1. Introduction
The fuzzy subring was introduced by Bhakat [
1], and many other mathematicians have devoted a lot of time to studying fuzzy subsystems of various algebraic structures. Group theory, ring theory, field theory, modules, vector spaces, lattices, and algebras over a field are example applications of algebraic structures. In abstract algebra, a field is a very beneficial part that takes different algebraic structures on them. It is a branch of modern algebra that has come to the forefront many years ago. A very useful area of mathematics, called field theory, has been employed extensively in cryptography, coding theory, combinatorial mathematics, cyber security, and the study of electronics circuits. McEliece [
2] introduced finite fields in computer and engineering studies. He discussed algebraic coding theory mathematically based on the theory of finite fields. He also treated coding theory courses in “Volkswagen” using finite fields in detail.
The fuzzy set theory is based on the principle of related rank-based affiliation, which is based on subjective experience and thinking. In 1965, Zadeh [
3] established fuzzy set and its fundamental algebraic results. The use of fuzzy set theory provides a powerful framework for addressing ambiguity and uncertainty in practical issues. As a result, crisp sets frequently lack the appropriate response and feedback for real-world situations with growing problems. A fuzzy subset
of a classical set
is define as
; thus,
is known as a supporting degree. By widening the function’s range from {0, 1} to [0, 1], it is undeniable that fuzzy sets are modifications of the characteristics value of traditional sets. Many scholars in various fields of agricultural science, mathematics, environmental science, and space science find fuzzy theory to be an attractive and engaging topic because a variety of fields, including coding theory, journey time histories, protein structure analysis, and medical diagnostic methods, adopt this special theory. Rosenfeld [
4] 1971 gave the novel concept of a fuzzy subgroup with a basic fundamental algebraic structure. Liu [
5] connected ring theory and fuzzy sets and established the notion of a fuzzy subring. Atanassov [
6] invented the theory of intuitionistic fuzzy sets and also provided the basic algebraic characteristics of this phenomenon. This concept has improved in effectiveness in the scientific community because it focuses on the degree of participation and non-participation within a unit interval; therefore,
represent the belonging degree and
represent the not belonging degree. This tenet is obviously a fundamental aspect of traditional fuzzy sets since it gives people more chances to present inaccurate information in order to address problems more effectively. The intuitionistic fuzzy theory’s most remarkable feature is that it has included the haziness and uncertainty of physical challenges and scientific problems better than the traditional fuzzy set accomplishes, for example in the areas of psychological analysis, decision-making, and strategies for a number of bio-informatics and computational biological-based problems. Decision-making involves studying and ranking a certain number of possibilities to determine how effective decision-makers are when all needs are continuously considered [
7,
8]. The notion of intuitionistic fuzzy subgroups was first suggested by Biswas [
9]. In order to examine non-associative rings and other mathematical properties, numerous mathematicians have developed intuitionistic fuzzy sets and hybrid power frameworks of fuzzy sets [
10,
11]. Malik and Mordeson [
12] introduced some fundamental characteristics of fuzzy subfield and provided an example of how to described a field expansions in terms of fuzzy subfield. Let
be a field; then, a fuzzy set
of
is a fuzzy subfield if
and
for all
, where 0 is an additive identity for all
Mordeson [
13] proposed the idea of fuzzy algebraic field extensions and established the circumstances under which a field extension has a singular maximum fuzzy field. All fuzzy intermediate fields with the sup property are used by Volf [
14] to characterize extensions; chained intermediate fields are described and show that any fuzzy intermediate field of such an extension has the sup property. Tang et al. [
15] introduced the idea of an intuitionistic fuzzy entropy derived symmetric implicational algorithm, and symmetric implicational principles and applications. Yang et al. [
16] used the N-base encoding method for the representation of particles and designed a particle update mechanism based on the Hamming distance and a fuzzy learning strategy, which can be performed in the discrete space.
In 2013, Yager [
17,
18] created the notion of a Pythagorean fuzzy subset
, where the squares of belonging and not belonging degrees add up to a range [0,1]. Therefore,
. Our understanding of problem solving in decision making has significantly benefited from
. Peng and Yang [
19] suggested the importance of two operations, division and subtraction, and discussed each of their features in order to clarify
. After that, the boundness, idempotency, and homogeneity properties of
analytic functions were examined. Li and Lu [
20] defined the
normalised Hamming distance, and
normalised Hausdorff distance by extending the Hamming distance and the Hausdorff distance with
. Ejegwa [
21] adopted
because they have many different applications and are extended intuitionistic fuzzy sets. It is crucial to consider how inventive such sets are in addressing the problem of career placements. He also talked about choosing a good profession based on academic ability and demonstrated how to do this using the suggested method. Ejegwa [
22] developed the technique of the max–min–max composite relation for
. The application of the enhanced composite relation for
in medical diagnostics was examined using a hypothetical medical database. Yager [
23] presented the novel notion of a q-rung orthopair fuzzy set (
) in 2017. Both the intuitionistic fuzzy set and the
are generalized versions of these. The entire sum of the qth powers of supporting and non-supporting is to range [0,1] in the
. As q increases, a wider variety of valid orthopairs become available, allowing for a conceptually much more expansive discussion of the membership score.
By utilising various significant tools, Ali [
24] restructured the
. He also described the fundamental algebraic operations under action of widely used in classification problems of
and presented the orbit-based
with the aid of illustrations and examples. If supporting score
and non-supporting score
is to bounded by 1,
:
. For
, Peng [
25] investigated the connection between the measurements of inclusion, likeness, distance, and entropy. He also demonstrated the validity of the similarity measure, which was then used for pattern recognition, density estimation, and clinical issues. Wang et al. [
26] developed the ten similarity measures by examining at the roles of belonging degree, not belonging degree, and indeterminacy belonging degree among the
based on cotangent and the traditional cosine similarity measurements. Additionally,
were employed to handle multi-objective decision. The
subgroup was a novel idea introduced by Asima and Razaq [
27], who also developed several significant findings. We expand on our examination of
by creating a new notion for
subfield and by establishing some fresh findings under its influence. The
is capable of solving a wide range of field theory issues. For their next research projects, mathematicians will find this theory helpful.
In this article, we present the
subfield. In
Section 2, we illustrate some fundamental mathematical properties of the
subfield. In
Section 3, we initiate the novel concepts of the
subfield and discuss its criteria and basics properties. Also, we show that every Pythagorean fuzzy subfield is a
subfield of certain field. Moreover, we establish some important basic theorems of
subfield and an example is provided to demonstrate the suitability and effectiveness of the initiated model. In
Section 4, we establish the images and pre-images of
subfield under field homomorphism. In
Section 5, we bring our proposed ideolody to a strong conclusion.
3. The q-Rung Orthopair Fuzzy Subfield
In this section, we define the subfield, and some fundamental algebraic attributes of the subfield are examined.
Definition 7. Assume that is a field; then, a of is known as a subfield of if the given axioms hold:
- 1.
;
- 2.
, for all , where 0 is an additive identity;
- 3.
;
- 4.
;
- 5.
;
- 6.
.
Definition 8. Suppose that is a fuzzy subfield of field for any i and in , is defined by for all u in F, and in F is called a fuzzy (i, c) co-set of .
Definition 9. Let be a fuzzy subfield of field and ; then, , order of V, is defined as , where 0 is an additive and 1 is a multiplicative identity element of .
Theorem 1. Let be a subfield of ; then, the following axioms hold:
- 1.
and for all ;
- 2.
and , ;
- 3.
and for all ;
- 4.
and for all .
Proof. Suppose that
, then
and for all . Therefore, and , which implies that and . Thus, and for all
□
Theorem 2. Let be a subfield of ; then, the following axioms hold:
- 1.
gives and gives for all ϑ and
- 2.
gives and gives for all ϑ and , where 0 and 1 are additive and multiplicative identity elements, respectively, in .
Proof. Suppose that , and are the additive and multiplicative identity elements, respectively, in
□
Example 2. Let be a field, where and is not a Pythagoran fuzzy subfield over , defined asandClearly, L is a q-rung orthopair fuzzy subfield of for but it is not a pythagorean fuzzy subfield of as Theorem 3. If is a subfield of field if and only if
- 1.
and for all ,
- 2.
and for all .
Proof. Let
E be a
subfield of field
and all
.
Conversely, if
,
,
and
Replace
by
; then, we obtain
and
for all
Then,
Hence,
E is a fuzzy subfield of field
□
Theorem 4. Suppose that is a field and is a subfield of ; then, is a subfield of .
Proof. Assume that
; then,
Furthermore,
Thus,
and
Therefore, for all
, using
,
and
. So,
These above expressions show that
M is a
subfield of
. □
Theorem 5. of is a subfield of if and only if and for all
Proof. Let be a subfield of . Then for all , and . Conversely, assume that and for all Then, . Therefore, . Similarly, . Therefore, . Next, , which means that . In similar fashion, we have . Then, obviously, and also hold. This implies that . Therefore, the expressions above demonstrate that is a subfield of . □
Theorem 6. Let and be two subfields of Then, is a subfield of .
Proof. Let
and
be two
subfields of
. Then for all
, we have
Moreover,
Then,
Similarly, we have
Now, we conclude that
for all
Hence, this conclude the proof that
is a
subfield of
. □
Theorem 7. Let be a subfield of . Then, for all and
Proof. We employ the mathematical induction technique to demonstrate this theorem. Let ; then, . As a result, the inequality is true for . Assume that the inequality holds for ; we have . Then, . From mathematical induction, we have for all In a similar fashion, . The result holds for ; so, we suggest that it holds true for such that . Then, . Now, we conclude this result for all Moreover, . Then, . Now, we have for all Similarly, we obtain . In the same fashion, we obtain the following result: . Then, . By using mathematical induction, we have for all □
Theorem 8. Let be a subfield of . If and for some , then and .
Proof. To obtain the desired result, we take arbitrary entities
as
; then, obviously,
. Consider
Since
, we conclude from Equation (
5) that
by Equations (
6) and (8), we obtain
Moreover, we obtain the results if
.
Since
, from Equation (
10), we have
by Equations (
11) and (12) we obtain
□
Theorem 9. Let be a subfield of field . Then,
- 1.
, and then, for all .
- 2.
If , then .
Proof. Let and be in .
Suppose that
; then,
Therefore, .
Moreover,
Therefore,
.
In similarly way,
Hence, this illustrates the proof.
□
Theorem 10. Let 0 and e represent the identity element of with respect to addition and multiplication, respectively, and be a subfield of . Therefore,
- 1.
If for some , then for all ;
- 2.
If for some , then for all ;
- 3.
If for some , then , for all ;
- 4.
If for some , then for all
Proof. Assume that
is a
subfield of
. We have
.
As
from Inequality (
13), we obtain
Thus,
. We have
From Equations (
14) and (
15), we obtain
Now,
Because
form the Inequality (
16), we have
Thus,
; then, we have
From Equations (
17) and (
18), we have
As
from the Inequality (
19), we conclude that
Thus,
. We have
From Equations (
20) and (
21), we obtain
□
Theorem 11. Suppose that 0 and 1 represent additive and multiplicative identity elements of , respectively, and is a subfield of . Then, is a subfield of .
Proof. We know that
, so
is a non-empty subset of
. Suppose
. Then,
From Theorem 1, we have
. Then, obviously, we have
. Now, we show that
for
Then, it is clear that
.
Furthermore,
In a similar fashion,
From Theorem 1, we obviously establish
for all
. Hence, this concludes the proof. □
4. Homomorphism on q-Rung Orthopair Fuzzy Subfield
In this part, the impact of field homomorphism on the subfield is investigated, which provides some fundamentally major findings under field homomorphism.
Theorem 12. Suppose that and are two subfields. Let δ be a surjective homomorphism from to and is a subfield of . Then, is a subfield of .
Proof. Let
be an onto homomorphism; then,
. Let
be entities of
. Suppose
,
and
.
is a subfield of . □
Theorem 13. Let is a bijective homomorphism and is a subring of such that is a subfield of
Proof. Consider
, then
. Now,
Similarly in case of non-membership,
In other words,
Let again
then
□
Theorem 14. If is surjective homomorphism, , are two subfields and be ideal of then is ideal of .
Proof. Suppose
be
ideal of
, then
and
for all
. Let
. Then there exist some elements
for this
. Then,
In the same way, we can show that and Hence, all the axiom of ideal are hold. So, is ideal of . □
Theorem 15. Let , are two subfield. Let δ be bijective homomorphism and be ideal of . Then is a subfield of .
Proof. be
ideal of
then
and
for all
. Suppose that
.then there exist some elements
for this
. Let
be elements of
. Suppose
,
and
.