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Abstract
In this paper, we determined a new characterisation of the derived nilpotent (Engel) Lie ring of fuzzy hyperrings by fuzzy strongly regular relation (
). Moreover, we proved that for a fuzzy hyperring S, the quotient
(
) was a nilpotent (Engel) Lie ring. Also, we introduced the notion of an ζ-role of a fuzzy hyperring and investigated its essential properties. Basically, we stated a necessary and sufficient condition for transitivity of ζ. Also, we studied the relationship between the strongly regular relation and ζ-role of a given fuzzy hyperring.
1. Introduction
The notion of nilpotency is the most critical concepts in the study of groups [Citation1]. Nilpotent groups arise in Galois theory, as well as in the classification of groups. By Galois theory, specific problems in field theory replace with problems in group theory, that is in some sense better understood. The certain generalisation of nilpotent groups is Engel groups [Citation2–4].
Hyperstructure theory introduced by Marty 1934, he defined hypergroups, investigated their properties, and applied them to groups and relational algebraic functions (see [Citation5]). In 1971, Rosenfeld used the concept of fuzzy sets to introduce fuzzy groups(see [Citation6]). Since then, many researchers extended the concepts of abstract algebra to the fuzzy sets (see [Citation6–14, Citation19–21]). The main tool in the definition of fuzzy hyperstructures is the ‘fuzzy hyperoperation’, where the non-empty set replaces the nonzero fuzzy set. Vougiouklis studied the fundamental relations on hyperrings [Citation15]. In [Citation16], Nozari and Fahimi introduced fundamental relation and commutative fundamental relation on fuzzy hyperrings.
In this paper, we define the and
relations, which are defined in fuzzy hyperrings. These relations connect the fuzzy hyperrins with (Engel)nilpotent-Lie rings. Especially, we introduce the notion of an ζ-role of a fuzzy hyperring and investigate its basic properties. Moreover, we study the relationship between the strongly regular relation and ζ-role of a given fuzzy hyperring. Also, we obtain Engel Lie rings from fuzzy hyperrings, by a FSR.
2. Preliminaries
First we recall some definitions and theorems.
Definition 2.1
Let H be a non-empty set and be the set of all fuzzy subsets of H. Suppose
, where 0 is the zero fuzzy set. Then H equieped with a fuzzy hyperoperation
, is called a fuzzy semihypergroup if for any
, where for any
, and
,
If A, B are non-empty subsets of H and , then for any
we have
If a fuzzy semihypergroup
satisfies in the condition
, for any
, where
is the characteristic function of H, then the pair
is called a fuzzy hypergroup.
Definition 2.2
[Citation8]
A non-empty set , where + and · are two fuzzy hyperoperations, is called a fuzzy hyperring (we write by
) if for any
, the following axioms are valid:
,
,
,
a + b = b + a,
and
,
where for fuzzy subsets of a fuzzy hypergroupoid
we have
The fuzzy hyperring
is called commutative if for any
,
. Also,
is called a subfuzzy hyperring of S if for any
and
, the following axioms are valid:
if
, then
,
,
if
, then
.
Definition 2.3
[Citation14]
Let ρ be an equivalence relation on a fuzzy semihypergroup . For any
, we define two relations
and
on
as follows:
if for any
, there exists
such that
and if
, then
and if
, then
.
if every element
such that
is ρ equivalent to every
such that
.
Definition 2.4
[Citation14]
An equivalence relation ρ on a fuzzy semihypergroup is called a fuzzy (strongly) regular (we write by F(S)R) if
and
, then
Definition 2.5
[Citation16]
Consider be a FHR and φ be a F(S)R on both
and
. Then φ is called a F(S)R on S.
Let and
be two FHR and g be a map from S to
. Then for any
we have
Theorem 2.6
[Citation16]
Let be a FHR and φ be an equivalence relation on S. If for every
,
then
(i) | the relation φ is a FR on | ||||
(ii) | the relation φ is a FSR on |
Definition 2.7
[Citation16]
Let be a FHR. A (commutative) fundamental relation on S is the smallest equivalence relations such that the quotient structure
is a (coommutative) ring.
Assume is a FHR and relation γ on S is defined as follows:
Then
, the transitive closure of γ, is a fundamental relation on S (See [Citation16]).
We recall that for a set X, a one to one function from X onto X is called a permutation on X. We denote the set of all permutations of X by . If
, then we write
instead of
.
forms a group.
is the symmetric group on X (see [Citation1]).
Consider S be a FHR. Nozari and Fahimi in [Citation16], introduced the relation ϵ on S as follows:
where
. The quotient
, where
is the transitive closure of ϵ, is a commutative ring (see [Citation14]).
Definition 2.8
[Citation17]
Let be a vector space over a field F. Consider the operation
defined by
. Then L is called a Lie algebra if the following axioms are satisfied:
.
Also, is called a Lie ideal if
for any
and
.
Let be a ring. We can introduce the Lie structure on R by defining the Lie product
for
. This Lie structure, denoted by
, is called the associated Lie ring of R. Also, the Lie ring
is called nilpotent of class r if we have
where for
,
is the Lie ideal of
generated by all elements of the form
with
and
. R is called nilpotent of lenth r if its Lie ring is nilpotent of lenth r. Clearly
.
The n-th Lie centre of is defined by
where commutator of weight n(
) is defined by
(see [Citation18]).
We can show that is a (unitary) subring of
and
If
for some integer m, then
is nilpotent. The smallest such integer is called the class of
(see [Citation18]).
Now, in this paper, we define two new relations and
on FHR S such that
and
are nilpotent, and Engel Lie rings, respectively.
Note. From now one, let or S be a fuzzy hyperring (i.e. FHR), unless otherwise stated.
3. Nilpotent Fundamental Relation ![](//:0)
![](//:0)
In this section, we present that combining a new FSR with the fuzzy hyperring is a nilpotent Lie ring.
Note. Let . For simplify, we use xy instead of
and
instead of:
Also, we use
instead of:
Definition 3.1
We consider and for any
,
For any , we let
and for any
and
, we define
as follows:
where
i.e.
Now, suppose that
. If
, then
for some
. Thus,
(I)
(I)
such that
Put
and
. So
and
(II)
(II)
Thus, by (I) and (II), for
,
. If
and so
Hence,
. Therefore,
is symmetric.
Define for any ,
. Then
is reflexive. Hence
, the transitive closure of
, is an equivalence relation.
Theorem 3.2
[Citation8]
Let be a ring. Then
, where
are defined as follows is a fuzzy hyperring.
Example 3.3
Let . By Theorem 3.2,
is a FHR. Then,
, and
Let r = 0, x = 0 and
. Then
and
Thus,
.
Lemma 3.4
Let ϱ be a permutation of ,
and
Thus
is a permutation of
Proof.
We show that is one to one. For this let
and
. By the definition of
for the case
and s = k + 1, we have
. Since
, we have
and so
which is a contradiction. For
and r = k + 1, we have
, which is a contradiction. Finally, for
, we have
, and so r = s. Therefore,
is one to one. It is clear that,
is onto. Therefore,
is a permutation of
.
Theorem 3.5
For each if
, then:
and
and
Proof.
(i) If , then there exists
such that
. Then there exist
,
and there exists
; (
), such that
Let
and
such that
and
. Then
Let p = x, then
. Also
Let q = y, then
.
We set , and we define
By Lemma 3.4,
is a permutation of
. Thus for
and any
;
such that
and
if
. Therefore, for any
such that
and
, we have
. Thus
. In the simillar way, we can show that
.
Now, if , then there exists
and
such that
By the above result we have
and so
. By the simillar way,
.
(ii) By the same manipulation we can prove and
Corollary 3.6
For any , the relation
is a FSR.
Proof.
Let and
,
. Then by Theorem 3.5, we have
and
. Then
. Similarly, we have
.
Proposition 3.7
For any , we have
.
Proof.
Let , then there exist
,
such that
We show that for any
,
(I)
(I)
Since
, there exists
such that
and so
. Such that
Now, let
and
,
and
. Since
, we have
. Hence, for
such that
Therefore,
.
Consequently, by (I) we have . Then
.
Corollary 3.8
If S is a commutative FHR, then .
Proof.
It is clear that, . It is enough to show that if S is a commutative, then
. For this, let
. Then for some
,
such that
and
.
Since S is commutative, for any , each element
can commute with others, and so we can consider
such that
then
which implies that
and so
.
Example 3.9
Let S be FHR as Example 3.3. Then, by Corollary 3.8, we have and so
.
Definition 3.10
Let be a Lie ring. We define
and
Remark 3.11
Let φ be a FSR on S. Then, by Theorem 2.6, without loss of generality, one can assume that is a Lie ring.
Theorem 3.12
is a nilpotent Lie ring of class at most n + 1.
Proof.
We show that for a FSR φ on S and any ,
(I)
(I)
The proof is based on induction on k. Put
and
, for any
. If k = 0, then
. Now, put
where
and
, so there exists
,
and
such that
and
. Then by Theorm 2.6,
and
. Therefore
. By induction hypotheses we have
. Hence,
.
Conversely, let . Then by Definition 3.10,
, where
and
. So induction hypotheses implies that
, where
and
. Since
, we have
and
, for some
. By Definition 2.2, for
, we have
, then there exists
such that
. Thus
, and so
. Hence, by
,
and Theorem 2.6, we have
. Thus,
, where
and
. Hence,
Therefore
.
Now, let and
, for any
. By induction on i, we show that
. Let i = 0, and
,
and
. For any
such that
. Assume that
Then
, and
. Then
. Therefore,
, and so by Theorem 2.6,
. Then
. Thus,
. Hence,
. Therefore,
. Now let
and
. Then for any
we have
. Hence for any
,
, and so by definition of
we get
. Therefore,
.
Now, let i = n + 1. Then , that is
. Hence R is nilpotent of class at most n + 1.
The relation which is defined in Definition 3.1, can also be introduced for rings.
Note that every ring is a FHR.
Here, we introduce the relation, which is defined on a finite FHR. This relation connects the FHRs with nilpotent Lie rings.
Definition 3.13
Let S be a finite FHR. Then we define the relations ζ and on S by
Theorem 3.14
Consider S be a finite FHR. Then the relation is a nilpotent fundamental relation.
Proof.
We show that is a FSR on S such that
is a nilpotent Lie ring, and if φ is a FSR on S such that
is a nilpotent Lie ring (of class n + 1), then
.
Since , by Corollary 3.6, it is easy to see that
is a FSR on S. Since S is finite, by Proposition 3.7, there exists
such that
. Thus
, for some
and so by Theorem 3.12,
is a nilpotent Lie ring.
Suppose φ is a FSR on S such that is a nilpotent Lie ring of class n + 1. We show that for any
Let
. Then there exists
such that
. Then
such that
Hence, by Theorem 2.6, for
,
Since
is a nilpotent Lie ring of class n + 1, by the proof of Theorem 3.12, we have
, and so
. Then
. Thus, for
and
, we get
. Moreover, if
, then
commutes with others. Also, if
then
, and so
. Consequently,
, which implies that
. Therefore,
.
Now, we can prove that . For this, let
and
. Then for any
,
and so there exist
(
) such that
. Then we have
. Hence,
.
4. ζ-Role of a Fuzzy Hyperring
In this section, we introduce the concept of ζ-role and we determine necessary and sufficient conditions such that the relation ζ be transitive.
. For simplify we use
instead of the following assumption:
,
Definition 4.1
Let . Then we say that X is an ζ-role of S if the following implication holds:
where
Note that by Definition 4.1, there exists such that
and for any
, we have
.
Remark 4.2
Let φ be a FSR on S, be the class of z module φ. Then
is an ζ-role of S.
Proof.
Let , there exists
such that
. If for
we have
, then φ is a FSR implies that
. Thus,
, is a contradiction.
Theorem 4.3
Let and
. Then the following conditions are equivalent:
(i) | X is an ζ-role of S, | ||||
(ii) | if | ||||
(iii) | if |
Proof.
If
is a pair of
, such that
and
, then for any
,
such that
and
. If
, then
and so by Definition 4.1,
, which is a contradiction. Thus
.
Suppose that
is a pair of
such that
and
. Then there exists
(
) such that
. By (ii) (k-times) we obtain
.
Let
and
,
such that
. Then for any
and any
and any
such that
, we have
. Thus
and by (iii), we obtain
, which is a contradiction. Therefore, X is an ζ-role of S.
Theorem 4.4
For any ,
is an ζ-role of S if and only if ζ is transitive.
Proof.
Suppose
. Then there exists
(
) such that
. Since
, for any
, is an ζ-role, by Theorem 4.3, we have
. Thus
and so
. Hence
. Therefore, ζ is transitive.
Suppose
,
and
. Since ζ is transitive, we conclude that
, and so by Theorem 4.3,
is a ζ-role of S.
Remark 4.5
Let θ be a FSR on S and . Then
is an ζ-role of S.
Definition 4.6
Consider A be a non-empty subset of S. The intersection of any ζ-role of S, which contains A is called ζ-closure of A in S. We denote the ζ-closure of A by .
. We use
instead of the following assumption:
Theorem 4.7
Let ,
and
(i) | |||||
(ii) |
| ||||
(iii) |
|
Proof.
By Definition 4.6, it is enough to prove is an ζ-role of S. If
and B is an ζ-role, then
. Suppose there exists
such that
, and
such that
, if
. Since
, there exists
such that
and
. Now, if there exists
such that
, then
. Thus,
which is a contradiction. Thus for any
,
and so
is an ζ-role of S. We prove the second role by induction on n. We have
. Suppose
. We show that
. If
, then
such that
and
, for some
. Since
we have
and
. Moreover, B is an ζ-role of S, and
, then
.
Theorem 4.8
If , then
.
Proof.
It is easy to see that for any ,
. By Theorem 4.7, we get
and
. Suppose
holds for n and
, then
such that
and
, for some
. By the hypotheses of induction,
and so for
we have
, then
for some
. Thus,
for some
. Hence,
, and so
. Therefore,
.
Lemma 4.9
For any and
we have
(i) |
| ||||
(ii) |
|
Proof.
(i) Let . Then by Theorem 4.7, we have
Now, we prove by induction on n. Suppose
. Then
(ii) We prove by induction on n. It is easy to see that
if and only if
. Suppose
. Then
such that
Clearly,
. Also,
and
then
. By hypotheses of induction, since
we have
and so by (i),
. Therefore,
if and only if
.
Theorem 4.10
Let .Then
if and only if
is an equivalence relation on S.
Proof.
By Theorem 4.7, and by Definition 4.6, we have
, thus
. Then E is reflexive. For transitivity, let
such that
and yEz. Then by Theorem 4.7,
and
. Thus, for any ζ-role P which contains z i.e.
, we have
and since
, we have
. Then
and so
. Thus for any ζ-role of S, we have
, i.e.
. Hence, by Theorem 4.7, xEz. Then E is transitive. The symmetrically of E follows directly from Lemma 4.9.
Theorem 4.11
For all , aEb if and only if
.
Proof.
Let
. Then for any
, there exists
, such that
. Then
such that
so
. Thus, by Lemma 4.9,
. Hence
and so
.
If xEy, then
for some
. Then
such that
and for some
we have
. Thus,
. Hence continuing this method
such that
and
. Then
. Therefore,
Remark 4.12
By Theorem 2.6, if is a FHR, then
is a ring. We define
by
, where
is the canonical projection.
Lemma 4.13
If M is an ζ-role of S, then .
Proof.
We know . If
, then there exists
such that
. For any
,
and so
. Thus,
. Since M is an ζ role of S,
and
, then by Theorem 4.3, we have
.
Theorem 4.14
is a fuzzy subhyperring of S, which is also an ζ-role of S.
Proof.
It is clear that, . Let
and for any
,
and
(
). Then by Theorem 2.6,
(
). Then
, and so
. We show that for any
,
. Let
. Then, by
, there exists
such that
. Therefore, by Theorem 2.6,
. Then
. Thus
. Consequently,
(I)
(I)
Hence,
is a fuzzy subhyperring of S. Now we prove that
(II)
(II)
For this let
,
and
. Then by Theorem 2.6,
and so
. Conversely, let
. Then
. Since
, there is
such that
and so
(since by Remark 2.6,
is a ring and
). Then
and so
. Thus by
, we have
. Also, since
we have
which by (I), (II) implies that
, and so
is a ζ-role of S.
By Lemma 4.13 and Theorem 4.14, we have the following corollary.
Corollary 4.15
.
5. Engel Lie Rings Derived from Fuzzy Hyperrings
In this section, continuing our previous work, we define a FSR, on a FHR S such that the quotient is an Engel Lie ring. Let s be a fixed element of S, unless we notify.
Definition 5.1
We define , and for
,
Now, let and
be the diagonal relation on S. For every integer
define
as follows:
i.e.
Consider
. Then
is symmetric (Similar to Definition
). Define for any
,
, thus
is reflexive. Then
, the transitive closure of
, is an equivalence relation on S.
In the same way of the Corollary 3.6, we have the following theorem.
Theorem 5.2
For any the relation
is a FSR.
Corollary 5.3
For a fixed element we have
.
Proof.
Let . Then there exists
such that
and so
Then
Now, let
and
,
and
. By
, we have
. Thus, for
such that
Hence,
and so
. By the similar way,
.
Similar to the Corollary 3.8, we have:
Corollary 5.4
Let S be a commutative FHR. Then .
Proposition 5.5
For any we have
.
Proof.
The proof is similar to Proposition 3.7.
Definition 5.6
Let be a Lie ring and
. Then,
is an n-Engel if for any
. Also, for a fixed element
we define the n-th Lie centre of
as follows:
We can conclude that, is an n-Engel if and only if for any
.
Also, let and
. Now by Definition 5.6, we have the following theorem.
Theorem 5.7
Let and
be the Lie ideal generated by
. Then Lie ring
is an n-Engel if and only if for any
,
.
Proof.
(⇒) Let y be a fixed element of n-Engel Lie ring and
. Then
for some
, and so
for some
. Then,
. Continuing this we have
, for some
. By hypotheses for any
, we have
. Hence, z = 0 and so
.
Let for any
,
and x be arbitary elements of
. By
, we have
. Continuing this we have
, and so
, which implies that
is an n-Engel Lie ring.
Remark 5.8
Let φ be a FSR on S. Then by Theorem 2.6, without loss of generality, we can assume that is a Lie ring.
Theorem 5.9
If φ is a FSR on S, then for any ,
Proof.
Put and
, for all
. If k = 0, then
. Now, we show that
Put
, where
, so there exist
and
such that
and
. Then by Theorem 2.6,
and
. Thus
. By induction hypotheses we have
. Hence
.
Conversely, let . Then
, where
. So hypotheses of induction implies that
, where
. Let
,
and
. For
we have,
. Then there exists
such that
. Thus
and so
. Hence, by Theorem 2.6,
. Thus,
Then,
Therefore
Theorem 5.10
is an
-Engel Lie ring.
Proof.
For any let
and
. By induction on i, we show that
. Let i = 0, then
. Let
. Then
. By hypotheses, we have
. Hence
, and so
. Thus,
. If i = n + 1, then
and so
. Therefore, R is an
-Engel Lie ring.
6. Conclusion
The fundamental relations on hyperrings were studied by Vougiouklis. Then, commutative fundamental relations on fuzzy hyperrings were introduced by Nozari and Fahimi. Now in this paper, first the smallest equivalence relation on a FHR S, such that the set of equivalence classes was a nilpotent Lie ring, was introduced. Then, the relation
was defined, as the FSR, so that the quotient would be an Engel Lie ring. Finally, theses two different relations were compared.
List of Abbreviations
, the set of all fuzzy subsets of H,
, a fuzzy hyperoperations,
, a fuzzy regular,
, a fuzzy strongly regular,
, a fuzzy (strongly) regular.
Disclosure Statement
No potential conflict of interest was reported by the author(s).
Additional information
Notes on contributors
E. Mohammadzadeh
E. Mohammadzadeh is assistant professor at the Payame Noor University, Iran. She has published more than 20 papers in the international journals. Her main scientific interests are algebraic logic, algebraic hyperstructures and fuzzy algebras.
R. A. Borzooei
R. A. Borzooei is full professor at the Shahid Beheshti University, Tehran, Iran. He is currently, Editor In-Chief and founder of “Iranian Journal of Fuzzy Systems” and “Journal of Algebraic Hyperstructures and Logical”, editorial board of six international journals. He has published more than 330 papers in the international journals. His main scientific interests are algebraic logics, ordered algebraic structures, algebraic hyperstructures, fuzzy algebras and fuzzy graphs.
F. Mohammadzadeh
F. Mohammadzadeh is assistant professor at the Payame Noor University, Tehran, Iran. Her main scientific interests are algebraic hyperstructures and fuzzy algebras.
S. S. Ahn
S. S. Ahn is full professor at the Dongguk University, Korea. She works in the Department of Mathematics Education from 1993 to present. She has published more than 150 papers in the international journals. Her main scientific interests are logical algebras, ordered algebras, algebraic hyperstructures, fuzzy algebras and fuzzy graphs.
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