1. Introduction
Vagueness in many real-life problems cannot be approached by the use of classical sets. In 1965, Zadeh [
1] generalized the classical set by introducing the fuzzy set. In a fuzzy set, an element’s membership is a non-negative real number that can attain one as a maximum value. The new concept was studied by many scholars; in 1983, it was extended by Atanassov [
2] to the intuitionistic fuzzy set (IFS). In an IFS, an element has membership and non-membership grades that are real numbers in the real unit interval where the sum is, again, in the real unit interval. Since then, many extensions of fuzzy sets have been proposed and applied to different real-life problems. For example, Yager [
3] introduced Pythagorean fuzzy sets. In 2019, Riaz and Hashmi [
4] generalized fuzzy sets into linear diophantine fuzzy sets (LDFS). Some related work can be found in [
5,
6].
In 1971, Rosenfeld [
7] pointed to a link between fuzzy sets and algebraic structures. He introduced the concept of fuzzy subgroups of a group and studied some of their properties. This work was important in the field of mathematics as it introduced a new field of research: fuzzy algebraic structure. After that, fuzzification of almost all algebraic structures was introduced. In particular, fuzzy sets of fields and vector spaces were studied [
8,
9]. Fuzzy algebraic structures were generalized to linear diophantine fuzzy algebraic structures, and they were first studied in 2021 [
10] by Kamaci. He studied LDFSs of different algebraic structures, such as groups, rings, and fields. Other scholars applied LDFSs to different algebraic structures [
11,
12,
13]. For more details, we refer to [
14,
15,
16]. Several different approaches to other fuzzy set extensions can be found in the literature. Molodtsov [
17] dealt with uncertainties occurring in natural and social sciences and initiated the soft set theory. In an attempt to study different existing mathematical structures in the context of the soft set theory, the notion of a soft point plays a significant role [
18]. The results in the field of fuzzy structures are enormous and the literature related to this issue is extensive. For illustration, let’s say for example [
19,
20,
21].
This paper discusses LDFSs of fields and vector spaces, and the remaining part of it is organized as follows. In
Section 2, we present some basic concepts related to LDFSs that are used throughout the paper. Our main results are presented in
Section 3 and
Section 4. In
Section 3, we introduce the notion of LDF subfields of a field; we investigate some of the properties that are essential in
Section 4, where we introduce the concept of a LDF subspace of a vector space. Moreover, we present some examples and highlight some properties. Moreover, we discuss the relationship between LDF subspaces and subspaces through level subsets. Finally, by starting with a LDF subspace
D of a vector space, we describe a LDF subspace of the quotient vector space
.
3. LDF Subfields of a Field
The concept of LDF subfields of a field was introduced in [
10]. In this section, we elaborate some properties of this concept that are used in
Section 4.
Definition 6. Let K be a field and F a LDFS of K. Then F is a LDF subfield of K if the following conditions hold for all .
- (1)
;
- (2)
;
- (3)
;
- (4)
.
Proposition 2. Let K be a field and F a LDF of K. Then the following statements are true.
- (1)
.
- (2)
.
- (3)
for all .
- (4)
for all .
- (5)
.
Proof. The proof is straightforward. □
Theorem 1. Let K be a field and F a LDFS of K. Then F is a LDF subfield of K if and only if the following conditions hold for all .
- (1)
for all ;
- (2)
for all .
Example 1. Let be the field of real numbers and D be the LDFS of defined as follows. One can easily see that D is a LDF subfield of .
Theorem 2. Let K be a field and F a LDFS of K. Then F is a LDF subfield of K if and only if every non-void level set of F is either or a subfield of K.
Proof. Let F be a LDF subfield of K and . Having implies that and having implies that . Let , then as .
Let . We show that the conditions of Definition 6 hold. If , we are done. Without loss of generality, suppose that and that , having implies that and, hence, and . Moreover, having and , a subfield of K implies that . Thus, and . □
Corollary 1. Let K be a field and F a LDFS of K with . Then F is a LDF subfield of K if and only if every non-void level set of F is a subfield of K.
Proof. The proof follows from Theorem 2. □
Corollary 2. Let p be a prime number and be the field of integers modulo p. Then F is a LDF subfield of if and only if it has one of the following forms, or , where for all and for some . Here, satisfying and for all .
Proof. The proof follows from Theorem 2 and with no proper subfields. □
4. LDF Subspaces of a Vector Space
In this section, we introduce and study the concept of the LDF subspaces of a vector space.
Definition 7. Let be a vector space over a field K, D a LDFS of , and F a LDF subfield of K. Then D is a LDF subspace of if the following conditions hold for all .
- (1)
;
- (2)
.
Example 2. Let be a vector space over a field K and let be any constant LDFSs of , respectively. Then D is a LDF subspace of .
Example 3. Let K be a field. By considering K as a vector space over K, then every LDF subfield of K is a LDF subspace of K.
Proposition 3. Let be a vector space over a field K, D a LDF subspace of , and F a LDF subfield of K. If and , then .
Proof. We have that . Since , it follows that . In a similar manner, we can prove that . Proposition 1 completes the proof. □
Example 4. Let be the field of real numbers and let be the vector space of all couples of real numbers over K. Define the LDFSs of as follows. One can easily see that F is a LDF subfield of K and D is a LDF subspace of .
Proposition 4. Let be a vector space over a field K, D a LDFS of , and F a LDF subfield of K. If D is a LDF subspace of , then the following condition holds for all . Proof. The proof is straightforward. □
Theorem 3. Let be a vector space over a field K, LDFSs of , and F a LDF subfield of K. Then is a LDF subspace of .
Proof. The proof is straightforward. □
Remark 2. Let be a vector space over a field K, LDFSs of , and F a LDF subfield of K. Then is not necessarily a LDF subspace of . We present an illustrative example.
Example 5. Let be the field of integers modulo 2 and let be the vector space over consisting of all couples with entries from . Define the LDFS F of as and the LDFSs of are defined as follows: It is clear that are LDF subspaces of . We show that is not a LDF subspace of . This is obvious as but .
In [
9], Kumar proved that for a fuzzy set
on a non-empty set
X with
, if
then
or
.
Example 6. Let be the field of integers modulo 2 and let be the vector space over consisting of all couples with entries from . Define the LDFS F of as and the LDFSs of is defined as follows. For all , Then for all . Thus, is a LDF subspace of but and .
Theorem 4. Let be a vector space over K, F a LDF subfield of K with , and D a LDF subspace of . For all with , is a subspace of over the subfield .
Proof. We have that and as and . Let and . Having D as a LDF subspace of implies that . The latter implies that . □
Corollary 3. Let be a vector space over K, F the LDF subfield of K with for all , and D a LDF subspace of . For all , is a subspace .
Proposition 5. Let be a vector space over K, F a LDF subfield of K, and D a LDFS of . If for all then D is a LDF subspace of if and only if for all .
Proof. Let D be a LDFS of . Proposition 4 asserts that as for all .
Let for all . By setting , we have . Having implies that . □
In what follows, our results are based on the condition of Proposition 5.
Proposition 6. Let be a vector space over K and D a LDF subspace of . Then for all .
Theorem 5. Let be vector spaces over a field K and be LDF subspaces of , respectively. Then is a LDF subspace of .
Proof. The proof is straightforward. □
Definition 8. Let be a vector space over a field K and A a subspace of . We define the characteristic function D of A as follows: Theorem 6. Let be a vector space over a field K and A a subset of . Then A is a subspace of if and only if its characteristic function is a LDF subspace of .
Proof. Let A be a subspace of , , and . We consider the following cases for and . If then . If then or . The latter implies that and, hence, the characteristic function D is a LDF subspace of .
Let with characteristic function D, , and . Then . Since A is a subspace of , it follows that . The latter implies that and, hence, A is a subspace of . □
Next, we present some results related to level subspaces.
Proposition 7. Let be a vector space over a field K, D a LDFS of and . Then .
Proof. The proof is straightforward. □
Theorem 7. Let be a vector space over a field K and D a LDFS of . Then D is a LDF subspace of if and only if is a subspace of for all .
Proof. Let D be a LDF subspace of , , and . Then and, hence, is a subspace of as .
Let with . Then . Having a subspace of implies that and, hence, . □
Proposition 8. Let be a vector space over a field K and A a subspace of . Then A is a level subspace of a LDF subspace of .
Proof. Let
A be a subspace of
and
D be the LDFS of
, defined as follows.
where
,
,
,
. One can easily see that
D is a LDF subspace of
and that
. □
Next, we discuss the LDF subspaces of the quotient vector space .
Definition 9. Let be a vector space over a field K and D a LDFS of . Then .
Lemma 1. Let be a vector space over a field K and D a LDF subspace of . For , if then .
Proof. Let . Then by Proposition 6. On the other hand, having implies that . Thus, . □
Lemma 2. Let be a vector space over a field K and D a LDF subspace of . Then is a subspace of .
Proof. Let and . Then . The latter implies that and, hence, is a subspace of . □
Corollary 4. Let be a vector space over a field K and D a LDF subspace of . Then is a vector space over .
Proof. The proof follows from Lemma 2. □
Proposition 9. Let be a vector space over a field K, , and D a LDF subspace of . Then .
Proof. The proof follows from having for all . □
Theorem 8. Let be a vector space over a field K and D a LDF subspace of . Then is a LDF subspace of . Here, is the LDFS of defined as follows. Proof. Let D be a LDF subspace of . First, we show that is well-defined on . Let . Then and hence, . Lemma 1 asserts that . Thus, is well-defined. Let and . Then . Having D, a LDF subspace of implies that . The latter implies that and, hence, is a LDF subspace of . □
Example 7. Let be the field of integers modulo 2 and let be the vector space over consisting of all triples with entries from . Define the LDFS D of as follows. For , One can easily see that D is a LDF subspace of . Having and using Theorem 8, then is a LDF subspace of . Here, is defined as follows. Proposition 10. Let be a vector space over a field K, D a LDF subspace of , and . Then .
Proof. The proof is straightforward. □
Proposition 11. Let be a vector space over a field K and D a LDF subspace of . Then D is the constant LDFS of if and only if is the constant LDFS of .
Proof. If D is the constant LDFS of then it is clear that is the constant LDFS of .
Let be the constant LDFS of . Then for all . The latter implies that for all and, hence, D is the constant LDFS of . □