1. Introduction and Motivation
The implementation of traditional fuzzy sets presented by Zadeh [
1] in the field of algebraic structures has brought great success to the study of fuzzy algebra. A few mathematical researchers, such as Rosenfeld [
2], Mordeson and Malik [
3], Akram et al. [
4], Mandal [
5], Shabir and Mahmood [
6] and Zhan [
7], have obtained many wonderful and useful results on fuzzy sets. Nevertheless, the membership degrees of elements in conventional fuzzy sets are all limited to the interval [0,1], which leads to a great difficulty in explaining the distinction between the irrelevant elements and the opposite elements in fuzzy sets. Zhang [
8] presented the theory of the bipolar fuzzy (BF) set, which has a range of
, to avoid this problem. Recently, keeping the results of BF sets under consideration, many researchers have been using BF sets to represent algebraic structures [
9,
10,
11,
12]. This theory provides the difference between positive and negative aspects of the same situation. In real life, we observe many bipolar concepts, good and bad effects of medicines, thickness and thinness of fluid, honesty and dishonesty. For more applications, see [
13,
14,
15,
16,
17,
18].
Pawlak, in 1982 [
19], initiated the theory of rough sets. Based on the known attributes, a rough set contains all the information. Rough sets are characterized by means of upper and lower approximation. This theory is a useful tool in the areas of artificial intelligence, such as pattern recognition, learning algorithms, inductive reasoning, automatic classification, etc. In addition, it has many applications in measurement theory, classification theory, taxonomy, cluster analysis, etc. In the medical field, a serious challenge is abdominal pain in children. There are many reasons for this disease and it is hard to detect the main cause. This theory helps the doctors to detect the cause through discharge observations.
In 1934, Vandiver studied the structure of semirings [
20]. A large amount of work has been conducted on this mathematical algebraic structure in the fields of medical science, social science, engineering, arts, economics and environmental science. From an algebraic point of view, semirings provide the most natural common generalization of the theories of rings and bounded distributive lattices, and the techniques used in analyzing them are taken from both areas.
Fuzzy semirings, introduced by Ahsan et al. [
21], have drawn widespread interest from scholars. In addition, a large amount of related findings have emerged by Ahsan et al. [
22]. Many experts have investigated roughness in algebra and fuzzy algebraic structures [
23,
24,
25,
26]. Hosseini et al. studied the generalizations of roughness or T-roughness in fuzzy algebra [
27]. In 2020, Bashir et al. studied the roughness of fuzzy ideals with three-dimensional congruence relations of ternary semigroups [
28]. In 2019, Shabir et al. [
29] worked on BF ideals of regular semigroups. Moreover, Bashir et al. extended the work of [
29] to regular ordered ternary semigroups [
30] and regular ternary semirings [
31]. However, approximation of BF sets has not been commonly used in semirings so far, to our knowledge. Therefore, consideration of a new framework of approximation of BF subsemirings (resp. ideals) is reasonable and necessary. We address the principles of RBF subsemirings (resp. ideals) in this article and analyze related properties by extending [
29,
32].
This paper is arranged as follows. In
Section 1, we introduce all of the terms used in this paper. In
Section 2, we discuss the ideas of rough sets and BF sets and present some basic definitions. The main section of this paper is
Section 3, where we will present RBF ideals, bi-ideals and interior ideals of semirings and their related theorems on the basis of CRs and CCRs. In the last section, a comparative study and conclusions are given.
The list of acronyms used in the research article is given in
Table 1.
2. Preliminaries
A nonempty set is known as a semiring if is a commutative semigroup, is a semigroup and for every . By a subset, we always mean a nonempty one. A subset P of a semiring is known as a subsemiring of if P is itself a semiring under addition and multiplication as defined in . A subset P of a semiring is known as a left (resp. right) ideal of if is a groupoid and If P is both a right and left ideal, then it is said to be an ideal of . A subset P of a semiring is called a bi-ideal if P is a subsemiring of and A subset P of a semiring is known as an interior ideal if P is a subsemiring and
In BF subset
of
,
presents the satisfaction degree of
z to the correlated characteristic of
and
is the satisfaction degree of
z to the somewhat opposite characteristic of
. The fuzzy set presents only positive aspects of a situation with membership function
. The difference between the fuzzy set and the BF set is shown by the following example. Let
be a set of workers of a company. Define a fuzzy set on
A with fuzzy property “honesty”, the workers
and
y having property “honesty” mapped to
, as shown by the bar graph in
Figure 1.
Other workers have no membership degree in range
, as they are not honest. In the fuzzy set, we can cover only positive aspects of any situation. We cannot deal with negative aspects of situations. To facilitate, we deal with such problems with the BF set. The property “dishonesty” is opposite to “honesty”. The workers
and
z are mapped to
with property “dishonesty”. In such a way, the BF set gives information about all elements, as shown in
Figure 2.
Throughout this article,
represents a semiring. Let
. Then,
if and only if
and
for every
, and
if and only if
and
. Let
. Then,
, the sum of
and
, is defined as
and
for
. For
, define the BF subset
of
as
and
for
Definition 1 ([
29]).
Let ρ Then, ρ is known as a BF subsemiring of if for every - (i)
, ;
- (ii)
, ;
- (iii)
, .
Definition 2 ([
29]).
Let ρ. Then, ρ is known as a BF left (resp. right) ideal of if for every - (i)
, ;
- (ii)
, resp. ,
If ρ is both a BF left and right ideal, then it is known as a BF ideal of .
Definition 3 ([
29]).
Let ρ. Then ρ is known as a BF bi-ideal of if for every :- (i)
, ;
- (ii)
, ;
- (iii)
, .
Definition 4. Let ρ Then, ρ is known as a BF interior ideal of if for every :
- (i)
, ;
- (ii)
, ;
- (iii)
, .
Let
W be a universe and
an equivalence relation on
W. Then
is known as an approximation space. The equivalence classes of
are the main constituents of the rough sets. Let
. The set
Y is called a definable subset of
W if it is the collection of some equivalence classes of the universal set
W; otherwise, it is not definable. The set
Y is approximated in the form of upper and lower approximations, which are given as:
where
is the equivalence class of
w for
The rough set is a pair if The set Y is a definable set if
3. Approximations of Bipolar Fuzzy Ideals in Semirings
This is the main section of our paper in which we present the concepts of RBF subsets of semirings and concentrate on their key properties. RBF subsemirings, RBF ideals, RBF bi-ideals and RBF interior ideals of the semirings are additionally discussed in this section. The lower and upper RBF approximations of
under the relation
are the BF subsets
and
of
, respectively, defined as:
where
If
, then
is
-definable; else,
is an RBF subset of
[
33]. An equivalence relation
on
which satisfies the additional condition that if
and
then
and
is known as a congruence relation (CR) on
. A CR
on
is complete if for every
and
[
32]. Some authors used the term full congruence instead of complete congruence in their research.
Theorem 1. If is a CCR on , then for any ξ,
Proof. Since the CR is complete on , for all Let . The following two cases arise for :
Case (i): If for then . Thus, and
Case (ii): If
for
then
where
and
Similarly, . Hence, □
Theorem 2. If is a CR on , then for any ξ,
Proof. As is a CR on , for all Let . The following two cases arise for :
Case (i): If for then it is obvious.
Case (ii): If
for
then we have
Similarly, Hence, □
Theorem 3. If is a CCR on , then for any ξ,
Proof. Since the CR
is complete on
,
for all
Let
and
. Consider
Similarly, Thus, □
Theorem 4. If is a CR on , then for any ξ,
Proof. As
is a CR on
,
for all
Let
Then, for any
, consider
Similarly, Hence, □
Definition 5. Let be a CR on and . Then, ξ is a lower (resp. upper) RBF subsemiring of if is a BF subsemiring of .
A BF subset ξ of which is both a lower and upper RBF subsemiring of is known as an RBF subsemiring of .
Theorem 5. If is a CR on , then each BF subsemiring of is an upper RBF subsemiring of .
Proof. For all
, consider
Similarly,
In addition,
Similarly, Thus, is a BF subsemiring of . Therefore, is an upper RBF subsemiring of . □
Theorem 6. If is a CCR on , then each BF subsemiring of is a lower RBF subsemiring of .
Proof. Let
be a BF subsemiring of
. Now, for all
, consider
Similarly,
In addition,
Similarly, Thus, is a BF subsemiring of . Therefore, is a lower RBF subsemiring of . □
The example defined below illustrates that Theorem 6 does not hold if the CR is not complete.
Example 1. Let be a semiring with the addition “+” and multiplication “· ” given in Table 2 and Table 3. Consider a binary relation
on
. Then,
is a CR on
, defining the congruence classes
,
,
and
, and
is not complete, since
. We take a BF subset
of
, as below.
Then,
is a BF subsemiring of
. Now,
It can be vindicated by simple calculations that is also a BF subsemiring of , whereas is not, as
Definition 6. If is a CR on and , then ξ is a lower (resp. upper) RBF left (resp. right) ideal of if resp. is a BF left (resp. right) ideal of .
Theorem 7. If is a CR on , then each BF left (resp. right) ideal of is an upper RBF left (resp. right) ideal of .
Proof. Similarly,
In addition,
Similarly, This implies that is a BF left ideal. Therefore, is an upper RBF left ideal of . Similarly, the case of a BF right ideal can be verified. □
Theorem 8. Let be a CCR on . Then, each BF left (resp. right) ideal of is a lower RBF left (resp. right) ideal of .
Proof. Similarly,
In addition,
Similarly, Thus, is a BF left ideal. Therefore, is a lower RBF left ideal of . Similarly, the case of a BF right ideal of can be verified. □
The example defined below illustrates that Theorem 8 does not hold if the CR is not complete.
Example 2. Let be a semiring with the addition “+” and multiplication “·” defined in Table 4 and Table 5. Consider a binary relation on . Then, is a CR on . Defining the congruence classes and , is not complete. Therefore, .
We take a BF subset
of
, as below.
Then,
is a BF left ideal of
. Now,
It can be vindicated by simple calculations that is also a BF left ideal of , whereas is not, as
Definition 7. If is a CR on and , then ξ is a lower (resp. upper) RBF bi-ideal of if is a BF bi-ideal of .
A BF subset ξ of which is both a lower and upper RBF bi-ideal is called an RBF bi-ideal of .
Theorem 9. For a CR on , each BF bi-ideal is an upper RBF bi-ideal of .
Proof. A BF bi-ideal
is also a BF subsemiring of
. We have by Theorem 5 that
is a BF subsemiring of
. Now, for
, consider
and
Thus,
is a BF bi-ideal of
. Therefore,
is an upper RBF bi-ideal of
. □
Theorem 10. For a CCR on , each BF bi-ideal is a lower RBF bi-ideal of .
Proof. A BF bi-ideal
is also a BF subsemiring of
. In addition, by Theorem 6,
is a BF subsemiring of
. Now, for
, consider
and
This implies
is a BF bi-ideal of
. Therefore,
is a lower RBF bi-ideal of
. □
The example defined below illustrates that Theorem 10 does not hold if CR is not complete.
Example 3. Let be the semiring and be the CR on , as defined in Example, which is not complete, and define the congruence classes .
We take a BF subset ξ of , as below. Then, ξ is a BF bi-ideal of . Then, It can be seen by simple calculations that is also a BF bi-ideal, whereas is not, as Definition 8. For a CR on and , ξ is a lower (resp. upper) RBF interior ideal of if is a BF interior ideal of .
If ξ is both a lower and upper RBF interior ideal, then it is called an RBF interior ideal of .
Theorem 11. For a CR on , each BF interior ideal, is an upper RBF interior ideal of .
Proof. Since a BF interior ideal
is also a BF subsemiring of
, by Theorem 5,
is a BF subsemiring of
. Now, for all
,
Similarly, This implies is a BF interior ideal of . Therefore, is an upper RBF interior ideal of . □
Theorem 12. Consider a CCR on . Then, each BF interior ideal is a lower RBF interior ideal of .
Proof. We have by Theorem 6 that
is a BF subsemiring of
. Consider, for every
,
Similarly, Thus, is a BF interior ideal of . Therefore, is a lower RBF interior ideal of . □
The example defined below illustrates that Theorem 12 does not hold if the CR is not complete.
Example 4. Let be the semiring and be the CR on , as defined in Example, which is not complete, and define the congruence classes .
We take a BF subset
of
, as below.
Thus,
is a BF interior ideal of
. Now,
It can be seen by simple calculations that
is also a BF interior ideal of
, whereas
is not, because