1. Introduction
Rough set theory is asserted by Pawlak [
1] as a tool to search structural relationships within uncertain and imprecise data. Pawlak refers to a pair consisting of a universal set and an indiscernibility relation (equivalence relation) on this set as a rough approximation space. The lower and upper approximations are definable subsets of the universal set. They can characterize any subset of the universal set, reveal hidden patterns within information systems and express these patterns as decision rules. The lower approximation contains objects that definitely belong to the subset, while the upper approximation contains objects that possibly belong to the subset. The difference between the upper and lower approximations is called the boundary region. If the boundary region is empty, the subset is crisp; otherwise, it is rough. As fundamental concepts, the approximations in Pawlak’s approximation space are constructed via an equivalence relation.
Because of the various restrictions of equivalence relations, many extensions have replaced equivalence relations with notions such as neighborhood systems, general binary relations, covering-based systems, Boolean algebras, etc. [
2,
3,
4]. Yao [
5,
6] started to redefine the approximation operators and their related notions from neighborhoods. Since the neighborhoods are very efficient at mitigating uncertainty and address a variety of practical applications, researchers have been interested in the novel sorts of granular computing derived from rough neighborhoods [
7,
8,
9,
10,
11]. Examining a particular problem choice of hypothesis is crucial in mathematical research. To start under some conditions enhances the theory’s applicability to real-world problems. It leads to more precise computations and improves decision making and data analysis efficiency. However, generalizations are to develop stronger theorems and address a broad spectrum of problems.
Fuzzy set theory is a theory on uncertainties through gradation. Since its introduction by Zadeh in 1965 [
12], it has complemented numerous theories examining datasets with uncertainties and has been the subject of extensive research. Considering fuzzy relations instead of crisp binary relations, Dubois and Prade [
13] introduced fuzzy rough sets. As one of the various fuzzy generalizations of rough approximations, Wu et al. [
14], following Radzikowska and Kerre [
15], redefined
-fuzzy rough sets, where I is an implication and T is a t-norm. Recent research includes studies on approximation operators defined by fuzzy neighborhoods using fuzzy coverings. In 2022, Yang extended fuzzy coverings to two universal sets [
4].
Algebraic structures are the fundamental concepts of mathematics that have a significant role in disciplines such as information science, theoretical physics, computer science, engineering, coding theory, etc., showcasing their broad applicability. This situation motivates researchers to analyze algebraic aspects of the various extensions of theories modeling uncertainties. Following Zadeh’s [
12] introduction to fuzzy sets, Rosenfeld [
16] pioneered the study of fuzzy subgroups within group theory. This initial groundwork has encouraged significant interest in extending abstract algebraic concepts to the fuzzy setting [
17,
18]. As group theory is one of the algebraic structures widely used and discussed in mathematics and its applications, we focus on an extension of group theory in this paper.
A hyperbinary operation, which generalizes binary operations, is a function defined on the Cartesian product of a non-empty set with itself, taking values in the non-empty subsets of that set. A non-empty set equipped with a hyperoperation is called a hyperstructure or a hypergroupoid. By satisfying some properties, hyperbinary operations can define other algebraic hyperstructures such as hypersemigroups, hypergroups and hyperrings. After Marty [
19] initiated the concept of the hypergroup, both the theories and applications of the hyperstructures have made achievements in pure and applied mathematics and computer science [
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30]. A canonical hypergroup is a natural generalization of the concept of an abelian group [
31,
32]. The class of hypergroups compasses a broader range of structures from groups, making the investigation of fuzzy hypergroups particularly compelling [
25,
31]. Numerous extensions of this topic to hyperalgebraic structures are available in the literature [
23,
25].
Algebraic concepts of rough set theory include algebraic structures. Rough sets have various compatible extensions on algebraic structures. Biswas and Nanda [
33] proposed the rough subgroups as the first rough application of algebra. Kuroki [
34] introduced rough ideals in semigroups using congruences instead of equivalence relations. Mordeson applied rough sets to rings [
35]. Rough sets were extended to two universal sets by Yao in 1996 [
5]. Davvaz applied rough sets, extended to two universal sets, to groups using his proposed set-valued homomorphisms [
36]. Yamak et al. applied these concepts to rings and modules [
37,
38]. Rough sets have extensions on algebraic structures with fuzzy equivalence relations in fuzzy sets. Li et al. (2007) studied TL-fuzzy rough ideals in rings [
39]. Li and Yin (2012) defined the
-lower and
-upper fuzzy rough approximation operators on semigroups [
40]. Wu et al. [
14] extended fuzzy rough sets to a two-universe
-fuzzy model, then Ekiz et al. applied
-
-fuzzy rough sets to semigroups, groups and rings as an extension of
-fuzzy rough sets to lattices [
41,
42,
43]. Some extensions of rough set theory were studied on hyperstructures [
44,
45].
This paper discusses applying
-fuzzy rough sets as a basic rough set model to hypergroups, extending beyond their application to groups. The importance of this paper is that it proposes a method for extending
-fuzzy rough sets to hyperalgebraic structures. In this paper, we introduce a concept of fuzzy set-valued homomorphisms as an extension of the fuzzy congruences of hypergroups [
23] and examine some of their features. While fuzzy congruence is a kind of fuzzy relation on a single hypergroup, fuzzy set-valued homomorphism is a kind of fuzzy set-valued mapping from a hypergroup to the fuzzy sets of another hypergroup. Herein, we generate fuzzy set-valued homomorphisms from fuzzy congruences. This paper extends Ameri’s study [
23] to two different hypergroups as universal sets. In this way, it becomes possible to apply the results of [
43] for hypergroups and investigate them in this broader context. In this paper, we also constitute the generalized fuzzy rough approximations by a fuzzy set-valued homomorphism on canonical hypergroups and examine some related properties. Some other similar research topics may be to define the fuzzy set-valued homomorphisms in various hyperalgebraic systems such as semihypergroups and hyperrings and analyze their properties. In this way, extensions of other hyperalgebraic structures may be possible in which fuzzy set-valued homomorphies can apply to the generalized fuzzy rough structures. For instance, [
24,
27,
28] can be extended in a fuzzy sense and [
40,
41,
42] in a hyper sense.
2. Preliminaries
Here, we provide a concise overview of the essential notions and preliminary evidence necessary for the subsequent results.
2.1. Fuzzy Logical Operators
Let
. Then
denotes
and
denotes
. On the unit-interval
, an associative and commutative binary relation
is a triangular norm (or
-norm) if it is an increasing function and satisfies the boundary condition
for all
. The minimum
-norm
and the drastic product
-norm
, respectively, are:
Let be a -norm. Then, for all , . denotes the set .
On the unit-interval , an associative and commutative binary relation is a triangular conorm (or -conorm) if it is an increasing function and satisfies the boundary condition for all . The maximum operator is the smallest -conorm on [0, 1].
A decreasing mapping
is a negator if it is providing the conditions
. The negator
for all
is the standard negator. An implication is a binary relation on
which is decreasing with respect to the first component, increasing concerning the second component and providing the conditions
. An
-implication based on a
-conorm
and a negator
is the implication
for all
, and
-implication (residual implication) based on a
-norm
is the implication
for all
(see [
40,
46,
47]).
2.2. Fuzzy Sets and Relations
Let and be non-empty sets as universes of discourses and .
A fuzzy subset of
is any function from
into
(see [
12]).
and
, respectively, denote the class of all subsets and fuzzy subsets of
.
denotes all of the non-empty subsets of
. Let
. Then the set
is
-cut (or level) subsets of
. Let
. Then
is a fuzzy subset of
with the value
if
and
elsewhere. Then
is called the characteristic function of
.
Let , and be a function. Then and are and . For any fuzzy subset , is a fuzzy subset of such that for all .
Let and be a -norm, -conorm and an implication, respectively. Then and for all .
A fuzzy subset is a fuzzy relation from to . is the degree of relation between and , where . If , then is a fuzzy relation on . Let be a function and . Then denotes a fuzzy set of .
Let and be functions. Then the fuzzy set-valued mappings and , respectively, are and for all . The set-valued function is for all .
The
- and
-compositions of the fuzzy set-valued functions
and
are the fuzzy set-valued functions
and
;
for all
(see [
17]).
Let and be a set-valued mapping. Then, in this paper, and are referred to as connected, if for all .
2.3. Generalized Fuzzy Rough Approximation Spaces
Let
be a set-valued mapping,
be a
-norm and
be an implication on the unit-interval. For any fuzzy subset
of
, the
-upper and
-lower fuzzy rough approximations of
are fuzzy sets of
,
and
, respectively. Their membership functions are
The triple
is a generalized fuzzy approximation space. The operators
and
from
to
are
-upper and
-lower fuzzy rough approximation operators of
, respectively. The pair
is a
-fuzzy rough set of
with respect to
(see [
14,
40,
41,
42,
43]). If
is a fuzzy relation on
, then
is a fuzzy approximation space.
2.4. Hypergroups
An algebraic hyperstructure (or hypergroupoid) is a non-empty set together with hyperoperation, i.e., a map .
A hyperstructure is a semihypergroup if, for all , . means and means . If and , then , and .
is a subsemihypergroup of if and in this case is a supersemihypergroup of .
Let be a semihypergroup. Then is a hypergroup if it satisfies the reproduction axiom: . An element in a semihypergroup is identity if, for all , . An element in a semihypergroup is zero element if .
A semihypergroup is a canonical hypergroup if
- (i)
for all ,
- (ii)
there exists such that for all ( is scalar identity),
- (iii)
there exists a unique element for any such that ( is the opposite of and denotes it),
- (iv)
implies and for every .
A canonical subhypergroup of a canonical hypergroup
is a non-empty subset
of
containing the scalar identity of
if it forms a canonical hypergroup under the hyperoperation
on
(see [
19,
20,
25,
31,
32,
48]).
Example 1. Let be a non-empty set.
- (i)
Let be an abelian group and a mapping be defined by for all . Then is a canonical hypergroup.
- (ii)
If is an abelian group and is an equivalence relation in which has classes , then for all , we define . Then is a canonical hypergroup [32].
Example 2. Let with the following multiplication table (Figure 1 ): Then is a canonical hypergroup with the scalar element .
Definition 1 ([
22,
25,
29])
. Let be hypergroups. Then a function is- (i)
an inclusion homomorphism if for all ;
- (ii)
a strong homomorphism (or good homomorphism) if for all .
2.5. T-Fuzzy Subhypergroups
Definition 2. Let be a canonical hypergroup and be a -norm. Then a fuzzy subset of is a -fuzzy subhypergroup of if the following conditions hold:
- (i)
,
- (ii)
for all (see [23,49]).
Definition 3 ([
23])
. Let be two fuzzy subsets of semihypergroup . Then is the fuzzy subset of , Definition 4 ([
23])
. Let be a canonical hypergroup. Then a fuzzy binary relation on is for all .
- (ii)
Fuzzy strong compatible if
for all .
2.6. Set-Valued Homomorphisms of Canonical Hypergroups
Definition 5 ([
28])
. Let and be canonical hypergroups. Then a mapping from which preserves the operations and for all is called a set-valued homomorphism of canonical hypergroups. A set-valued homomorphism of canonical hypergroups is called strong if for all .
3. T-Fuzzy Set-Valued Homomorphisms of Canonical Hypergroups
In this section, we introduce the concept of fuzzy set-valued homomorphism of hypergroups and demonstrate some of their features.
Definition 6. Let be two fuzzy subsets of the semihypergroup and be a -norm. Then we define the fuzzy subset of byfor all . Clearly is a binary relation on and is called the -product of and . If is a canonical hypergroup, then we have since for all .
Definition 7. Let be canonical hypergroups and be a mapping. Then
- (i)
is a -fuzzy set-valued homomorphism of semihypergroups if, for all , .
- (ii)
is a strong -fuzzy set-valued homomorphism of semihypergroups if, for all , .
- (iii)
is a completely strong -fuzzy set-valued homomorphism of semihypergroups if, for all , .
denotes the set of all the -fuzzy set-valued homomorphisms, denotes the set of all the strong -fuzzy set-valued homomorphisms, and denotes the set of all the completely strong -fuzzy set-valued homomorphisms from to . If , then a -fuzzy set-valued homomorphism is called a fuzzy set-valued homomorphism and the set of all the fuzzy set-valued homomorphisms from to is denoted by . If , then the statement also applies analogously to and .
Definition 8. Let and be canonical hypergroups and be a mapping satisfying for all . Then
- (i)
is a -fuzzy set-valued homomorphism of canonical hypergroups.
- (ii)
is a strong -fuzzy set-valued homomorphism of canonical hypergroups.
- (iii)
is a completely strong -fuzzy set-valued homomorphism of canonical hypergroups.
Example 3. Let be a Klein- group and the canonical hypergroup in Example 1 (i) and let be the canonical hypergroup in Example 2. Then defined by Figure 2 is a completely strong -fuzzy set-valued homomorphim of canonical hypergroups but it is not a -fuzzy set-valued homomorphism since Example 4. Let and be canonical hypergroups and . are defined by the following:
Then .
Proposition 1. Let be canonical hypergroups and . Then for all .
Proof. Straightforward. □
Theorem 1. Let be an inclusion homomorphism of semihypergroups. Then if .
Proof. Let and let for any . Then there exist such that and , and . Thus and . So since is an inclusion homomorphism. There exist such that . Then . Thus . □
Example 5. Let be the canonical hypergroup in Example 2 and be defined as for all . Then is an inclusion homomorphism of semihypergroups and the mapping is a -fuzzy set-valued homomorphism but is not strong since
Theorem 2. Let be a set-valued homomorphism, and . Then defined by for all is a -fuzzy set-valued homomorphism.
Proof. Let
and let
for any
. Then there exists
such that
and
, and
. Thus
and
. Hence
since
is a set-valued homomorphism. Thus, there exists an
such that
. Then
Hence is a -fuzzy set-valued homomorphism. □
The following example demonstrates that the fuzzy set-valued homomorphism obtained by the method in Theorem 2 may not be strong.
Example 6. Let be the canonical hypergroup in Example 2 and be defined as for all . Then is a strong set-valued homomorphism and the mapping defined by for all is not a strong -fuzzy set-valued homomorphism since The following theorem shows that the fuzzy congruences are a special type of the set-valued homomorphisms.
Theorem 3. Let be a semihypergroup, be a fuzzy (strong) compatible on and be the connected T-fuzzy set-valued mapping. Then is a (strong) -fuzzy set-valued homomorphism.
Proof. Let
. Since
is fuzzy compatible, then, for any
, we have
Hence
is a
-fuzzy set-valued homomorphism since
for all
. Let
be a fuzzy strong compatible on
. Then, for any
, we have
Hence
is a strong
-fuzzy set-valued homomorphism since
for all
. □
Example 7. - (i)
in Example 5 is a -fuzzy set-valued homomorphism. Moreover, its connected fuzzy relation is fuzzy compatible. However, it is not strong fuzzy compatible since it is not a strong -fuzzy set-valued homomorphism.
- (ii)
Even though the connected fuzzy relation of in Example 6 is fuzzy compatible, is not a strong -fuzzy set-valued homomorphism.
Theorem 4. Let and be canonical hypergroups and let be a fuzzy set-valued mapping and be the set-valued mapping defined by for all .
- (i)
If and , then is a set-valued homomorphism.
- (ii)
If is completely strong, then for all .
- (iii)
If is a set-valued homomorphism for all , then .
Proof. Let .
- (i)
Let . Then there exist and such that . Hence and . Thus and . So since . Therefore, . So since . Hence there exists an element such that . Thus . So since . Thus is obtained. Let . Thus and so . Hence . So . Since . Hence . Thus . So . Therefore is obtained. Finally, is a set-valued homomorphism.
- (ii)
Let . Then for all . Hence . So . Therefore . Thus since is completely strong. Then . There exist such that and . Thus . Hence and . Then and . So since . Finally, for all .
- (iii)
Let and let for any such that . Thus and . Hence and . So . Since is a set-valued homomorphism, then . Hence . Thus there exists an such that . Then . Hence, is obtained. Thus, we have . Therefore is verified. Let and be . Then . So . Hence . Thus . Hence . Since is a set-valued homomorphism, then . Hence , i.e., . Thus . Therefore . Finally, is obtained. □
Theorem 5. Let be canonical hypergroups, and . Then if and .
Proof. Let
and
. We have
Theorem 6. Let be canonical hypergroups and and be inclusion homomorphisms.
- (i)
If , then .
- (ii)
If , then .
Proof. Let and . Thus
Definition 9. Let be canonical hypergroups and . Then
- (i)
the kernel of is a fuzzy subset of denoted by such that for all .
- (ii)
the image of is a fuzzy subset of denoted by such that for all (see [43]).
Theorem 7. Let . Then is a -fuzzy subhypergroup of if is strong.
Proof. Let
. Then
is a T-fuzzy subhypergroup of
since
□
Theorem 8. Let . Then is a -fuzzy subhypergroup of .
Proof. Thus is obtained and clearly . Then is a -fuzzy subhypergroup of . □
Theorem 9. Let be canonical hypergroups and . Then is a -fuzzy subhypergroup of .
Proof. Also, since . Thus is a -fuzzy subhypergroup of . □
Definition 10. Let be semihypergroups be a function and and be set-valued mappings. Then the set-valued function isand the set-valued function is for all and .
Theorem 10. Let be a good homomorphismand , be set-valued homomorphisms. Then:
- (i)
is a set-valued homomorphism if is a bijection.
- (ii)
is a set-valued homomorphism.
Proof. - (i)
Let . Then and . Hence
where
,
. Thus
is a set-valued homomorphism.
- (ii)
Let . Then
where
such that
since
. Hence
is a set-valued homomorphism. □
Theorem 11. Let define for all . Then if .
Proof. Let
and
. Then
We have
. We also have
since
Therefore .
The following example shows that the above theorem may not be true for some implications. □
Example 8. Let be the canonical hypergroup in Example 2 and be in Example 4. Let be an -implicator based on . Then since .
Theorem 12. Let . Then we have
- (i)
- (ii)
Proof. - (i)
Let
and
. Then we have
since
- (ii)
It is proved similarly to part (i). □
4. Generalized Fuzzy Rough Approximations on Canonical Hypergroups
In this section, some properties provided for groups [
43] are tested for hypergroups. Thus, we investigate some features of the generalized fuzzy rough approximation spaces that occurred by a
-fuzzy set-valued homomorphism of canonical hypergroups.
Theorem 13. Let . Thenfor all if .
Proof. Example 9 shows that Theorem 13 may not be true for the -lower fuzzy rough approximations. □
Example 9. Let and be considered as in Example 3, be considered as in Example 4, with the values and , and let fuzzy subhypergroups and of be taken as follows (Figure 3):
Then we have
where
is the
-implication of
. Moreover, we have
where
is the
-implication based on the standard negator.
Corollary 1. . If is a -fuzzy subsemihypergroup of , thenfor all .
Proof. Let
. Then, we have
for all
by Theorem 13. Therefore, we have
since
. □
Theorem 14. Let . If is a -fuzzy subhypergroup of , then is a -fuzzy subhypergroup of .
Proof. We have
for all
by Corollary 34. Let
. Then
Hence is a -fuzzy subhypergroup of . □
Theorem 15. Let . Then if .
Proof. Let
. Then
since
Example 10 shows that Theorem 14 and Theorem 15 may be false for the -lower fuzzy rough approximations. □
Example 10. Let be considered as in Example 2, , be the -implication of and be defined asfor all . Then . Let be considered as in Example 9. Hence since . However, even though is a -fuzzy subhypergroup of , is not a -fuzzy subhypergroup of since .
Theorem 16. Let be canonical hypergroups, be an implication and let and for a -norm . Then
- (i)
for all ,
- (ii)
,
- (iii)
,
- (iv)
,
- (v)
,
- (vi)
,
- (vii)
,
- (viii)
,
- (ix)
.
Proof. Straightforward. □
Theorem 17. Let a bijection be a good homomorphismand be a set-valued homomorphism. Then Proof. Let
. Then
. Hence
and
where
. □
Theorem 18. Let be a mapping, and be a set-valued homomorphism. Then:
- (i)
.
- (ii)
.
- (iii)
and if is surjective.
Proof. - (i)
- (ii)
- (iii)
It follows immediately from (i) and (ii). □
Theorem 19. Let . Then and if .
Proof. Straightforward. □
Theorem 20. Let . Then and .
Proof. Straightforward. □
Theorem 21. Let and . Then
- (i)
,
- (ii)
if is an -implication based on the -norm .
Proof. Let . Then we have