1. Introduction
The concept of an “ordered semiring” was first used by Gan and Jiang [
1] in connection to a semiring with a compatible partial order relation. They also proposed the idea of ideals in ordered semirings. Good et al. [
2] developed the concept of bi-ideals in semigroups. Following that, Lajos et al. [
3] established bi-ideals in associative rings. Bi-ideals of ordered semirings were described and characterized in terms of regularity, and the relationship between bi-ideals and quasi-ideals was characterized by Palakawong et al. [
4]. Senarat et al. [
5] developed the terms-ordered
k-bi-ideal, strong-prime-ordered
k-bi-ideal, and prime-ordered
k-bi-ideal in ordered semirings. By expanding on the idea of bi-ideals of ordered semirings, Davvaz et al. [
6] introduced the concept of bi-hyperideals in ordered semi-hyperrings. The notions of
-bi-hyperideals and Prime
-bi-hyperideals were established and inter-related properties were considered by Omidi and Davvaz [
7]. The characterization of ordered h-regular semirings was considered by Anjum et al. [
8]. In [
9], Patchakhieo and Pibaljommee characterized ordered k-regular semirings using ordered k-ideals. The ordered intra-k-regular semirings have been introduced and defined in different ways by Ayutthaya and Pibaljommee [
10]. Omidi and Davvaz [
7] considered the concepts of
-bi-hyperideals and Prime
-bi-hyperideals and established inter-related features. Anjum et al. [
8] proposed characterizing ordered h-regular semirings. By using ordered k-ideals, Patchakhieo and Pibaljommee described ordered k-regular semirings in [
9]. The ordered intra-k-regular semirings have been presented and characterized in various ways by Ayutthaya and Pibaljommee [
10].
Fuzzy sets to semirings were initially discussed by Ahsan et al. in [
11] and Kuroki [
12] applied the idea to semigroups. Mandal [
13] pioneered the study of ideals and interior ideals in ordered semirings, as well as their characterizations in the sense of regularity. He developed the concepts of fuzzy bi-ideals and fuzzy quasi-ideals in ordered semirings in [
14]. Gao et al. [
15] presented semisimple fuzzy ordered semirings and weakly regular fuzzy ordered semirings in terms of different kinds of fuzzy ideals. Saba et al. [
16] initiated the study of ordered semirings based on single-valued neutrosophic sets. Several characterizations of regular and intra-regular ordered semigroups in terms of
-fuzzy generalized bi-ideals were presented by Jun et al. [
17], who also proposed the idea of
-fuzzy generalized bi-ideal in ordered semigroups. Similar semiring concepts, such as
-fuzzy bi-ideals on semirings, were investigated by Hedayati [
18]. Additionally, other ideas connected to our research in several domains have been examined in [
19,
20,
21,
22,
23,
24,
25].
In this study, we describe a novel form of fuzzy ideal in ordered semirings. The concept of
-fuzzy bi-ideal is presented. We show that any fuzzy bi-ideal is the
-fuzzy bi-ideal, but the converse assertion is invalid, and an example is shown. A criterion for an
-fuzzy bi-ideal to be a fuzzy bi-ideal is given. Furthermore, some correspondences between bi-ideal and
-fuzzy bi-ideal are included. Furthermore, regularly ordered semirings are described in terms of
-fuzzy bi-ideals and their
-lower parts. The structure of the paper is as follows:
Section 2 highlights some of the ideas and properties of ordered semirings, ideals, fuzzy subsets, and fuzzy subsemirings that are necessary to generate our key results.
Section 3 focuses on the concept of the
-fuzzy bi-ideal of ordered semirings.
Section 4 examines the
-lower part of the
-fuzzy bi-ideal.
Section 5 contains instructions for some potential future research work.
2. Preliminaries
An ordered semiring is a semiring with compatible order relation , i.e., and , .
If , , then is said to be additively commutative. An element is an absorbing zero if and , .
For , we define For , is defined as .
A subset ( of is said to be a sub-semiring if and . Additionally, refers to the left (resp. right) ideal of if and , and . If it is both the left and right ideals of , it is referred to as an ideal. A sub-semiring P of is called a bi-ideal (in brief, BI) of if and .
A mapping
is said to be fuzzy set (in brief,
FS) of
. For the
FSs and
of
,
and
are described as:
and
For
, the
characteristic function is defined as:
Define ⪯ on the set
of all FSs of
by
If
such that
, then
,
and
. We represent by
the
FS of
given by
.
Let . Then ; ; .
A FS is called a:
Fuzzy subsemiring of if and , .
Fuzzy left (resp. right) ideal (in brief, FL(R)I) of if and (resp. ), .
Fuzzy ideal of if is both fuzzy right and left ideals of .
Fuzzy bi-ideal (in brief, FBI) if it is fuzzy subsemiring and and , .
3. -Fuzzy Bi-Ideals of Ordered Semirings
In this section, the concept of -fuzzy bi-ideals of is introduced.
Let
and
. The
ordered fuzzy point (OFP)
of
is defined by
For
,
represents for
. Thus
.
Definition 1. An OFP of Υ is said to be -quasi-coincident with a FS of Υ for , denoted as , and defined as:For the OFP , we define - (1)
, if ;
- (2)
, if or ;
- (3)
, if does not hold for ;
for .
Definition 2. A FS of Υ is said to be an -fuzzy bi-ideal (in brief, -FBI) of Υ if:
- (1)
, ,
- (2)
and ,
- (3)
and , and
- (4)
, , .
and .
Example 1. On , define the opertaions and order relation asThen is an ordered semiring. Define an FS of Υ as is the -FBI of Υ and can be easily verified. Lemma 1. Each FBI of Υ is the -FBI of Υ.
Proof. Straightforward. □
Remark 1. In general, the converse of Lemma 1 does not hold. It is illustrated by the following example:
Example 2. Define operations and ordered relations on as follows:Then, is an ordered semiring. Define the FS of Υ asIt can be easily verified that is the -fuzzy bi-deal of Υ but not an FBI of Υ as follows: . Theorem 1. An FS is an -FBI of Υ⇔
- (1)
- (2)
,
- (3)
, and
- (4)
,
.
Proof. (⇒) Let such that . If , then ∃ such that . So , but , which is a contradiction. Therefore . Next, if , for some , then , for some . Thus, , but , which is a contradiction. Therefore, . Similarly, , . Again, if , for some , then for some . Thus, , but , again a contradiction. Consequently, .
(⇐) Take any and such that and . Then, , and it follows that . If , then implies . Again, if , then . Thus, , so . Therefore, . Again, take any and . Then, and . Therefore, . Now, if , then implies . Again, if , then . Therefore, implies that . Therefore, . Similarly, for any and . Further, take any and , . Then and . Therefore, . Now if , then implies . If , then . Thus i.e., . Therefore, , as required. □
Theorem 2. If ) is -FBI of Υ with , . Then is an FBI of Υ.
Proof. Suppose that
such that
. Since
is
-
FBI,
. By hypothesis,
; thus, it implies
. Again, for any
, we have
and
Since
and
, so
and
Finally, take any
. Since
is
-
FBI, by Theorem 1 and the hypothesis
as required. □
Theorem 3. Let . Then Ω is a BI of Υ⇔, an -FBI.
Proof. Straightforward. □
Theorem 4. An FS is the -FBI of Υ⇔, a BI of Υ.
Proof. (⇒) Let and be such that . Then, . By Theorem 1, . Therefore, . Let , where . Then and . By Theorem 1, . Therefore, . Similarly, for . Let and . Then, and . So, by Theorem 1, . Thus . Therefore . Hence is a BI.
(⇐) Take any with . If , then for some , . So , but , which is a contradiction. Thus , with . Again, if , for some , then , for some . Thus, , but , a contradiction. Therefore, , . Similarly, , . Further, if , for some . Then, such that implies , but , again a contradiction. Therefore , , as required. □
Example 3. Define the operations and order relation ≤ on in the following ways:Then, is an ordered semiring. Now define an FS of Υ as and . Therefore,By Theorem 4, is an -FBI of Υ as is a BI of Υ, , with and . Definition 3. Let . The setwhere is said to be an -level subset of . Theorem 5. Let such that implies . Then. is an -FBI of Υ⇔, the -level subset of is a bi-deal of Υ.
Proof. (⇒) Take any and such that . As , we have implies or . By hypothesis, we have or . Thus, . Therefore, . Next, take any . Then, ; that is, or and or .
- Case (i).
Let
and
. If
; then,
and, so,
. If
, then
and so
. Hence,
.
- Case (ii).
Let
and
. If
, then
that is,
, and thus
. If
, then
and so
. Hence,
.
- Case (iii).
Let and . Proof is analogous to case proof (ii).
- Case (iv).
Let and . Proof is analogous to previous two cases.
Thus for all cases, we have , and thus . Similarly, for any and , we have and . Hence, is a BI of .
(⇐) Let , for some . Then, such that . Thus, it follows that but , which is a contradiction, and hence . Let for some . Then ∃ such that . Thus, it follows that but , which is a contradiction. Therefore, , . Similarly, , . Next, suppose that for some . It follows that but which is again a contradiction. Thus , as required. □
4. Lower Part of -FBI
The concept of the lower part of the -FBI of is defined and characterized.
Definition 4. The -lower part of is defined as and . The -lower part of the characteristic function is defined for as Definition 5. Let . Define , , and as follows: and . Lemma 2. . Then,
- (1)
and ;
- (2)
If , and , then and ;
- (3)
If , and , then and ;
- (4)
;
- (5)
;
- (6)
;
- (7)
.
Proof. Straightforward. □
Lemma 3. Let . Then,
- (1)
;
- (2)
;
- (3)
;
- (4)
.
Proof. Straightforward. □
Lemma 4. If is the -FBI of Υ, then is an FBI of Υ.
Proof. Let be such that . Then, . Thus, it implies , and, so, . Next suppose that . Since is an -FBI of . It follows that , and hence, . Similarly, , . Let ; we have . Then , and so . Therefore, is an FBI of . □
Lemma 5. Let . Then, Ω is a BI of Υ⇔, the -FBI of Υ.
Proof. Let and be such that and . Then, and . Therefore, . As is a BI of , . Thus . If , then , so we have . If , then . So . Similarly, and imply . Therefore, . Let and be such that . Then, , . Since is a BI of , we have . Thus, . If , then . Therefore . Again, if , then . So . Thus, , as required.
Let and such that . Then . Since is an -FBI of , and , we have . Thus, and so . Let . Then, and . Since is an -FBI of , we have and also . Thus, it implies and . Therefore, . Let and . Then and . Now, . Hence . Therefore . Hence, is a BI of . □
Theorem 6. Let . Then is an -FBI of Υ⇔
- (1)
,
- (2)
,
- (3)
, and
- (4)
.
Proof. (⇒) Suppose that
is an
-
FBI of
. If
, then
. Suppose that
. Then, we have
Thus,
. Similarly,
. Again,
, then
. Suppose that
. Then, we have
Therefore,
.
(⇐) Let
. Then, by hypothesis, we have
Similarly, by hypothesis,
.
We also have
as required. □
Theorem 7. The following statements are equivalent in Υ:
- (1)
Υ is regular.
- (1)
for any -FBI of Υ.
Proof. Assume that
is an
-
FBI of
. If
, then, as
is regular, ∃
such that
. Now, we have
Thus,
.
. Let
B be a
BI of
. Then, by Lemma 5,
is an
-
FBI of
. Thus, by hypothesis, we have
So
. Since
B is
BI, so
. Thus
. Hence, by ([
9] Lemma 2.2),
is regular. □
Theorem 8. The following statements are equivalent in Υ:
- (1)
Υ is regular and intra-regular.
- (2)
for any -FBI of Υ.
Proof. (⇒) Suppose that
is an
-
FBI of
. As
is regular and intra-regular,
and
. Therefore,
. We have
Thus
. Since
is an
-
FBI, so
. Hence
.
. Let
B be a
BI of
. Then, by Lemma 5,
is an
-
FBI of
. Thus by hypothesis, we have
Therefore,
. Since
B is
BI, so
. Thus
. Hence, by ([
9], Theorem 3.12)
is regular. □
Definition 6. Let and . Define the following of Υ as Lemma 6. Let be the -FBI of Υ. Then () is the BI of Υ.
Proof. Let . As , we have . Take any . Then, and . Since is the -FBI of , , so . By a similar argument, .
Next, take any and . Then and . By hypothesis, . Therefore, . Thus . Additionally, for any and such that , we have . Hence, is a BI of . □
Definition 7. An ordered semiring Υ is called -fuzzy bi-simple if every -FBI is constant. That is, ; we have , for each -FBI of Υ.
Theorem 9. The ordered semiring Υ is bi-simple ⇔ it is -fuzzy bi-simple.
Proof. (⇒) Let be the -FBI of and . By Lemma 6, is an left ideal of . As is bi-simple, . So . Thus, . Therefore, . Similarly, . Thus, , as required.
(⇐) Assume that I is the proper BI of . By Lemma 5, is the -FBI of . As is -fuzzy bi-simple, , . Let and . Then, . As , we have . Therefore, , which implies that . Thus, , and hence is bi-simple. □