting of the element x of U is symbolized by $[x]_{\sigma }$. These classes work as the fundamental building blocks of the information.

Definition 1

[47] Let U be a finite and non-empty universe, and $\sigma \subseteq U\times U$ be an ER. With the help of this ER $\sigma$, a subset S of U can be characterized through a pair of lower and upper rough approximations

$$\begin{aligned} {\underline{\sigma }}(S)&= \big \{x\in U : [x]_{\sigma }\subseteq S \big \}, \end{aligned}$$

(1.1)

$$\begin{aligned} {\overline{\sigma }}(S)&= \big \{x\in U : [x]_{\sigma }\cap S \ne \emptyset \big \}. \end{aligned}$$

(1.2)

The pair $\big ( {\underline{\sigma }}(S),{\overline{\sigma }}(S) \big )$ is called an RS of S in U.

FSs and BFSs

The theory of FSs measures the uncertainty of phenomena related to the objects through a mapping, known as the MF.

Definition 2

[61] A FS $\xi$ in U is characterized by a MF $\xi : U \longrightarrow \left[ 0, 1\right]$. Thus, an FS $\xi$ assigns to each element $x\in U$, a membership value $\xi (x)$ specifying the degree to which x belongs to $\xi$.

The idea of BFS was introduced by Zhang [62] as a generalization of the FSs to resolve a specific class of DM problems. In the BFSs, the MDs are described by a pair of MFs: a positive MF (PMF) and a negative MF (NMF).

Definition 3

[62] A BFS $\mu$ in U is a structure of the form

$$\begin{aligned} \mu = \big \{\big (x,\mu ^{P}(x), \mu ^{N}(x) \big ): x \in U \big \}, \end{aligned}$$

(1.3)

where $\mu ^{P}: U \longrightarrow \left[ 0, 1\right]$ and $\mu ^{N}: U \longrightarrow \left[ -1, 0\right]$ are the PMF and the NMF, respectively.

If $\mu$ is defined by a property A and $x\in U$, then the PMF $\mu ^{P}(x)$ indicates the degree of fulfillment of x to the property A, while the NMF $\mu ^{N}(x)$ demonstrates the fulfillment degree of x to some implicit counter property of A. If $\mu ^{P}(x) = 0 = \mu ^{N}(x)$, then the object x is irrelevant to the property A.

The collection of all BFSs in U is represented by BF(U). We write $\mu (x) = \big (\mu ^{P}(x), \mu ^{N}(x)\big )$ for $\big (x, \mu ^{P}(x), \mu ^{N}(x)\big )$.

Lee [35] defined some basic operations on the BFSs.

Definition 4

[35] Let $\mu , \nu \in BF(U)$. Then, $\mu$ is contained in $\nu$, that is, $\mu \subseteq \nu$, if $\mu ^{P}(x)\le \nu ^{P}(x)$ and $\mu ^{N}(x)\ge \nu ^{N}(x)$ for all $x\in U$. Clearly, $\mu = \nu$ if and only if $\mu \subseteq \nu$ and $\nu \subseteq \mu$.

Definition 5

[35] The whole BFS in U is represented by $I_{U} = \big (I^{P}, I^{N} \big )$, where $I^{P}(x) = 1$ and $I^{N}(x) = 0$ for all $x \in U$. The null BFS in U is expressed as $O_{U}= \big (O^{P}, O^{N}\big )$, where $O^{P}(x) = 0$ and $O^{N}(x) = -1$ for all $x \in U$. Thus, $I_{U}(x) = (1,0)$ and $O_{U}(x)= (0,-1)$ for all $x\in U$.

Definition 6

[35] Let $\mu , \nu \in BF(U)$. Then

$$\begin{aligned} \mu \cup \nu = \big \{\big (x,\mu ^{P}(x)\vee \nu ^{P}(x), \mu ^{N}(x)\wedge \nu ^{N}(x)\big ): x\in U \big \}, \nonumber \\ \end{aligned}$$

(1.4)

$$\begin{aligned} \mu \cap \nu = \big \{\big (x,\mu ^{P}(x)\wedge \nu ^{P}(x), \mu ^{N}(x)\vee \nu ^{N}(x)\big ): x\in U \big \}.\nonumber \\ \end{aligned}$$

(1.5)

Definition 7

[35] The complement $\mu ^c$ of $\mu \in BF(U)$ is characterized as

$$\begin{aligned} \mu ^c = \big \{ \big (x, 1-\mu ^{P}(x), -1-\mu ^{N}(x) \big ): x\in U \big \}. \end{aligned}$$

(1.6)

Ideals and rough ideals in SGs

Here, we review some definitions of ideals and rough ideals in SGs and their characterizations.

Recall that, an SG $\varUpsilon$ is a non-empty set on which an associative binary operation is defined. Take a non-empty subset Q of a SG $\varUpsilon$. Then

• Q is called a subsemigroup (SSG) of $\varUpsilon$, if $ab\in Q$ for each $a, b\in Q$ (that is, $QQ \subseteq Q$).

• Q is called a left ideal of $\varUpsilon$, if $xa\in Q$ (that is, $\varUpsilon Q \subseteq Q$) and Q is the right ideal of $\varUpsilon$, if $ax \in Q$ (that is, $Q\varUpsilon \subseteq Q$) for each $a\in Q$ and $x\in \varUpsilon$.

• Q is an ideal of $\varUpsilon ,$ if it is both, the left and right ideal of $\varUpsilon$.

• Q is called an interior ideal of $\varUpsilon$, if $xay \in Q$ for each $a\in Q$ and $x,y\in \varUpsilon$ (that is, $\varUpsilon Q\varUpsilon \subseteq Q$).

• Q is called a bi-ideal of $\varUpsilon$, if Q is an SSG of $\varUpsilon$ and $axb\in Q$ for each $a,b\in Q$, and $x\in \varUpsilon$ (that is, $Q\varUpsilon Q\subseteq Q$).

Definition 8

A cng-R $\Re$ on an SG $\varUpsilon$ is an ER $\Re$ on $\varUpsilon$ which is right and left compatible $\big (\hbox {that is}, (x,y)\in \Re$ implies that $(ax, ay), (xa, ya)\in \Re$ for each $a, x, y \in \varUpsilon \big )$. Let $[x]_{\Re }$ denote the $\Re$-cng-class of $x\in \varUpsilon$. For a cng-R $\Re$ on $\varUpsilon$, we generally have $[x]_{\Re }[y]_{\Re }\subseteq [xy]_{\Re }$ for each $x, y\in \varUpsilon$. A cng-R $\Re$ on $\varUpsilon$ is complete, if $[x]_{\Re }[y]_{\Re }=[xy]_{\Re }$ for each $x, y \in \varUpsilon$.

This can be observed in the subsequent example.

Table 1 Multiplication table for SG

Example 1

Let $\varUpsilon =\{s,t,u,v\}$ represent an SG whose table of binary operation is given below.

We take two cng-Rs $\Re _{1}$ and $\Re _{2}$ on $\varUpsilon$, defined below

$$\begin{aligned} \Re _{1}= & {} \big \{(s,s),(t,t),(u,u),(v,v),(u,v),(v,u) \big \}, \\ \Re _{2}= & {} \big \{(s,s),(t,t),(u,u),(v,v),(u,t),(t,u)\big \}. \end{aligned}$$

Then, $\Re _{1}$ defines the cng-classes $\{s\},\{t\}$ and $\{u,v\},$ while $\Re _{2}$ defines the cng-classes $\{s\},\{t,u\}$ and $\{v\}.$ It can be easily checked that, $[x]_{\Re _{1}}[y]_{\Re _{1}}=[xy]_{\Re _{1}}$ for each $x,y\in \varUpsilon$. That is, $\Re _{1}$ is a complete cng-R on $\varUpsilon$. While, $[v]_{\Re _{2}}[v]_{\Re _{2}}\subsetneq [vv]_{\Re _{2}}$ for $v\in \varUpsilon$, because $[v]_{\Re _{2}} = \{v\}$, so $[v]_{\Re _{2}}[v]_{\Re _{2}} = \{vv\} = \{u\}$ and $[vv]_{\Re _{2}}=[u]_{\Re _{2}} = \{t,u\}$. This means that $\Re _{2}$ is not a complete cng-R.

Rough subsets in SGs were defined by Kuroki [33], which are given below. He also gave some characterizations of the lower and upper approximations of these subsets under the cng-R defined on the SG.

Definition 9

[33] Let $\varUpsilon$ be an SG and $\Re$ be a cng-R on $\varUpsilon$. The lower and upper approximations of a subset Q of $\varUpsilon$ are the subsets ${\underline{Q}}$ and ${\overline{Q}}$ of $\varUpsilon$, respectively, which are defined as follows:

$$\begin{aligned} {\underline{Q}}&= \big \{ x \in \varUpsilon : [x]_{\Re } \subseteq Q \big \}, \end{aligned}$$

(1.6)

$$\begin{aligned} {\overline{Q}}&= \big \{ x \in \varUpsilon : [x]_{\Re }\cap Q\ne \phi \big \}. \end{aligned}$$

(1.7)

A subset Q of $\varUpsilon$ is said to be a rough subset of the SG $\varUpsilon$, if ${\underline{Q}} \ne {\overline{Q}}$; otherwise, Q is a definable subset of $\varUpsilon$. The subset Q of an SG $\varUpsilon$ is called a lower (an upper) rough SSG of $\varUpsilon$, if ${\underline{Q}}$ $({\overline{Q}})$ is an SSG of $\varUpsilon$. The subset Q of $\varUpsilon$ is a lower rough left (right, two-sided, interior, bi-) ideal of $\varUpsilon$ if ${\underline{Q}}$ is a left (right, two-sided, interior, bi-) ideal of $\varUpsilon$ and Q is an upper rough left (right, two-sided, interior, bi-) ideal of $\varUpsilon$ if ${\overline{Q}}$ is a left (right, two-sided, interior, bi-) ideal of $\varUpsilon$. The subset Q of $\varUpsilon$ is called a rough left (right, two-sided, interior, bi-) ideal of $\varUpsilon$, if it is both, lower and upper left (right, two-sided, interior, bi-) ideal of $\varUpsilon$.

Theorem 1

[33] Let $\Re$ be a cng-R on an SG $\varUpsilon$. Then, we have the subsequent assentations.

• 1. Each SSG of $\varUpsilon$ is an upper rough SSG of $\varUpsilon$.

• 2. Each left (right, bi-) ideal of $\varUpsilon$ is an upper rough left (right, bi-) ideal of $\varUpsilon$.

Theorem 2

[33] Let $\Re$ be a complete cng-R on a SG $\varUpsilon$. Then, we have the subsequent assentations.

• 1. Each SSG Q of $\varUpsilon$ is a lower rough SSG of $\varUpsilon$, if ${\underline{Q}}\ne \phi$.

• 2. Each left (right, bi-) ideal Q of $\varUpsilon$ is a lower rough left (right, bi-) ideal of $\varUpsilon$, if ${\underline{Q}}\ne \phi$.

BF ideals in SGs

The BF ideals in SGs were defined by Kim et al. [31]. Yaqoob [59] studied BF ideals in LA-SGs. This subsection reviews some definitions of the BF ideals in an SG.

Definition 10

[59] Let $\varUpsilon$ be an SG and let $\lambda , \mu \in BFS(\varUpsilon )$. The composition $\lambda \circ \mu$ of $\lambda$ and $\mu$ in $\varUpsilon$ is defined as

$$\begin{aligned} \lambda \circ \mu = \big (\lambda ^{P}\circ \mu ^{P}, \lambda ^{N}\circ \mu ^{N} \big ), \end{aligned}$$

where

$$\begin{aligned}{} & {} (\lambda ^{P}\circ \mu ^{P})(s)\\{} & {} =\left\{ \begin{array}{ll} \underset{s=mn}{\bigvee }\ \big (\lambda ^{P}(m)\wedge \mu ^{P}(n) \big ) &{} \text {if }s=mn \text { for some }m,n\in \varUpsilon \\ 0 &{} \text {otherwise} \end{array} \right. \end{aligned}$$

and

$$\begin{aligned}{} & {} (\lambda ^{N}\circ \mu ^{N})(s)\\{} & {} =\left\{ \begin{array}{ll} \underset{s=mn}{\bigwedge }\ \big (\lambda ^{N}(m)\vee \mu ^{N}(n)\big ) &{} \text {if }s=mn \text { for some }m,n\in \varUpsilon \\ 0 &{} \text {otherwise} \end{array} \right. \end{aligned}$$

for each $s\in \varUpsilon$.

Definition 11

[31] A BFS $\lambda$ in a SG $\varUpsilon$ is called a BF-SSG of $\varUpsilon$ if for each $m, n\in \varUpsilon$

$$\begin{aligned} \lambda ^{P}(mn)\ge \lambda ^{P}(m)\wedge \lambda ^{P}(n)\ \text {and}\ \lambda ^{N}(mn)\le \lambda ^{N}(m)\vee \lambda ^{N}(n). \end{aligned}$$

Definition 12

[31] A BFS $\lambda$ in an SG $\varUpsilon$ is called a BF left (or right) ideal of $\varUpsilon$ if $\lambda ^{P}(mn) \ge \lambda ^{P}(n)$ and $\lambda ^{N}(mn)\le \lambda ^{N}(n)$ (or $\lambda ^{P}(mn)\ge \lambda ^{P}(m)$ and $\lambda ^{N}(mn)\le \lambda ^{N}(n)$) for each $m,n\in \varUpsilon$.

A BFS $\lambda$ in a $\varUpsilon$ is called a BF ideal of $\varUpsilon$ if it is both, a BF left and a BF right ideal of $\varUpsilon$, that is, $\lambda ^{P}(mn)\ge \lambda ^{P}(m)\vee \lambda ^{P}(n)$ and $\lambda ^{N}(mn)\le \lambda ^{N}(m)\wedge \lambda ^{N}(n)$ for each $m, n \in \varUpsilon$.

Definition 13

[59] A BFS $\lambda$ in an SG $\varUpsilon$ is called a BF interior ideal of $\varUpsilon$ if for each $s, t, u \in \varUpsilon$

$$\begin{aligned} \lambda ^{P}(stu)\ge \lambda ^{P}(t)\text { and }\lambda ^{N}(stu) \le \lambda ^{N}(t). \end{aligned}$$

Definition 14

[31] A BF-SSG $\lambda$ of an SG $\varUpsilon$ is called a BF bi-ideal of $\varUpsilon$ if for each $s, t, u \in \varUpsilon$

$$\begin{aligned} \lambda ^{P}(stu)\ge \lambda ^{P}(s)\wedge \lambda ^{P}(u)\text { and } \lambda ^{N}(stu)\le \lambda ^{N}(s)\vee \lambda ^{N}(u). \end{aligned}$$

Example 2

Recall the SG $\varUpsilon =\{s,t,u,v\}$ as established in Example 1. Take some BFSs in $\varUpsilon$, defined below

$$\begin{aligned} \lambda _{1}= & {} \big \{(s,0.3,-0.4),(t,0.4,-0.3),\\{} & {} (u,0.6,-0.1),(v,0.6,-0.1) \big \}, \\ \lambda _{2}= & {} \big \{(s,0.2,-0.2),(t,0.4,-0.4),\\{} & {} (u,0.5,-0.5),(v,0.5,-0.5) \big \}, \\ \lambda _{3}= & {} \big \{(s,0.3,-0.1),(t,0.4,-0.2),\\{} & {} (u,0.7,-0.2),(v,0.7,-0.2) \big \}, \\ \lambda _{4}= & {} \big \{(s,0.1,-0.1),(t,0.3,-0.3),\\{} & {} (u,0.4,-0.4),(v,0.4,-0.4) \big \}. \end{aligned}$$

Simple calculations confirm that $\lambda _{1}$ is a BF-SSG, $\lambda _{2}$ is a BF left ideal, $\lambda _{3}$ is a BF interior ideal, and $\lambda _{4}$ is a BF bi-ideal of $\varUpsilon$.

Bipolar fuzzy sets in semigroups

In this section, we link the BF-SSGs to the SSGs of an SG and the BF left (right, two-sided, interior, bi-) ideals in SGs to the left (right, two-sided, interior, bi-) ideals of SGs, using the $\alpha$-level P-cuts (positive cut) and $\alpha$-level N-cuts (negative cut) of the BFS.

Definition 15

Let $\lambda \in BFS(\varUpsilon )$. Then, the $\alpha -$ level P-cut and the $\alpha -$ level N-cut of $\lambda$ in U are, respectively, defined as

$$\begin{aligned} \lambda _{\alpha }= & {} \big \{u\in U:\lambda ^{P}(u)\ge \alpha \big \}, \end{aligned}$$

(3.1)

$$\begin{aligned} \lambda ^{\alpha }= & {} \{u\in U:\lambda ^{N}(u)\le - \alpha \}, \end{aligned}$$

(3.2)

respectively, for each $0 \lvertneqq \alpha \le 1$. Then, we have the following results.

Theorem 3

Let $\varUpsilon$ be an SG and $\lambda \in BFS(\varUpsilon )$. Then, $\lambda$ is a BF-SSG of $\varUpsilon$ if and only if $\lambda _{\alpha }$ and $\lambda ^{\alpha }$, if non-empty, are SSGs of $\varUpsilon$, for each $\alpha \in [0,1]$.

Proof

Let $\lambda$ be a BF-SSG of $\varUpsilon$, and let $a, b\in \lambda _{\alpha }$. Then, $\lambda ^{p}(a)\geqslant \alpha$ and $\lambda ^{p}(b)\geqslant \alpha$. Since $\lambda$ is a BF-SSGs of $\varUpsilon$, so

$$\begin{aligned} \lambda ^{P}(ab)\geqslant \lambda ^{P}(a)\wedge \lambda ^{P}(b) \geqslant \alpha , \end{aligned}$$

which implies that $ab\in \lambda _{\alpha }$. Therefore, $\lambda _{\alpha }$ is an SSG of $\varUpsilon$ for each $\alpha$. Similarly, $\lambda ^{\alpha }$ is an SSG of $\varUpsilon$ for each $\alpha$.

Conversely, let $\lambda _{\alpha }$ and $\lambda ^{\alpha }$ be non-empty SSGs of $\varUpsilon$ for each $\alpha \in [0, 1],$ and let $a, b\in \varUpsilon$. Denote $\lambda ^{P}(a)\wedge \lambda ^{P}(b)$ by $\alpha _{\circ }\in [0, 1]$. Then surely, $\lambda ^{P}(a),\lambda ^{P}(b)\geqslant \alpha _{\circ }$, and so, $a, b\in \lambda _{\alpha _{\circ }}$. However, $\lambda _{\alpha _{\circ }}$ is an SSG of $\varUpsilon$, so $ab\in \lambda _{\alpha _{\circ }}$; which yields $\lambda ^{P}(ab)\geqslant \alpha _{\circ }$. That is

$$\begin{aligned} \lambda ^{P}(ab)\geqslant \lambda ^{P}(a)\wedge \lambda ^{P}(b). \end{aligned}$$

(3.3)

Now, denote $\lambda ^{N}(a)\vee \lambda ^{N}(b)$ by $-\alpha _{1},$ where $\alpha _{1}\in [0,1]$. Then, $\lambda ^{N}(a),\lambda ^{N}(b)\leqslant -\alpha _{1}$, and so, $a,b\in \lambda ^{\alpha _{1}}$. However, $\lambda ^{\alpha _{1}}$ is an SSG of $\varUpsilon$, so $ab\in \lambda ^{\alpha _{1}}$; which yields $\lambda ^{N}(ab)\leqslant -\alpha _{1}.$ That is

$$\begin{aligned} \lambda ^{N}(ab)\leqslant \lambda ^{N}(a)\vee \lambda ^{N}(b). \end{aligned}$$

(3.4)

Assertions (3.3) and (3.4) prove that $\lambda$ is a BF-SSG of $\varUpsilon$. $\square$

Theorem 4

Let $\varUpsilon$ be an SG and $\lambda \in BFS(\varUpsilon )$. Then, $\lambda$ is a BF left (right, two-sided) ideal of $\varUpsilon$ if and only if $\lambda _{\alpha }$ and $\lambda ^{\alpha },$ if non-empty, are left (right, two-sided) ideals of $\varUpsilon$, for each $\alpha \in [0,1]$.

Proof

Let $\lambda$ be a BF left ideal of $\varUpsilon$ and let $x\in \varUpsilon$. Then, for any $\alpha \in [0,1]$ and for each $a\in \lambda _{\alpha }$, we have

$$\begin{aligned} \lambda ^{P}(xa)\geqslant \lambda ^{P}(a)\geqslant \alpha ; \end{aligned}$$

which implies that $xa\in \lambda _{\alpha }$ for each $x\in \varUpsilon$ and $a\in \lambda _{\alpha }$. That is, $\lambda _{\alpha }$ is a left ideal of $\varUpsilon$ for each $\alpha \in [0,1]$. Now, for each $a\in \lambda ^{\alpha },$ we have

$$\begin{aligned} \lambda ^{N}(xa)\leqslant \lambda ^{N}(a)\leqslant -\alpha ; \end{aligned}$$

which implies that $xa\in \lambda ^{\alpha }$ for each $x\in \varUpsilon$ and $a\in \lambda ^{\alpha }$. That is, $\lambda ^{\alpha }$ is also a left ideal of $\varUpsilon$ for each $\alpha \in [0,1]$.

Conversely, let $\lambda _{\alpha }$ and $\lambda ^{\alpha }$ be non-empty left ideals of $\varUpsilon$ for each $\alpha \in [0, 1]$ and let $a, b\in \varUpsilon$. Denote $\lambda ^{P}(b)$ by $\alpha _{\circ }\in [0, 1]$. Then surely, $b\in \lambda _{\alpha _{\circ }}.$ Thus, $ab\in \lambda _{\alpha _{\circ }},$ as $\lambda _{\alpha _{\circ }}$ is a left ideal of $\varUpsilon$; which gives $\lambda ^{P}(ab)\geqslant \alpha _{\circ }$. That is

$$\begin{aligned} \lambda ^{P}(ab)\geqslant \lambda ^{P}(b). \end{aligned}$$

(3.5)

Now, denote $\lambda ^{N}(b)$ by $-\alpha _{1},$ where $\alpha _{1}\in [0,1]$. Then, $b\in \lambda ^{\alpha _{1}}$ and $\lambda ^{\alpha _{1}}$ is a left ideal of $\varUpsilon$. Thus, $ab\in \lambda ^{\alpha _{1}}$; which gives $\lambda ^{N}(ab)\leqslant -\alpha _{1}$. That is

$$\begin{aligned} \lambda ^{N}(ab)\leqslant \lambda ^{N}(b). \end{aligned}$$

(3.6)

The expressions (3.5) and (3.6) prove that $\lambda$ is a BF left ideal of $\varUpsilon$.

Similar is the proof when $\lambda$ is a BF right (or two-sided) ideal of $\varUpsilon$. $\square$

Theorem 5

Let $\varUpsilon$ be an SG and $\lambda \in BFS(\varUpsilon )$. Then, $\lambda$ is a BF interior ideal of $\varUpsilon$ if and only if $\lambda _{\alpha }$ and $\lambda ^{\alpha },$ if non-empty, are interior ideals of $\varUpsilon$, for each $\alpha \in [0,1]$.

Proof

Let $\lambda$ be a BF interior ideal of $\varUpsilon$ and let $x,y\in \varUpsilon$. Then, for any $\alpha \in [0,1]$ and for each $a\in \lambda _{\alpha }$, we have

$$\begin{aligned} \lambda ^{P}(xay)\geqslant \lambda ^{P}(a)\geqslant \alpha ; \end{aligned}$$

which implies that $xay\in \lambda _{\alpha }$ for each $x,y\in \varUpsilon$ and $a\in \lambda _{\alpha }$. That is, $\lambda _{\alpha }$ is an interior ideal of $\varUpsilon$ for each $\alpha \in [0,1]$. Now, for each $a\in \lambda ^{\alpha },$ we have

$$\begin{aligned} \lambda ^{N}(xay)\leqslant \lambda ^{N}(a)\leqslant -\alpha ; \end{aligned}$$

which implies that $xay\in \lambda ^{\alpha }$ for each $x,y\in \varUpsilon$ and $a\in \lambda ^{\alpha }$. That is, $\lambda ^{\alpha }$ is also an interior ideal of $\varUpsilon$ for each $\alpha \in [0,1]$.

Conversely, let $\lambda _{\alpha }$ and $\lambda ^{\alpha }$ be non-empty interior ideals of $\varUpsilon$ for each $\alpha \in [0, 1]$ and let $a, b, c\in \varUpsilon$. Denote $\lambda ^{P}(b)$ by $\alpha _{\circ }\in [0, 1]$. Then surely, $b \in \lambda _{\alpha _{\circ }}.$ Thus, $abc\in \lambda _{\alpha _{\circ }}$, as $\lambda _{\alpha _{\circ }}$ is interior ideal of $\varUpsilon$. Which gives $\lambda ^{P}(abc)\geqslant \alpha _{\circ }$. That is

$$\begin{aligned} \lambda ^{P}(abc)\geqslant \lambda ^{P}(b). \end{aligned}$$

(3.7)

Now, denote $\lambda ^{N}(b)$ by $-\alpha _{1},$ where $\alpha _{1}\in [0,1]$. Surely, $b\in \lambda ^{\alpha _{1}}$ and $\lambda ^{\alpha _{1}}$ is an interior ideal of $\varUpsilon$. Thus, $abc\in \lambda ^{\alpha _{1}}$ for each $a,c\in \varUpsilon$; which gives $\lambda ^{N}(abc)\leqslant -\alpha _{1}$. That is

$$\begin{aligned} \lambda ^{N}(abc)\leqslant \lambda ^{N}(b). \end{aligned}$$

(3.8)

The expressions (3.7) and (3.8) prove that $\lambda$ is a BF interior ideal of $\varUpsilon$. $\square$

Theorem 6

Let $\varUpsilon$ be an SG and $\lambda \in BFS(\varUpsilon )$. Then, $\lambda$ is a BF bi-ideal of $\varUpsilon$ if and only if $\lambda _{\alpha }$ and $\lambda ^{\alpha }$, if they are non-empty, are bi-ideals of $\varUpsilon$, for each $\alpha \in [0,1]$.

Proof

Let $\lambda$ be a BF bi-ideal of $\varUpsilon$. Then, $\lambda$ is also a BF-SSG of $\varUpsilon$. Hence, $\lambda _{\alpha }$ and $\lambda ^{\alpha }$, if non-empty, are SSGs of $\varUpsilon$, for each $\alpha \in [0, 1]$, by Theorem 3. Now, let $a, c\in \lambda _{\alpha }$. Then, $\lambda ^{P}(a)\geqslant \alpha$ and $\lambda ^{P}(c)\geqslant \alpha$. Since $\lambda$ is a BF bi-ideal of $\varUpsilon$, so for each $b \in \varUpsilon$, we have

$$\begin{aligned} \lambda ^{P}(abc)\geqslant \lambda ^{P}(a)\wedge \lambda ^{P}(c)\geqslant \alpha ; \end{aligned}$$

which implies $abc\in \lambda _{\alpha }$ for each $a,c\in \lambda _{\alpha }$ and $b\in \varUpsilon$. Therefore, $\lambda _{\alpha }$ is a bi-ideal of $\varUpsilon$ for each $\alpha$. Similarly, $\lambda ^{\alpha }$ is a bi-ideal of $\varUpsilon$ for each $\alpha$.

Conversely, let $\lambda _{\alpha }$ and $\lambda ^{\alpha }$ be non-empty bi-ideals of $\varUpsilon$ for each $\alpha \in [0, 1]$, and let $a, b, c \in \varUpsilon$. Denote $\lambda ^{P}(a)\wedge \lambda ^{P}(c)$ by $\alpha _{\circ }\in [0,1]$. Then surely, $\lambda ^{P}(a),\lambda ^{P}(c)\geqslant \alpha _{\circ }$, and so $a, c \in \lambda _{\alpha _{\circ }}$. However, $\lambda _{\alpha _{\circ }}$ is a bi-ideal of $\varUpsilon$. Therefore, $abc\in \lambda _{\alpha _{\circ }}$ for each $b\in \varUpsilon$; which yields $\lambda ^{P}(abc)\geqslant \alpha _{\circ }$. Or

$$\begin{aligned} \lambda ^{P}(abc)\geqslant \lambda ^{P}(a)\wedge \lambda ^{P}(c). \end{aligned}$$

(3.9)

Now, denote $\lambda ^{N}(a)\vee \lambda ^{N}(c)$ by $-\alpha _{1},$ where $\alpha _{1}\in [0,1]$. Then, $\lambda ^{N}(a),\lambda ^{N}(c)\leqslant -\alpha _{1}$, and so, $a,c\in \lambda ^{\alpha _{1}}$. However, $\lambda ^{\alpha _{1}}$ is a bi-ideal of $\varUpsilon$. Therefore, $abc\in \lambda ^{\alpha _{1}}$ for each $b\in \varUpsilon$; which yields $\lambda ^{N}(abc)\leqslant -\alpha _{1}$. Or

$$\begin{aligned} \lambda ^{N}(abc)\leqslant \lambda ^{N}(a)\vee \lambda ^{N}(c). \end{aligned}$$

(3.10)

Assertions (3.9) and (3.10) prove that $\lambda$ is a BF bi-ideal of $\varUpsilon$. $\square$

Rough bipolar fuzzy sets in semigroups

The RBFSs in SGs are defined with the help of lower and upper RBF approximations of the BFSs in the SG $\varUpsilon$, on which a cng-R $\Re$ is defined. These approximations are defined and discussed in this section. We also define the RBF SSG of $\varUpsilon$.

Definition 16

Let $\Re$ be a cng-R on a SG $\varUpsilon$ and let $\lambda \in BFS(\varUpsilon )$. The lower and upper RBF approximations of $\lambda$ w.r.t $\Re$ are the BFSs ${\underline{\Re }}(\lambda )$ and ${\overline{\Re }}(\lambda )$ in $\varUpsilon$, respectively, defined for each $s\in \varUpsilon$ as

$$\begin{aligned} {\underline{\Re }}(\lambda )= & {} \big \{(s, {\underline{\Re }}\lambda ^{P}(s), {\underline{\Re }}\lambda ^{N}(s)): s \in \varUpsilon \big \}, \end{aligned}$$

(4.1)

$$\begin{aligned} {\overline{\Re }}(\lambda )= & {} \big \{(s, {\overline{\Re }}\lambda ^{P}(s), {\overline{\Re }}\lambda ^{N}(s)): s \in \varUpsilon \big \}, \end{aligned}$$

(4.2)

where

$$\begin{aligned} {\underline{\Re }}\lambda ^{P}(s)= & {} \underset{t\in [s]_{\Re }}{\wedge }\lambda ^{P}(t), \,\,\,\,{\underline{\Re }}\lambda ^{N}(s) = \underset{t\in [s]_{\Re }}{\vee }\lambda ^{N}(t), \\ {\overline{\Re }}\lambda ^{P}(s)= & {} \underset{t\in [s]_{\Re }}{\vee } \lambda ^{P}(t), \,\,\,\, {\overline{\Re }}\lambda ^{N}(s)=\underset{ t\in [s]_{\Re }}{\wedge }\lambda ^{N}(t). \end{aligned}$$

If ${\underline{\Re }}(\lambda )={\overline{\Re }}(\lambda )$, then $\lambda$ is said to be $\Re$-definable; otherwise, $\lambda$ is an RBFS in $\varUpsilon .$

In [40], some characterizations of RBFSs in a non-empty set U having an ER $\Re$ were presented. These characterizations are also valid when the set U is replaced by the SG $\varUpsilon$ and the ER $\Re$ on U is replaced by the cng-R $\Re$ on $\varUpsilon$. Therefore, the results in [40] also hold for the lower and upper RBF approximations of the BFSs in the SG $\varUpsilon$, given in Definition 16. Some other results are as follows.

Theorem 7

Let $\Re$ be a cng-R on a SG $\varUpsilon$. Then, $\overline{\Re }(\lambda )\circ {\overline{\Re }}(\nu )\subseteq \overline{\Re }(\lambda \circ \nu )$ holds for each $\lambda ,\nu \in BFS(\varUpsilon )$.

Proof

Since $\Re$ is a cng-R on $\varUpsilon$, so $[x]_{\Re }[y]_{\Re }\subseteq [xy]_{\Re }$ for each $x, y\in \varUpsilon$. Let $\lambda , \nu \in BFS(\varUpsilon )$. We have

$$\begin{aligned} {\overline{\Re }}(\lambda )\circ {\overline{\Re }}(\nu )= & {} ({\overline{\Re }} \lambda ^{P}\circ {\overline{\Re }}\nu ^{P},{\overline{\Re }}\lambda ^{N}\circ {\overline{\Re }}\nu ^{N}), \\ {\overline{\Re }}(\lambda \circ \nu )= & {} ({\overline{\Re }}(\lambda ^{P}\circ \nu ^{P}),{\overline{\Re }}(\lambda ^{N}\circ \nu ^{N})). \end{aligned}$$

Take any $s\in \varUpsilon$. If there exist elements $x,y\in \varUpsilon$, such that $s=xy,$ then we have

$$\begin{aligned}{} & {} ({\overline{\Re }}\lambda ^{P}\circ {\overline{\Re }}\nu ^{P})(s) =\underset{ s=xy}{\vee }({\overline{\Re }}\lambda ^{P}(x)\wedge {\overline{\Re }}\nu ^{P}(y)) \\{} & {} \quad =\underset{s=xy}{\vee }\left( (\underset{a\in [x]_{\Re }}{\vee }\lambda ^{P}(a)\right) \wedge \left( \underset{b\in [y]_{\Re }}{\vee }\nu ^{P}(b))\right) \\{} & {} \quad =\underset{s=xy}{\vee }\left( \underset{a\in [x]_{\Re },\text { }b\in [y]_{\Re }}{\vee }(\lambda ^{P}(a)\wedge \nu ^{P}(b))\right) \\{} & {} \quad \le \underset{s=xy}{\vee }\left( \underset{ab\in [xy]_{\Re }}{\vee } (\lambda ^{P}(a)\wedge \nu ^{P}(b))\right) ,\text {since }ab\in [x]_{\Re }[y]_{\Re }\\{} & {} \quad \subseteq [xy]_{\Re } \\{} & {} \quad =\underset{ab\in [s]_{\Re }}{\vee }(\lambda ^{P}(a)\wedge \nu ^{P}(b)),\qquad \qquad \, \, \, \, \ ,\, \, \text {since }xy=s \\{} & {} \quad =\underset{t\in [s]_{\Re },\text { }t=ab}{\vee }(\lambda ^{P}(a)\wedge \nu ^{P}(b)) \\{} & {} \quad =\underset{t\in [s]_{\Re }}{\vee }(\underset{t=ab}{\vee }(\lambda ^{P}(a)\wedge \nu ^{P}(b))) \\{} & {} \quad =\underset{t\in [s]_{\Re }}{\vee }(\lambda ^{P}\circ \nu ^{P})(t)= {\overline{\Re }}(\lambda ^{P}\circ \nu ^{P})(s). \end{aligned}$$

If there are no such elements $x,y\in \varUpsilon$, such that $s=xy,$ then

$$\begin{aligned} ({\overline{\Re }}\lambda ^{P}\circ {\overline{\Re }}\nu ^{P})(s)=0\le {\overline{\Re }}(\lambda ^{P}\circ \nu ^{P})(s). \end{aligned}$$

Similarly, for each $s\in \varUpsilon$, we have

$$\begin{aligned} ({\overline{\Re }}\lambda ^{N}\circ {\overline{\Re }}\nu ^{N})(s)\ge {\overline{\Re }}(\lambda ^{N}\circ \nu ^{N})(s). \end{aligned}$$

Thus, by Definition 3, we have

$$\begin{aligned} {\overline{\Re }}(\lambda )\circ {\overline{\Re }}(\nu )\subseteq {\overline{\Re }}(\lambda \circ \nu ) \end{aligned}$$

for each $\lambda ,\nu \in BFS(\varUpsilon )$. $\square$

Theorem 8

Let $\Re$ be a complete cng-R on an SG $\varUpsilon$. Then, ${\underline{\Re }}(\lambda )\circ {\underline{\Re }}(\nu )\subseteq {\underline{\Re }}(\lambda \circ \nu )$ holds for each $\lambda , \nu \in BFS(\varUpsilon )$.

Proof

Since $\Re$ is a complete cng-R on $\varUpsilon$, so $[x]_{\Re }[y]_{\Re } = [xy]_{\Re }$ for each $x, y \in \varUpsilon$. Let $\lambda ,\nu \in BFS(\varUpsilon )$. We have

$$\begin{aligned} {\underline{\Re }}(\lambda )\circ {\underline{\Re }}(\nu )= & {} ({\underline{\Re }} \lambda ^{P}\circ {\underline{\Re }}\nu ^{P}, {\underline{\Re }}\lambda ^{N}\circ {\underline{\Re }}\nu ^{N}),\\ {\underline{\Re }}(\lambda \circ \nu )= & {} ({\underline{\Re }}(\lambda ^{P}\circ \nu ^{P}), {\underline{\Re }}(\lambda ^{N}\circ \nu ^{N})). \end{aligned}$$

Take any $s\in \varUpsilon$. If there exist elements $x,y\in \varUpsilon$, such that $s = xy,$ then we have

$$\begin{aligned}{} & {} ({\underline{\Re }}\lambda ^{P}\circ {\underline{\Re }}\nu ^{P})(s) =\underset{s=xy}{\vee }({\underline{\Re }}\lambda ^{P}(x)\wedge {\underline{\Re }}\nu ^{P}(y))\\= & {} \underset{s=xy}{\vee }\left( (\underset{a\in [x]_{\Re }}{\wedge }\lambda ^{P}(a))\wedge (\underset{b\in [y]_{\Re }}{\wedge }\nu ^{P}(b))\right) \\= & {} \underset{s=xy}{\vee }\left( \underset{a\in [x]_{\Re }, \text { }b\in [y]_{\Re }}{\wedge }(\lambda ^{P}(a)\wedge \nu ^{P}(b))\right) \\\le & {} \underset{s=xy}{\vee }\left( \underset{a\in [x]_{\Re }, \text { }b\in [y]_{\Re }}{\wedge }\underset{ab=t_{1}t_{2}}{\vee }(\lambda ^{P}(t_{1})\wedge \nu ^{P}(t_{2}))\right) ,\\{} & {} \text {where }t_{1},t_{2}\in \varUpsilon \\= & {} \underset{s=xy}{\vee }\left( \underset{a\in [x]_{\Re },\text { }b\in [y]_{\Re }}{\wedge }(\lambda ^{P}\circ \nu ^{P})(ab)\right) \\= & {} \underset{s=xy}{\vee }\left( \underset{ab\in [xy]_{\Re }}{\wedge } (\lambda ^{P}\circ \nu ^{P})(ab)\right) ,\\{} & {} \ \text {since }ab\in [x]_{\Re }[y]_{\Re }=[xy]_{\Re } \\= & {} \underset{s=xy}{\vee }({\underline{\Re }}(\lambda ^{P}\circ \nu ^{P})(xy)) = {\underline{\Re }}(\lambda ^{P}\circ \nu ^{P})(s). \end{aligned}$$

If there are no such elements $x, y \in \varUpsilon$, such that $s = xy,$ then

$$\begin{aligned} ({\underline{\Re }}\lambda ^{P}\circ {\underline{\Re }}\nu ^{P})(s)=0\le {\underline{\Re }}(\lambda ^{P}\circ \nu ^{P})(s). \end{aligned}$$

Similarly, for each $s\in \varUpsilon$, we have

$$\begin{aligned} ({\underline{\Re }}\lambda ^{N}\circ {\underline{\Re }}\nu ^{N})(s)\ge {\underline{\Re }}(\lambda ^{N}\circ \nu ^{N})(s). \end{aligned}$$

Thus, by Definition 3, we get

$$\begin{aligned} {\underline{\Re }}(\lambda )\circ {\underline{\Re }}(\nu )\subseteq {\underline{\Re }}(\lambda \circ \nu ) \end{aligned}$$

for each $\lambda ,\nu \in BFS(\varUpsilon )$. $\square$

Definition 17

Let $\Re$ be a cng-R on an SG $\varUpsilon$ and let $\lambda \in BFS(\varUpsilon )$. Then, $\lambda$ is a lower (or upper) RBF SSG of $\varUpsilon$, if ${\underline{\Re }}(\lambda )$ (or ${\overline{\Re }} (\lambda ))$ is a BF-SSG of $\varUpsilon$.

A BFS $\lambda$ in $\varUpsilon$, which is both, lower and upper RBF SSG of $\varUpsilon$, is called an RBF SSG of $\varUpsilon$.

Theorem 9

Let $\Re$ be a cng-R on a SG $\varUpsilon$. Then, each BF-SSG of $\varUpsilon$ is an upper RBF SSG of $\varUpsilon$.

Proof

Take a BF-SSG $\lambda$ of $\varUpsilon$. Then, $\lambda ^{P}(ab)\ge \lambda ^{P}(a)\wedge \lambda ^{P}(b)$ and $\lambda ^{N}(ab)\le \lambda ^{N}(a)\vee \lambda ^{N}(b)$ for each $a,b\in \varUpsilon$. Now, for each $x,y\in \varUpsilon$, we have

$$\begin{aligned} {\overline{\Re }}\lambda ^{P}(xy)= & {} \underset{s\in [xy]_{\Re }}{\vee } \lambda ^{P}(s) \\\ge & {} \underset{s\in [x]_{\Re }[y]_{\Re }}{\vee }\lambda ^{P}(s), \,\,\,\,\,\,\,\,\, \text { since }[x]_{\Re }[y]_{\Re }\subseteq [xy]_{\Re } \\= & {} \underset{ab\in [x]_{\Re }[y]_{\Re }}{\vee }\lambda ^{P}(ab), \, \, \, \, \, \, \, \, \, \text {where }s=ab \\\ge & {} \underset{a\in [x]_{\Re },\text { }b\in [y]_{\Re }}{\vee }(\lambda ^{P}(a)\wedge \lambda ^{P}(b)) \\= & {} \left( \underset{a\in [x]_{\Re }}{\vee }\lambda ^{P}(a))\wedge (\underset{b\in [y]_{\Re }}{\vee }\lambda ^{P}(b)\right) \\= & {} {\overline{\Re }}\lambda ^{P}(x)\wedge {\overline{\Re }}\lambda ^{P}(y) \end{aligned}$$

and

$$\begin{aligned} {\overline{\Re }}\lambda ^{N}(xy)= & {} \underset{s\in [xy]_{\Re }}{\wedge }\lambda ^{N}(s) \\\le & {} \underset{s\in [x]_{\Re }[y]_{\Re }}{\wedge }\lambda ^{N}(s), \, \, \, \, \, \, \, \, \, \text {since }[x]_{\Re }[y]_{\Re }\subseteq [xy]_{\Re } \\= & {} \underset{ab\in [x]_{\Re }[y]_{\Re }}{\wedge }\lambda ^{N}(ab), \, \, \, \, \, \, \, \, \, \text {where }s=ab \\\le & {} \underset{a\in [x]_{\Re },\text { }b\in [y]_{\Re }}{ \wedge }(\lambda ^{N}(a)\vee \lambda ^{N}(b)) \\= & {} \left( \underset{a\in [x]_{\Re }}{\wedge }\lambda ^{N}(a)\right) \vee \left( \underset{b\in [y]_{\Re }}{\wedge }\lambda ^{N}(b)\right) \\= & {} {\overline{\Re }}\lambda ^{N}(x)\vee {\overline{\Re }}\lambda ^{N}(y). \end{aligned}$$

This verifies that ${\overline{\Re }}(\lambda )$ is a BF-SSG of $\varUpsilon$. Thus, $\lambda$ is an upper RBF SSG of $\varUpsilon$. $\square$

The converse statement of the Theorem 9 is invalid in general, as exhibited in the next example.

Table 2 Multiplication table for SG

Example 3

Let $\varUpsilon = \{a,b,c,d\}$ represent an SG with the following table of binary operation given in Table 2.

Consider a binary relation $\Re = \big \{(a,a),(b,b),(c,c),(d,d),(b,d),(d,b) \big \}$ on $\varUpsilon$. Then, $\Re$ is a cng-R on $\varUpsilon$, defining the cng-classes $\{a\},\{c\},\{b,d\}$. We take a BFS $\lambda$ in $\varUpsilon$, as follows:

$$\begin{aligned} \lambda= & {} \big \{(a,0.4,{-}0.1),(b,0.3,{-}0.2),\\{} & {} (c,0.4,{-}0.2),(d,0.5,-0.3) \big \}. \end{aligned}$$

The upper RBF approximation of $\lambda$ is calculated as

$$\begin{aligned} {\overline{\Re }}(\lambda )= & {} \big \{(a,0.4,{-}0.1),(b,0.5,{-}0.3),\\{} & {} (c,0.4,{-}0.2),(d,0.5,-0.3)\big \}. \end{aligned}$$

Simple calculations verify that ${\overline{\Re }}(\lambda )$ is a BF-SSG of $\varUpsilon$. However, $\lambda$ is not a BF-SSG of $\varUpsilon$, as

$$\begin{aligned} \lambda ^{P}(cc)= & {} \lambda ^{P}(b)=0.3 \\\ngeq & {} \lambda ^{P}(c)\wedge \lambda ^{P}(c) = 0.4. \end{aligned}$$

Theorem 10

Let $\Re$ be a complete cng-R on a SG $\varUpsilon$. Then, each BF-SSG of $\varUpsilon$ is a lower RBF SSG of $\varUpsilon$.

Proof

Let $\Re$ be a complete cng-R on $\varUpsilon$ and $\lambda$ be a BF-SSG of $\varUpsilon$. Now, for each $x, y\in \varUpsilon$, we have

$$\begin{aligned} {\underline{\Re }}\lambda ^{P}(xy)= & {} \underset{s\in [xy]_{\Re }}{ \wedge }\lambda ^{P}(s) \\= & {} \underset{s\in [x]_{\Re }[y]_{\Re }}{\wedge }\lambda ^{P}(s), \, \, \, \, \, \, \, \, \, \text {since }[x]_{\Re }[y]_{\Re }=[xy]_{\Re } \\= & {} \underset{ab\in [x]_{\Re }[y]_{\Re }}{\wedge }\lambda ^{P}(ab), \, \, \, \, \, \, \, \, \, \text {where }s=ab \\\ge & {} \underset{a\in [x]_{\Re },\text { }b\in [y]_{\Re }}{ \wedge }(\lambda ^{P}(a)\wedge \lambda ^{P}(b)) \\= & {} \left( \underset{a\in [x]_{\Re }}{\wedge }\lambda ^{P}(a)\right) \wedge \left( \underset{b\in [y]_{\Re }}{\wedge }\lambda ^{P}(b)\right) \\= & {} {\underline{\Re }}\lambda ^{P}(x)\wedge {\underline{\Re }}\lambda ^{P}(y) \end{aligned}$$

and

$$\begin{aligned} {\underline{\Re }}\lambda ^{N}(xy)= & {} \underset{s\in [xy]_{\Re }}{\vee } \lambda ^{N}(s) \\= & {} \underset{s\in [x]_{\Re }[y]_{\Re }}{\vee }\lambda ^{N}(s), \, \, \, \, \, \, \, \, \, \text {since }[x]_{\Re }[y]_{\Re }=[xy]_{\Re } \\= & {} \underset{ab\in [x]_{\Re }[y]_{\Re }}{\vee }\lambda ^{N}(ab), \, \, \, \, \, \, \, \, \, \text {where }s=ab \\\le & {} \underset{a\in [x]_{\Re },\text { }b\in [y]_{\Re }}{\vee }(\lambda ^{N}(a)\vee \lambda ^{N}(b)) \\= & {} \left( \underset{a\in [x]_{\Re }}{\vee }\lambda ^{N}(a)\right) \vee \left( \underset{ b\in [y]_{\Re }}{\vee }\lambda ^{N}(b)\right) \\= & {} {\underline{\Re }}\lambda ^{N}(x)\vee {\underline{\Re }}\lambda ^{N}(y). \end{aligned}$$

This verifies that ${\underline{\Re }}(\lambda )$ is a BF-SSG of $\varUpsilon$. Hence, $\lambda$ is a lower RBF SSG of $\varUpsilon$. $\square$

The converse statement of the Theorem 10 is not true generally, as exhibited in the next example.

Table 3 Multiplication table for SG

Example 4

Let us consider an SG $\varUpsilon =\{s,t,u,v\}$ whose binary operation is given in Table 3.

Take a cng-R $\Re$ on $\varUpsilon$, defining cng-classes $\{s\},\{t\}$ and $\{u,v\}$. Then, $\Re$ is a complete cng-R on $\varUpsilon$. We take a BFS $\lambda$ in $\varUpsilon$, as follows:

$$\begin{aligned} \lambda= & {} \{(s,0.3,-0.4),(t,0.4,-0.3),\\{} & {} (u,0.6,-0.2),(v,0.8,-0.1).\} \end{aligned}$$

The lower RBF approximation ${\underline{\Re }}(\lambda )$ of $\lambda$ is calculated as

$$\begin{aligned} {\underline{\Re }}(\lambda )= & {} \big \{(s,0.3,-0.4),(t,0.4,-0.3),\\{} & {} (u,0.6,-0.1),(v,0.6,-0.1) \big \}. \end{aligned}$$

Simple calculations verify that ${\underline{\Re }}(\lambda )$ is a BF-SSG of $\varUpsilon$. However, $\lambda$ is not a BF-SSG of $\varUpsilon$, as

$$\begin{aligned} \lambda ^{P}(vv)= & {} \lambda ^{P}(u)=0.6 \\\ngeq & {} \lambda ^{P}(v)\wedge \lambda ^{P}(v)=0.8. \end{aligned}$$

The next example demonstrates that Theorem 10 is invalid if the cng-R $\Re$ is not complete.

Example 5

Recall the SG $\varUpsilon =\{a,b,c,d\}$ and the cng-R $\Re$ on $\varUpsilon$, as established in Example 3. Then, $\Re$ is not complete. We take a BFS $\lambda$ in $\varUpsilon$, as follows:

$$\begin{aligned} \lambda= & {} \big \{(a,0.4,-0.2),(b,0.6,-0.4),\\{} & {} (c,0.4,-0.1),(d,0.3,-0.3) \big \}. \end{aligned}$$

Then, $\lambda$ is a BF-SSG of $\varUpsilon .$ The RBF approximations of $\lambda$ are calculated by Definition 16 as

$$\begin{aligned} {\overline{\Re }}(\lambda )= & {} \big \{(a,0.4,-0.2),(b,0.6,-0.4),\\{} & {} (c,0.4,-0.1),(d,0.6,-0.4) \big \}, \\ {\underline{\Re }}(\lambda )= & {} \big \{(a,0.4,-0.2),(b,0.3,-0.3),\\{} & {} (c,0.4,-0.1),(d,0.3,-0.3)\big \}. \end{aligned}$$

Simple calculations verify that ${\overline{\Re }}(\lambda )$ is also a BF-SSG of $\varUpsilon ,$ while ${\underline{\Re }}(\lambda )$ is not a BF-SSG of $\varUpsilon$, as

$$\begin{aligned} {\underline{\Re }}\lambda ^{P}(ac)= & {} {\underline{\Re }}\lambda ^{P}(b)=0.3 \\\ngeq & {} {\underline{\Re }}\lambda ^{P}(a)\wedge \underline{\Re }\lambda ^{P}(c)=0.4. \end{aligned}$$

Rough bipolar fuzzy ideals in semigroups

This section presents the notions of the RBF left ideal, RBF right ideal, RBF two-sided ideal, RBFI ideal, and RBF bi-ideal of $\varUpsilon$. We also explore some of their characteristics.

Definition 18

Let $\Re$ be a cng-R on an SG $\varUpsilon$ and let $\lambda \in BFS(\varUpsilon )$. Then, $\lambda$ is a lower RBF left (right, two-sided) ideal of $\varUpsilon$, if ${\underline{\Re }}(\lambda )$ is a BF left (right, two-sided) ideal of $\varUpsilon$.

Definition 19

Let $\Re$ be a cng-R on an SG $\varUpsilon$ and let $\lambda \in BFS(\varUpsilon )$. Then, $\lambda$ is an upper RBF left (right, two-sided) ideal of $\varUpsilon$, if ${\overline{\Re }}(\lambda )$ is a BF left (right, two-sided) ideal of $\varUpsilon$.

A BFS $\lambda$ in $\varUpsilon$, which is both, lower and upper RBF left (right, two-sided) ideal of $\varUpsilon$, is called an RBF left (right, two-sided) ideal of $\varUpsilon$.

Theorem 11

Let $\Re$ be a cng-R on an SG $\varUpsilon$. Then, ${\overline{\Re }}(\lambda \circ \nu )\subseteq {\overline{\Re }}(\lambda )\cap {\overline{\Re }}(\nu )$ holds for each BF right ideal $\lambda$ and BF left ideal $\nu$ of $\varUpsilon$.

Proof

Take a BF right ideal $\lambda$ and a BF left ideal $\nu$ of $\varUpsilon$. We have

$$\begin{aligned} {\overline{\Re }}(\lambda \circ \nu )= & {} ({\overline{\Re }}(\lambda ^{P}\circ \nu ^{P}),{\overline{\Re }}(\lambda ^{N}\circ \nu ^{N})), \\ {\overline{\Re }}(\lambda )\cap {\overline{\Re }}(\nu )= & {} ({\overline{\Re }} \lambda ^{P}\cap {\overline{\Re }}\nu ^{P},{\overline{\Re }}\lambda ^{N}\cup {\overline{\Re }}\nu ^{N}). \end{aligned}$$

Now, for each $s\in \varUpsilon$, we have

$$\begin{aligned}{} & {} {\overline{\Re }}(\lambda ^{P}\circ \nu ^{P})(s) =\underset{t\in [s]_{\Re }}{\vee }(\lambda ^{P}\circ \nu ^{P})(t) \\= & {} \underset{t\in [s]_{\Re }}{\vee }\underset{t=ab}{\vee }(\lambda ^{P}(a)\wedge \nu ^{P}(b)) \\\le & {} \underset{t\in [s]_{\Re }}{\vee }\underset{t=ab}{\vee }\ (\lambda ^{P}(ab)\wedge \nu ^{P}(ab)),\,\,\, \text { since }\lambda \text { is a BF right ideal} \\{} & {} \qquad \qquad \qquad \qquad \qquad \qquad \text { and }\nu \text { is a BF left ideal of }\varUpsilon . \\= & {} \underset{t\in [s]_{\Re }}{\vee }(\lambda ^{P}(t)\wedge \nu ^{P}(t)) \\\le & {} \underset{t\in [s]_{\Re }}{\vee }\underset{t^{\prime }\in [s]_{\Re }}{\vee }(\lambda ^{P}(t)\wedge \nu ^{P}(t^{\prime })) \\= & {} \left( \underset{t\in [s]_{\Re }}{\vee }\lambda ^{P}(t)\right) \wedge \left( \underset{t^{\prime }\in [s]_{\Re }}{\vee }\nu ^{P}(t^{\prime })\right) \\= & {} {\overline{\Re }}\lambda ^{P}(s)\wedge {\overline{\Re }}\nu ^{P}(s) \\= & {} ({\overline{\Re }}\lambda ^{P}\cap {\overline{\Re }}\nu ^{P})(s). \end{aligned}$$

Similarly, for each $s\in \varUpsilon$, we have

$$\begin{aligned} {\overline{\Re }}(\lambda ^{N}\circ \nu ^{N})(s)\ge (\overline{\Re }\lambda ^{N}\cup {\overline{\Re }}\nu ^{N})(s). \end{aligned}$$

Thus, Definition 3 gives

$$\begin{aligned} {\overline{\Re }}(\lambda \circ \nu )\subseteq {\overline{\Re }}(\lambda )\cap {\overline{\Re }}(\nu ) \end{aligned}$$

for each BF right ideal $\lambda$ and BF left ideal $\nu$ of $\varUpsilon$. $\square$

Theorem 12

Let $\Re$ be a cng-R on a SG $\varUpsilon$. Then, each BF left (right, two-sided) ideal of $\varUpsilon$ is an upper RBF left (right, two-sided) ideal of $\varUpsilon$.

Proof

Take a BF left ideal $\lambda$ of $\varUpsilon$. Then, $\lambda ^{P}(ab)\ge \lambda ^{P}(b)$ and $\lambda ^{N}(ab)\le \lambda ^{N}(b)$ for each $a,b\in \varUpsilon .$ Now, for each $x,y\in \varUpsilon$, we have

$$\begin{aligned} {\overline{\Re }}\lambda ^{P}(xy)= & {} \underset{s\in [xy]_{\Re }}{\vee } \lambda ^{P}(s) \\\ge & {} \underset{s\in [x]_{\Re }[y]_{\Re }}{\vee }\lambda ^{P}(s), \,\,\, \,\,\,\,\text {since }[x]_{\Re }[y]_{\Re }\subseteq [xy]_{\Re } \\= & {} \underset{ab\in [x]_{\Re }[y]_{\Re }}{\vee }\lambda ^{P}(ab), \, \, \, \, \, \, \, \, \, \text {where }s=ab \\\ge & {} \underset{a\in [x]_{\Re },\text { }b\in [y]_{\Re }}{\vee }\lambda ^{P}(b) \\= & {} \underset{b\in [y]_{\Re }}{\vee }\lambda ^{P}(b) \\= & {} {\overline{\Re }}\lambda ^{P}(y) \end{aligned}$$

and

$$\begin{aligned} {\overline{\Re }}\lambda ^{N}(xy)= & {} \underset{s\in [xy]_{\Re }}{\wedge }\lambda ^{N}(s) \\\le & {} \underset{s\in [x]_{\Re }[y]_{\Re }}{\wedge }\lambda ^{N}(s), \, \, \, \, \, \, \, \, \, \text {since }[x]_{\Re }[y]_{\Re }\subseteq [xy]_{\Re } \\= & {} \underset{ab\in [x]_{\Re }[y]_{\Re }}{\wedge }\lambda ^{N}(ab), \, \, \, \, \, \, \, \, \, \text {where }s=ab \\\le & {} \underset{a\in [x]_{\Re },\text { }b\in [y]_{\Re }}{ \wedge }\lambda ^{N}(b) \\= & {} \underset{b\in [y]_{\Re }}{\wedge }\lambda ^{N}(b) \\= & {} {\overline{\Re }}\lambda ^{N}(y). \end{aligned}$$

This verifies that ${\overline{\Re }}(\lambda )$ is a BF left ideal of $\varUpsilon$. Hence, $\lambda$ is an upper RBF left ideal of $\varUpsilon$.

Similarly, the cases of BF right ideal and BF (two-sided) ideal of $\varUpsilon$ can be verified. $\square$

The converse statement of the Theorem 12 is not valid generally, as shown in the next example.

Table 4 Multiplication table for SG

Example 6

Let $\varUpsilon =\{k,l,m,n\}$ represent an SG with binary operation given in Table 4.

Let $\Re$ be a cng-R on $\varUpsilon$, defining cng-classes $\{k,l,n\}$ and $\{m\}$. We take a BFS $\lambda$ in $\varUpsilon$, defined as follows:

$$\begin{aligned} \lambda= & {} \{(k,0.5,{-}0.1),(l,0.7,{-}0.1),\\{} & {} (m,0.6,{-}0.1),(n,0.8,-0.1)\}. \end{aligned}$$

The upper RBF approximation of $\lambda$ is calculated as

$$\begin{aligned} {\overline{\Re }}(\lambda )= & {} \big \{(k,0.8,-0.1),(l,0.8,-0.1),\\{} & {} (m,0.6,-0.1),(n,0.8,-0.1) \big \}. \end{aligned}$$

Simple calculations verify that ${\overline{\Re }}(\lambda )$ is a BF left ideal of $\varUpsilon$. However, $\lambda$ is not a BF left ideal of $\varUpsilon$, as

$$\begin{aligned} \lambda ^{P}(lm)= & {} \lambda ^{P}(k)=0.5 \\\ngeq & {} \lambda ^{P}(m)=0.6. \end{aligned}$$

Theorem 13

Let $\Re$ be a complete cng-R on a SG $\varUpsilon$. Then, each BF left (right, two-sided) ideal of $\varUpsilon$ is a lower RBF left (right, two-sided) ideal of $\varUpsilon$.

Proof

Let $\Re$ be a complete cng-R on $\varUpsilon$ and $\lambda$ be a BF left ideal of $\varUpsilon$. Now, for each $x, y\in \varUpsilon$, we have

$$\begin{aligned} {\underline{\Re }}\lambda ^{P}(xy)= & {} \underset{s\in [xy]_{\Re }}{ \wedge }\lambda ^{P}(s) \\= & {} \underset{s\in [x]_{\Re }[y]_{\Re }}{\wedge }\lambda ^{P}(s), \, \, \, \, \, \, \, \, \, \text {since }[x]_{\Re }[y]_{\Re }=[xy]_{\Re } \\= & {} \underset{ab\in [x]_{\Re }[y]_{\Re }}{\wedge }\lambda ^{P}(ab), \, \, \, \, \, \, \, \, \, \text {where }s=ab \\\ge & {} \underset{a\in [x]_{\Re },\text { }b\in [y]_{\Re }}{ \wedge }\lambda ^{P}(b) \\= & {} \underset{b\in [y]_{\Re }}{\wedge }\lambda ^{P}(b) \\= & {} {\underline{\Re }}\lambda ^{P}(y) \end{aligned}$$

and

$$\begin{aligned} {\underline{\Re }}\lambda ^{N}(xy)= & {} \underset{s\in [xy]_{\Re }}{\vee } \lambda ^{N}(s) \\= & {} \underset{s\in [x]_{\Re }[y]_{\Re }}{\vee }\lambda ^{N}(s), \, \, \, \, \, \, \, \, \, \text {since }[x]_{\Re }[y]_{\Re }=[xy]_{\Re } \\= & {} \underset{ab\in [x]_{\Re }[y]_{\Re }}{\vee }\lambda ^{N}(ab), \, \, \, \, \, \, \, \, \, \text {where }s=ab \\\le & {} \underset{a\in [x]_{\Re },\text { }b\in [y]_{\Re }}{\vee }\lambda ^{N}(b) \\= & {} \underset{b\in [y]_{\Re }}{\vee }\lambda ^{N}(b) \\= & {} {\underline{\Re }}\lambda ^{N}(y). \end{aligned}$$

This verifies that ${\underline{\Re }}(\lambda )$ is a BF left ideal of $\varUpsilon$. Thus, $\lambda$ is a lower RBF left ideal of $\varUpsilon$.

Similarly, the cases of BF right ideal and the BF two-sided ideal of $\varUpsilon$ can be verified. $\square$

The converse statement of the Theorem 13 is not valid generally, as exhibited in the following example.

Example 7

Recall the SG $\varUpsilon =\{s,t,u,v\}$ and the complete cng-R $\Re$ on $\varUpsilon$, as established in Example 4. We take a BFS $\lambda$ in $\varUpsilon$, as follows:

$$\begin{aligned} \lambda= & {} \big \{(s,0.2,-0.2),(t,0.4,-0.4),\\{} & {} (u,0.5,-0.6),(v,0.6,-0.5) \big \}. \end{aligned}$$

The lower RBF approximation ${\underline{\Re }}(\lambda )$ of $\lambda$ is calculated as

$$\begin{aligned} {\underline{\Re }}(\lambda )= & {} \big \{(s,0.2,-0.2),(t,0.4,-0.4),\\{} & {} (u,0.5,-0.5),(v,0.5,-0.5) \big \}. \end{aligned}$$

Simple calculations verify that ${\underline{\Re }}(\lambda )$ is a BF left ideal of $\varUpsilon$. However, $\lambda$ is not a BF left ideal of $\varUpsilon$, as

$$\begin{aligned} \lambda ^{P}(vv)= & {} \lambda ^{P}(u)=0.5 \\\ngeq & {} \lambda ^{P}(v)=0.6. \end{aligned}$$

The following example indicates that Theorem 13 is not valid if the cng-R $\Re$ is not complete.

Example 8

Let $\varUpsilon =\{a,b,c,d\}$ be the SG as established in Example 3, on which, we define a cng-R $\Re = \{(a,a),(b,b),(c,c),(d,d),(b,c),(c,b)\}.$ Then, $\Re$ defines the cng-classes $\{a\},\{b,c\},\{d\}$ and $\Re$ is not complete. We take a BFS $\lambda$ in $\varUpsilon$, as follows:

$$\begin{aligned} \lambda{} & {} = \big \{(a,0.5,-0.4),(b,0.6,-0.6),\\{} & {} (c,0.4,-0.5),(d,0.7,-0.7) \big \}. \end{aligned}$$

Then, $\lambda$ is a BF left ideal of $\varUpsilon$. The RBF approximations of $\lambda$ are calculated by Definition 16 as

$$\begin{aligned} {\overline{\Re }}(\lambda )= & {} \big \{(a,0.5,-0.4),(b,0.6,-0.6),\\{} & {} (c,0.6,-0.6),(d,0.7,-0.7) \big \}, \\ {\underline{\Re }}(\lambda )= & {} \big \{(a,0.5,-0.4),(b,0.4,-0.5),\\{} & {} (c,0.4,-0.5),(d,0.7,-0.7) \big \}. \end{aligned}$$

Simple calculations verify that ${\overline{\Re }}(\lambda )$ is also a BF left ideal of $\varUpsilon$, while ${\underline{\Re }}(\lambda )$ is not a BF left ideal of $\varUpsilon$, as

$$\begin{aligned} {\underline{\Re }}\lambda ^{P}(ba)= & {} {\underline{\Re }}\lambda ^{P}(b)=0.4 \\\ngeq & {} {\underline{\Re }}\lambda ^{P}(a)=0.5. \end{aligned}$$

Definition 20

Let $\Re$ be a cng-R on a SG $\varUpsilon$ and let $\lambda \in BFS(\varUpsilon )$. Then, $\lambda$ is a lower (or upper) RBFI ideal of $\varUpsilon$, if ${\underline{\Re }}(\lambda )$ (or ${\overline{\Re }} (\lambda ))$ is a BF interior ideal of $\varUpsilon$.

A BFS $\lambda$ in $\varUpsilon$, which is both, lower and upper RBFI ideal of $\varUpsilon$, is named as an RBFI ideal of $\varUpsilon$.

Theorem 14

Let $\Re$ be a cng-R on a SG $\varUpsilon$. Then, each BF interior ideal of $\varUpsilon$ is an upper RBFI ideal of $\varUpsilon$.

Proof

Take a BF interior ideal $\lambda$ of $\varUpsilon$. Then, $\lambda ^{P}(abc)\ge \lambda ^{P}(b)$ and $\lambda ^{N}(abc)\le \lambda ^{N}(b)$ for each $a,b,c\in \varUpsilon$. Now, for each $x,w,y\in \varUpsilon$, we have

$$\begin{aligned} {\overline{\Re }}\lambda ^{P}(xwy)= & {} \underset{s\in [xwy]_{\Re }}{\vee }\lambda ^{P}(s) \\\ge & {} \underset{s\in [x]_{\Re }[w]_{\Re }[y]_{\Re }}{\vee }\lambda ^{P}(s),\text { since }[x]_{\Re }[w]_{\Re }[y]_{\Re }\\\subseteq & {} [xwy]_{\Re } \\= & {} \underset{abc\in [x]_{\Re }[w]_{\Re }[y]_{\Re }}{\vee }\lambda ^{P}(abc),\text {where }s=abc \\\ge & {} \underset{b\in [w]_{\Re }}{\vee }\lambda ^{P}(b) \\= & {} {\overline{\Re }}\lambda ^{P}(w) \end{aligned}$$

and

$$\begin{aligned} {\overline{\Re }}\lambda ^{N}(xwy)= & {} \underset{s\in [xwy]_{\Re }}{ \wedge }\lambda ^{N}(s) \\\le & {} \underset{s\in [x]_{\Re }[w]_{\Re }[y]_{\Re }}{\wedge }\lambda ^{N}(s),\text { since }[x]_{\Re }[w]_{\Re }[y]_{\Re }\\\subseteq & {} [xwy]_{\Re } \\= & {} \underset{abc\in [x]_{\Re }[w]_{\Re }[y]_{\Re }}{\wedge }\lambda ^{N}(abc),\text {where }s=abc \\\le & {} \underset{b\in [w]_{\Re }}{\wedge }\lambda ^{N}(b) \\= & {} {\overline{\Re }}\lambda ^{N}(w). \end{aligned}$$

This verifies that ${\overline{\Re }}(\lambda )$ is a BF interior ideal of $\varUpsilon$. Hence, $\lambda$ is an upper RBFI ideal of $\varUpsilon$. $\square$

The following example indicates that the converse statement of Theorem 14 is not valid generally.

Example 9

Recall the SG $\varUpsilon = \{k,l,m,n\}$ and the cng-R $\Re$ on $\varUpsilon$, as established in Example 6. We take a BFS $\lambda$ in $\varUpsilon$, defined as follows:

$$\begin{aligned} \lambda= & {} \big \{(k,0.7,-0.2),(l,0.8,-0.2),\\{} & {} (m,0.4,-0.2),(n,0.9,-0.2) \big \}. \end{aligned}$$

The upper RBF approximation of $\lambda$ is calculated as

$$\begin{aligned} {\overline{\Re }}(\lambda )= & {} \big \{(k,0.9,-0.2),(l,0.9,-0.2),\\{} & {} (m,0.4,-0.2),(n,0.9,-0.2)\big \}. \end{aligned}$$

Simple calculations verify that ${\overline{\Re }}(\lambda )$ is a BF interior ideal of $\varUpsilon$. However, $\lambda$ is not a BF interior ideal of $\varUpsilon$, as

$$\begin{aligned} \lambda ^{P}(klm)= & {} \lambda ^{P}(k)=0.7 \\\ngeq & {} \lambda ^{P}(l)=0.8. \end{aligned}$$

Theorem 15

Let $\Re$ be a complete cng-R on a SG $\varUpsilon$. Then, each BF interior ideal of $\varUpsilon$ is a lower RBFI ideal of $\varUpsilon$.

Proof

Let $\Re$ be a complete cng-R on $\varUpsilon$ and $\lambda$ be a BF interior ideal of $\varUpsilon$. Now, for each $x, w, y \in \varUpsilon$, we have

$$\begin{aligned}{} & {} {\underline{\Re }}\lambda ^{P}(xwy) =\underset{s\in [xwy]_{\Re }}{ \wedge }\lambda ^{P}(s) \\= & {} \underset{s\in [x]_{\Re }[w]_{\Re }[y]_{\Re }}{\wedge }\lambda ^{P}(s),\text {since }[x]_{\Re }[w]_{\Re }[y]_{\Re }{=}[xwy]_{\Re } \\= & {} \underset{abc\in [x]_{\Re }[w]_{\Re }[y]_{\Re }}{\wedge }\lambda ^{P}(abc),\text {where }s{=}abc \\\ge & {} \underset{b\in [w]_{\Re }}{\wedge }\lambda ^{P}(b) \\= & {} {\underline{\Re }}\lambda ^{P}(w) \end{aligned}$$

and

$$\begin{aligned}{} & {} {\underline{\Re }}\lambda ^{N}(xwy) =\underset{s\in [xwy]_{\Re }}{ \vee }\lambda ^{N}(s) \\= & {} \underset{s\in [x]_{\Re }[w]_{\Re }[y]_{\Re }}{\vee }\lambda ^{N}(s), \, \, \, \, \, \, \, \, \, \text {since }[x]_{\Re }[w]_{\Re }[y]_{\Re }=[xwy]_{\Re } \\= & {} \underset{abc\in [x]_{\Re }[w]_{\Re }[y]_{\Re }}{\vee }\lambda ^{N}(abc), \, \, \, \, \, \, \, \, \, \text {where }s=abc \\\le & {} \underset{b\in [w]_{\Re }}{\vee }\lambda ^{N}(b) \\= & {} {\underline{\Re }}\lambda ^{N}(w). \end{aligned}$$

This verifies that ${\underline{\Re }}(\lambda )$ is a BF interior ideal of $\varUpsilon$. Hence, $\lambda$ is a lower RBFI ideal of $\varUpsilon$. $\square$

The converse statement of Theorem 15 is invalid generally, as exhibited in the next example.

Example 10

Recall the SG $\varUpsilon =\{s,t,u,v\}$ and the complete cng-R $\Re$ on $\varUpsilon$, as established in Example 4. We take a BFS $\lambda$ in $\varUpsilon$, as follows:

$$\begin{aligned} \lambda= & {} \big \{(s,0.3,-0.1),(t,0.4,-0.2),\\{} & {} (u,0.8,-0.3),(v,0.7,-0.2) \big \}. \end{aligned}$$

The lower RBF approximation ${\underline{\Re }}(\lambda )$ of $\lambda$ is calculated as

$$\begin{aligned} {\underline{\Re }}(\lambda )= & {} \big \{(s,0.3,-0.1),(t,0.4,-0.2),\\{} & {} (u,0.7,-0.2),(v,0.7,-0.2)\big \}. \end{aligned}$$

Simple calculations verify that ${\underline{\Re }}(\lambda )$ is a BF interior ideal of $\varUpsilon$. However, $\lambda$ is not a BF interior ideal of $\varUpsilon$, as

$$\begin{aligned} \lambda ^{P}(tuv)= & {} \lambda ^{P}(v)=0.7 \\\ngeq & {} \lambda ^{P}(u)=0.8. \end{aligned}$$

The next example reveals that Theorem 15 is invalid if the cng-R $\Re$ is not complete.

Example 11

Let $\varUpsilon = \{a,b,c,d\}$ be the SG as established in Example 3 and $\Re$ be the cng-R on $\varUpsilon$ as in Example 8, which is not complete and defines the cng-classes $\{a\},\{b,c\},\{d\}$. We take a BFS $\lambda$ in $\varUpsilon$, as follows:

$$\begin{aligned} \lambda= & {} \{(a,0.3,-0.4),(b,0.4,-0.5),\\{} & {} (c,0.2,-0.2),(d,0.7,-0.7)\}. \end{aligned}$$

Then, $\lambda$ is a BF interior ideal of $\varUpsilon$. The $\Re$-RBF approximations of $\lambda$ are calculated by Definition 16 as

$$\begin{aligned} {\overline{\Re }}(\lambda )= & {} \big \{(a,0.3,-0.4),(b,0.4,-0.5),\\{} & {} (c,0.4,-0.5),(d,0.7,-0.7) \big \}, \\ {\underline{\Re }}(\lambda )= & {} \big \{(a,0.3,-0.4),(b,0.2,-0.2),\\{} & {} (c,0.2,-0.2),(d,0.7,-0.7) \big \}. \end{aligned}$$

Simple calculations verify that ${\overline{\Re }}(\lambda )$ is also a BF interior ideal of $\varUpsilon$, while ${\underline{\Re }}(\lambda )$ is not a BF interior ideal of $\varUpsilon$, as

$$\begin{aligned} {\underline{\Re }}\lambda ^{P}(bac)= & {} {\underline{\Re }}\lambda ^{P}(b)=0.2 \\\ngeq & {} {\underline{\Re }}\lambda ^{P}(a)=0.3. \end{aligned}$$

Definition 21

Let $\Re$ be a cng-R on an SG $\varUpsilon$ and let $\lambda \in BFS(\varUpsilon )$. Then, $\lambda$ is a lower (or upper) RBF bi-ideal of $\varUpsilon$, if ${\underline{\Re }}(\lambda )$ (or ${\overline{\Re }}(\lambda ))$ is a BF bi-ideal of $\varUpsilon$.

A BFS $\lambda$ in $\varUpsilon$, which is both, lower and upper RBF bi-ideal of $\varUpsilon$, is said to be an RBF bi-ideal of $\varUpsilon$.

Theorem 16

Let $\Re$ be a cng-R on a SG $\varUpsilon$. Then, each BF bi-ideal of $\varUpsilon$ is an upper RBF bi-ideal of $\varUpsilon$.

Proof

Take a BF bi-ideal $\lambda$ of $\varUpsilon$. Then, $\lambda$ is also a BF-SSG of $\varUpsilon$. Which implies by Theorem 9, that, ${\overline{\Re }}(\lambda ) = ({\overline{\Re }}(\lambda ^{P}), {\overline{\Re }}(\lambda ^{N}))$ is a BF-SSG of $\varUpsilon$. Now, for each $x, w, y\in \varUpsilon$, we have

$$\begin{aligned} {\overline{\Re }}\lambda ^{P}(xwy)= & {} \underset{s\in [xwy]_{\Re }}{\vee }\lambda ^{P}(s) \\\ge & {} \underset{s\in [x]_{\Re }[w]_{\Re }[y]_{\Re }}{\vee }\lambda ^{P}(s),\text {since }[x]_{\Re }[w]_{\Re }[y]_{\Re }\\\subseteq & {} [xwy]_{\Re } \\= & {} \underset{abc\in [x]_{\Re }[w]_{\Re }[y]_{\Re }}{\vee }\lambda ^{P}(abc),\text { where }s=abc \\\ge & {} \underset{a\in [x]_{\Re },\text { }b\in [w]_{\Re },\text { }c\in [y]_{\Re }}{\vee }(\lambda ^{P}(a)\wedge \lambda ^{P}(c)) \\= & {} (\underset{a\in [x]_{\Re }}{\vee }\lambda ^{P}(a))\wedge (\underset{c\in [y]_{\Re }}{\vee }\lambda ^{P}(c)) \\= & {} {\overline{\Re }}\lambda ^{P}(x)\wedge {\overline{\Re }}\lambda ^{P}(y) \end{aligned}$$

and

$$\begin{aligned} {\overline{\Re }}\lambda ^{N}(xwy)= & {} \underset{s\in [xwy]_{\Re }}{ \wedge }\lambda ^{N}(s) \\\le & {} \underset{s\in [x]_{\Re }[w]_{\Re }[y]_{\Re }}{\wedge }\lambda ^{N}(s),\text { since }[x]_{\Re }[w]_{\Re }[y]_{\Re }\\\subseteq & {} [xwy]_{\Re } \\= & {} \underset{abc\in [x]_{\Re }[w]_{\Re }[y]_{\Re }}{\wedge }\lambda ^{N}(abc),\text { where }s=abc \\\le & {} \underset{a\in [x]_{\Re },\text { }b\in [w]_{\Re },\text { }c\in [y]_{\Re }}{\wedge }(\lambda ^{N}(a)\vee \lambda ^{N}(c)) \\= & {} (\underset{a\in [x]_{\Re }}{\wedge }\lambda ^{N}(a))\vee (\underset{c\in [y]_{\Re }}{\wedge }\lambda ^{N}(c)) \\= & {} {\overline{\Re }}\lambda ^{N}(x)\vee {\overline{\Re }}\lambda ^{N}(y). \end{aligned}$$

This verifies that ${\overline{\Re }}(\lambda )$ is a BF bi-ideal of $\varUpsilon$. Thus, $\lambda$ is an upper RBF bi-ideal of $\varUpsilon$. $\square$

The converse statement of Theorem 16 is invalid generally, as exhibited in the following example.

Example 12

Recall the SG $\varUpsilon = \{k,l,m,n\}$ and the cng-R $\Re$ on $\varUpsilon$, as established in Example 6. We take a BFS $\lambda$ in $\varUpsilon$, defined as follows:

$$\begin{aligned} \lambda= & {} \{(k,0.4,-0.3),(l,0.3,-0.3),\\{} & {} (m,0.1,-0.3),(n,0.2,-0.3)\}. \end{aligned}$$

The upper RBF approximation of $\lambda$ is calculated as

$$\begin{aligned} {\overline{\Re }}(\lambda )= & {} \big \{(k,0.4,-0.3),(l,0.4,-0.3),\\{} & {} (m,0.1,-0.3),(n,0.4,-0.3)\big \}. \end{aligned}$$

Simple calculations verify that ${\overline{\Re }}(\lambda )$ is a BF bi-ideal of $\varUpsilon$. However, $\lambda$ is not a BF bi-ideal of $\varUpsilon$, as

$$\begin{aligned} \lambda ^{P}(knl)= & {} \lambda ^{P}(n)=0.2 \\\ngeq & {} \lambda ^{P}(k)\wedge \lambda ^{P}(l)=0.3. \end{aligned}$$

Theorem 17

Let $\Re$ be a complete cng-R on a SG $\varUpsilon$. Then, each BF bi-ideal of $\varUpsilon$ is a lower RBF bi-ideal of $\varUpsilon$.

Proof

Let $\Re$ be a complete cng-R on $\varUpsilon$ and $\lambda$ be a BF bi-ideal of $\varUpsilon$. Then, $\lambda$ is also a BF-SSG of $\varUpsilon$. Which implies by Theorem 10 that ${\underline{\Re }} (\lambda )=({\underline{\Re }}(\lambda ^{P}),{\underline{\Re }}(\lambda ^{N}))$ is a BF-SSG of $\varUpsilon$. Now, for each $x, w, y\in \varUpsilon$, we have

$$\begin{aligned}{} & {} {\underline{\Re }}\lambda ^{P}(xwy) =\underset{s\in [xwy]_{\Re }}{ \wedge }\lambda ^{P}(s) \\= & {} \underset{s\in [x]_{\Re }[w]_{\Re }[y]_{\Re }}{\wedge }\lambda ^{P}(s),\text {since }[x]_{\Re }[w]_{\Re }[y]_{\Re }=[xwy]_{\Re } \\= & {} \underset{abc\in [x]_{\Re }[w]_{\Re }[y]_{\Re }}{\wedge }\lambda ^{P}(abc),\text { where }s=abc \\\ge & {} \underset{a\in [x]_{\Re },\text { }b\in [w]_{\Re },\text { }c\in [y]_{\Re }}{\wedge }(\lambda ^{P}(a)\wedge \lambda ^{P}(c)) \\= & {} \left( \underset{a\in [x]_{\Re }}{\wedge }\lambda ^{P}(a)\right) \wedge \left( \underset{c\in [y]_{\Re }}{\wedge }\lambda ^{P}(c)\right) \\= & {} {\underline{\Re }}\lambda ^{P}(x)\wedge {\underline{\Re }}\lambda ^{P}(y) \end{aligned}$$

and

$$\begin{aligned}{} & {} {\underline{\Re }}\lambda ^{N}(xwy) =\underset{s\in [xwy]_{\Re }}{ \vee }\lambda ^{N}(s) \\= & {} \underset{s\in [x]_{\Re }[w]_{\Re }[y]_{\Re }}{\vee }\lambda ^{N}(s), \, \, \, \, \, \, \, \, \, \text {since }[x]_{\Re }[w]_{\Re }[y]_{\Re }=[xwy]_{\Re } \\= & {} \underset{abc\in [x]_{\Re }[w]_{\Re }[y]_{\Re }}{\vee }\lambda ^{N}(abc), \, \, \, \, \, \, \, \, \, \text {where }s=abc \\\le & {} \underset{a\in [x]_{\Re },\text { }b\in [w]_{\Re },\text { }c\in [y]_{\Re }}{\vee }(\lambda ^{N}(a)\vee \lambda ^{N}(c)) \\= & {} \left( \underset{a\in [x]_{\Re }}{\vee }\lambda ^{N}(a)\right) \vee \left( \underset{ c\in [y]_{\Re }}{\vee }\lambda ^{N}(c)\right) \\= & {} {\underline{\Re }}\lambda ^{N}(x)\vee {\underline{\Re }}\lambda ^{N}(y). \end{aligned}$$

This verifies that ${\underline{\Re }}(\lambda )$ is a BF bi-ideal of $\varUpsilon$. Hence, $\lambda$ is a lower RBF bi-ideal of $\varUpsilon$. $\square$

The converse statement of Theorem 17 is invalid generally, as exhibited in the next example.

Example 13

Recall the SG $\varUpsilon = \{s,t,u,v\}$ and the complete cng-R $\Re$ on $\varUpsilon$, as established in Example 4. We take a BFS $\lambda$ in $\varUpsilon$, as follows:

$$\begin{aligned} \lambda= & {} \big \{(s,0.1,-0.1), (t,0.3,-0.3), \\{} & {} (u,0.4,-0.4), (v,0.5,-0.5) \big \}. \end{aligned}$$

The lower RBF approximation ${\underline{\Re }}(\lambda )$ of $\lambda$ is calculated as

$$\begin{aligned} {\underline{\Re }}(\lambda )= & {} \big \{(s,0.1,-0.1), (t,0.3,-0.3), \\{} & {} (u,0.4,-0.4), (v,0.4,-0.4) \big \}. \end{aligned}$$

Simple calculations verify that ${\underline{\Re }}(\lambda )$ is a BF bi-ideal of $\varUpsilon$. However, $\lambda$ is not a BF bi-ideal of $\varUpsilon$, as

$$\begin{aligned} \lambda ^{P}(vsv)= & {} \lambda ^{P}(u)=0.4 \\\ngeq & {} \lambda ^{P}(v)\wedge \lambda ^{P}(v)=0.5. \end{aligned}$$

The subsequent example indicates that Theorem 17 is invalid if the cng-R $\Re$ is not complete.

Example 14

Let $\varUpsilon = \{a,b,c,d\}$ be the SG as established in Example 3 and $\Re$ be the cng-R on $\varUpsilon$, which defines the cng-classes $\{a\},\{b,c\},\{d\}$. Then, $\Re$ is not complete. We take a BFS $\lambda$ in $\varUpsilon$, as follows:

$$\begin{aligned} \lambda= & {} \big \{(a,0.3,-0.2),(b,0.4,-0.3),\\{} & {} (c,0.2,-0.1),(d,0.5,-0.4) \big \}. \end{aligned}$$

Then, $\lambda$ is a BF bi-ideal of $\varUpsilon .$ The RBF approximations of $\lambda$ are calculated by Definition 16 as

$$\begin{aligned} {\overline{\Re }}(\lambda )= & {} \big \{(a,0.3,-0.2),(b,0.4,-0.3),\\{} & {} (c,0.4,-0.3),(d,0.7,-0.7) \big \},\\ {\underline{\Re }}(\lambda )= & {} \big \{(a,0.3,-0.2),(b,0.2,-0.1),\\{} & {} (c,0.2,-0.1),(d,0.7,-0.7) \big \}. \end{aligned}$$

Simple calculations verify that ${\overline{\Re }}(\lambda )$ is also a BF bi-ideal of $\varUpsilon ,$ while ${\underline{\Re }}(\lambda )$ is not a BF bi-ideal of $\varUpsilon ,$ as

$$\begin{aligned} {\underline{\Re }}\lambda ^{P}(aba)= & {} {\underline{\Re }}\lambda ^{P}(b)=0.2 \\\ngeq & {} {\underline{\Re }}\lambda ^{P}(a)\wedge \underline{\Re }\lambda ^{P}(a)=0.3. \end{aligned}$$

Comparative analysis and discussion

Rough bipolar fuzzy ideals in SGs are a novel concept that incorporates RS theory and bipolarity in the study of fuzzy ideals in SGs. This approach provides a more flexible and refined way of handling uncertainties in SG theory by investigating rough approximations and bipolarity in the construction of fuzzy ideals. This approach leads to more realistic and accurate models of SGs. When compared to other theories, rough bipolar fuzzy ideals in SGs offer several unique advantages:

• (1) One of the biggest advantages of rough bipolar fuzzy ideals is their flexibility in representing uncertainty and ambiguity. By combining RS theory and BFS theory, they can handle a wider range of data types and better represent the imprecision and vagueness that often arise in real-world applications.

• (2) FS theory is hybridized with ideal theory in SGs in many ways (see [1, 2, 21, 29, 50,51,52, 54]). Some authors have investigated roughness in different types of fuzzy ideals in SGs (see [10, 23]), but their work does not capture the bipolarity which is an essential aspect of the data and human thinking. On the other hand, the bipolar nature of rough bipolar fuzzy ideals offered and can capture both positive and negative data, as well as the trade-offs between them more naturally and understandably.

• (3) Some work on the bipolarity in fuzzy ideals can also be found in the literature [31, 59], but the roughness of the presented ideals is not investigated in these articles. Our work is a conglomeration of FS theory, RS theory, and the bipolarity of data applied to the ideal theory of SGs. In this article, we apply the concept of roughness to the bipolar fuzzy ideals in SGs, which is the distinctiveness and novelty of our work.

In summary, rough bipolar fuzzy ideals in SGs offer a more versatile and flexible approach than traditional theories to study fuzzy ideals in SGs. They are more flexible, provide a unified framework for analyzing different types of SGs, offer more accurate results, and represent a novel and advanced approach to study fuzzy ideals in SGs. Therefore, they can be considered superior to existing notions of fuzzy ideals in semigroups.

Conclusions

The RS theory has been more popular in recent years. It offers a novel theoretical framework for tackling complicated problems in an uncertain environment. Meanwhile, BFS theory is a generalization of FSs and plays a vital role in human decision-making. Positive information depicts what is considered possible, whereas negative information indicates what is considered impossible. In the present study, the notions of BFSs in SGs are presented. Then, the idea is extended to RBFSs in SGs by defining the lower and upper approximations of BFSs in SGs. Meanwhile, the notions of the RBF left (resp. right) ideals, RBF two-sided ideal, RBFI ideal, and RBF bi-ideal of SGs are presented. Some important properties associated with these ideas are also given. Several illustrations, where required, have been included to perceive the notions presented with ease.

Keeping aside the notional and theoretic features, the ideas unveiled have enough potential to be expanded in coping intelligently with day-to-day life situations including business and trade analysis, robotics, social sciences, life sciences, agricultural sciences, human resource management, pattern recognition, water management, medicine, economics, energy crisis problems, recruit problems, and many other fields of practical usage. In the future, we intend to focus on the roughness of bipolar fuzzy ideals of semigroups in the context of soft set theory. Also, we will give attention to the roughness of multi-polar fuzzy ideals and multi-polar hyperideals of semihypergroups.