Multi-Fuzzy Rings
ABDALLAH AL-HUSBAN1, MOWAFAQ OMAR AL-QADRI2, RANIA SAADEH3,
AHMAD QAZZA3, HEBA HAZZA ALMOMANI1
1Department of Mathematics, Irbid National University, JORDAN
2Department of Mathematics, Jerash University, JORDAN
3Department of Mathematics, Zarqa University, JORDAN
Abstract: In this article, we generalize the notion of a fuzzy space defined by Dib and Fathi for the multi-
membership function by examining and developing the concept for the multi-fuzzy binary operation. This
inspired us to study and consider the multi-fuzzy ring theory approach.
Key-Words: - Multi-fuzzy ring, multi-fuzzy subspaces, multi-fuzzy ring and multi-fuzzy subring.
Received: August 27, 2021. Revised: August 7, 2022. Accepted: September 12, 2022. Published: October 4, 2022.
1 Introduction
In 1965, Zadeh defined the concept of the fuzzy set
theory, and it was advanced in many mathematical
fields and applications to solve several examples
and complicated problems in medical sciences,
logic, control engineering, economics, etc [1]. By
utilizing the ordered sequences of the membership
function, Sebastian and Ramakrishnan found a new
type of multi-fuzzy set. The colour of pixels is one
of the problems that was explained by the notion of
multi-fuzzy sets since it provides new methods to
represent the problems. But we should note that is
difficult to represent some other problems using
another extension of a fuzzy set theory. The multi-
fuzzy extensions of functions and multi-fuzzy
subgroups were discussed by Sebastian and
Ramakrishnan, we considered that the multi-fuzzy
sets are an extension of the theories of fuzzy sets
and it’s called Atanassov intuitionistic fuzzy sets
and L-fuzzy sets [2-4]. In 2010 they also studied and
introduced some of the elementary properties of
multi-fuzzy subgroups [5-7]. We will discuss the
development of fuzzy group theory, starting from
1956, when L. A. Zadeh introduced the definition
and the concept of a fuzzy set as we mentioned
earlier then applied by Chang [8] in fuzzy
topological spaces to generalize and define some
fundamental definitions in general topology. In
1971, Azriel Rosenfeld defines some applications in
groupoids and groups from the elementary theory
[9]. Negoita and Ralescu use Rosenfeld’s definition
to consider a generalization where the unit interval
In 1960, Schweizer and Sklar [10] define
the triangular norm then in 1979, Anthony and
Sherwood used the Schweizer and Sklar definition
to redefine the fuzzy subgroup of a group[11-13].
Note that the Rosenfeld-Anthony-Sherwood
approach was used by many mathematicians to
define or investigate fuzzy group theory. Also, a
new approach was defined by Youssef and Dib for
defining the fuzzy groupoid and the fuzzy
subgroupoid [14-17]. Since the concept of a fuzzy
universal set is not defined, so we can't define the
fuzzy group and fuzzy subgroup. This was the
reason for Youssef and Dib to introduce and define
the fuzzy universal set and fuzzy binary operation.
In 1982, J. Liu introduced in his paper, the concept
of fuzzy rings and fuzzy ideals, then in 1985, the
notions of fuzzy ideals and fuzzy quotient ring
studied by Ren. In 2004, the concept of the
intuitionistic fuzzy subgroup was generalized by
Zhan and Tan [18], by adding restrictions for the
definition of the non-membership function. They
also introduced the definitions for the notion of a
fuzzy space and an intuitionistic fuzzy function in
their paper [18], they made a generalization of
Rosenfeld's fuzzy subgroup definition to define the
intuitionistic fuzzy subgroup. For any given
classical group, we can use the classical binary
operation to define the intuitionistic fuzzy subgroup.
The definition of the notion of an intuitionistic fuzzy
group was introduced by M. Fathi and AR Salleh in
2009 [19], by using the notion of intuitionistic fuzzy
space and fuzzy function. To introduce the notion of
the intuitionistic fuzzy group we will consider both
of their notion of intuitionistic fuzzy space and
intuitionistic fuzzy function [18]. In 2021, Jaradat
and Al-Husban introduced the concept of multi-
fuzzy group spaces on multi-fuzzy space. Notice
that the fuzzy set concept will find in many
mathematical fields [20-27]. As result, we will use
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2022.21.82
Abdallah Al-Husban, Mowafaq Omar Al-Qadri,
Rania Saadeh, Ahmad Qazza, Heba Hazza Almomani
E-ISSN: 2224-2880
701
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the multi-fuzzy space to create a new algebraic
system, known as the multi-fuzzy ring, by
integrating two mathematical fields on fuzzy sets
which are fuzzy algebra and multi-fuzzy set theory.
2 Preliminaries
In this section, we will mention many theorems and
definitions in the fuzzy set, which we will mainly
use in the third section.
Definition 1 [1] Consider the fuzzy set
where is the universe of discourse, then the set
will be characterized by the membership function
where:
Definition 2 [2], Consider where the set is
intuitionistic fuzzy set and is the universe of
discourse (a non-empty set) then we get the form:
consider the
functions is the degree of
membership and is the degree of
non-membership of each element to the set B
respectively, and for all
.
Definition 3 [6] Let be a positive integer number
and the set is multi-fuzzy set, where then
is a set of ordered sequences as follow:
where
The function
is called the multi-
membership function of multi-fuzzy set , where
is called the dimension of The set of all multi-
fuzzy sets of dimension in is denoted by
.
Remark 1 [6], Consider the following summation
where: ,
.
1- If then we get multi-fuzzy set of
dimensions , called Zadeh’s fuzzy set.
2- If and then we get
multi-fuzzy set of dimensions , called
Atanassov’s intuitionistic fuzzy set.
3- If , then the multi-fuzzy set
of dimensions is called a normalized
multi-fuzzy set.
Definition 4 [6], Let
and
be two
multi-fuzzy sets of dimensions in Then
satisfying this relations and operations:
(1) iff and
(2) iff and
(3)
(4)
(5)
Definition 5 Let be a nonempty set then
consider as a fuzzy
space, the ordered pair is called a fuzzy
element in
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Abdallah Al-Husban, Mowafaq Omar Al-Qadri,
Rania Saadeh, Ahmad Qazza, Heba Hazza Almomani
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Definition 6 [16], Let be a nonempty set. Then
the intuitionistic fuzzy space is the set of all ordered
triples that denoted by
with the
conditions and The intuitionistic
fuzzy element of defined as , and
the condition with is hold.
Definition 7 We can define the intuitionistic
fuzzy subgroup of the group , by consider an
intuitionistic fuzzy set
in a group with
the binary operation (∘) and satisfying the following
conditions:
Definition 8 [25], Let be a non-empty set. A
multi-fuzzy space is the set of all
ordered sequences
that is, where
The ordered
sequences is called a multi-fuzzy element
in the multi-fuzzy space Also, the
ordinary set of ordered sequences is a multi-fuzzy
space. The first component in the ordered sequence
indicates the ordinary element, while a list of
potential multi-membership values is indicated by
the second component. The Dib’s fuzzy group is
defined as A multi-fuzzy space of dimension , and
The Fathi’s intuitionistic fuzzy group is defined as a
multi-fuzzy space of dimension .
Definition 9 [6], Let be a multi-fuzzy subset of
the ordinary group with the ordinary binary
operation , then will be a multi-fuzzy
subgroup of if and only if satisfying the following
conditions:
(1)
(2) for all
Also, we can define Rosenfeld’s fuzzy subgroup if
the multi-fuzzy subgroup of dimension , and define
Zhan’s intuitionistic fuzzy subgroup if the multi-
fuzzy subgroup of dimension .
Definition 10 [25] Let denote the support of
that is A
multi-fuzzy subspace of the multi-fuzzy space
is the collection of all ordered sequences
where for some and
that contains besides the zero element
at least another one element.
Definition 11 [25] A multi-fuzzy binary operation
on the multi-fuzzy space
is a multi-fuzzy function from
with multi-co-
membership functions with the following
conditions:
(i) if
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Abdallah Al-Husban, Mowafaq Omar Al-Qadri,
Rania Saadeh, Ahmad Qazza, Heba Hazza Almomani
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(ii) are onto, that is,
Definition 12 [25], The multi-fuzzy group is
defined as a multi-fuzzy algebraic system
if and only if for every
the
following conditions are satisfied:
1-Associative:
that is
2-Existence of a multi-fuzzy identity element
for which
that is
3-Every multi-fuzzy element has an
inverse such that:
From and , it follows that
is a multi-fuzzy group over the multi-
fuzzy space
3 Multi-Fuzzy Ring
The main result of this paper is to define the concept
of the multi-fuzzy ring by using two multi-fuzzy
binary operations and adding them, to generate a
multi-fuzzy space with similar conditions to the
ordinary cases and multi-fuzzy.
Definition 13 A multi-fuzzy ring
is considered as a multi-fuzzy
space with two multi fuzzy binary
operations.
And satisfying the following conditions:
1. is a commutative multi-
fuzzy group,
2. is a multi-fuzzy semigroup,
3. The distributive laws
holds for all
Example 1 Let the set and defined a
multi-fuzzy binary operation
1-
2-
over the multi fuzzy space by taking the
set as follows:
.
3- .
4- .
Where refers addition modulo 3, refers to
multiplication modulo 3, where ( ) refers to
multiplication modulo 3.
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Abdallah Al-Husban, Mowafaq Omar Al-Qadri,
Rania Saadeh, Ahmad Qazza, Heba Hazza Almomani
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Thus is a multi-fuzzy ring
using multi-fuzzy spaces.
Example 2 Let the set and defined the
multi-fuzzy binary operation as follow:
1-
2- .
Over the multi-fuzzy spaces such that:
Where, the multi-fuzzy space with
defined a multi-fuzzy ring
Definition 14 The multi-fuzzy ring will be
commutative if and
we have
Definition 15 Considered the following
be a multi-fuzzy ring satisfying
the following:
(1) A multi fuzzy element in is
a multi-fuzzy unity denoted by if
for all . A multi
fuzzy ring having a unity is called a multi fuzzy
ring with unity.
(2) A multi fuzzy element in
is called a unit if
there exist a multi fuzzy element
such that
.
In the following theorem, we will demonstrate the
correspondence relationship between the multi-
fuzzy ring and the intuitionistic fuzzy ring.
Theorem 2 Associated to each multi-fuzzy ring
is an intuitionistic fuzzy ring
which is isomorphic to the
complex fuzzy ring by the
correspondence
Proof. Let be a multi fuzzy ring.
Restrict the multi fuzzy binary
operations , to
the intuitionistic fuzzy case for using the
isomorphic such that
,
respectively. Thus, the resultant structure
is an intuitionistic fuzzy ring in the
sense of Fithi.
4 Conclusion
In this paper, we create a new algebraic system by
generalizing and studying the Dib definitions of
fuzzy rings that are based on multi-fuzzy spaces. In
the future we attend to modify new definitions and
inequalities and introduce new theorems related to
them with more applications [28-36].
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Abdallah Al-Husban, Mowafaq Omar Al-Qadri,
Rania Saadeh, Ahmad Qazza, Heba Hazza Almomani
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DOI: 10.37394/23206.2022.21.82
Abdallah Al-Husban, Mowafaq Omar Al-Qadri,
Rania Saadeh, Ahmad Qazza, Heba Hazza Almomani
E-ISSN: 2224-2880
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