On the Algebraic Foundations, Morphisms and Aggregation Operators of Q̃–Fuzzy Soft Subrings

On the Algebraic Foundations, Morphisms and Aggregation Operators of QFuzzy Soft Subrings

Geethalakshmi M (1), Karpagam (2) , S. Bala Krishnan (3) , C. Gurubaran (4), M. Mohamed Althaf (5)

Mudassir Muhammed (6)

(1) Gnanam School of Business, Thanjavur, Tamil Nadu, India

(2) SRM Valliammai Engineering College, Kattankulathur, Tamil Nadu, India

(3,6) University of Technology and Applied Sciences, Nizwa, Oman.

(4,5) Jamal Mohamed College, Trichirapalli, Tamil Nadu. India

Abstract – This paper presents a unified algebraic framework for Q-fuzzy soft subrings. It extends the classical notions of fuzzy subrings and Q-fuzzy subrings to the broader setting of Q-fuzzy soft subsets, where membership values are defined on the interval [, ). We introduce and systematically investigate Q-fuzzy soft subrings under a generalized triangular norm (T-norm) framework. Fundamental algebraic properties, structural characterizations, and relationships with existing fuzzy and soft algebraic structures are established. Emphasis is placed on the study of morphisms, including homomorphisms, inverse images, direct images, and isomorphisms, to examine the preservation of algebraic structures within the Q-fuzzy soft environment. Furthermore, aggregation operators induced by T-norms and related fuzzy connectives are analysed, and their roles in constructing and combining Q-fuzzy soft subrings are explored. Several new results concerning closure properties, intersections, lattice-theoretic aspects, and algebraic invariance are proved and illustrated with suitable examples. The proposed framework generalizes existing fuzzy and soft algebraic models and provides a foundation for further applications in algebraic systems, decision- making, information processing, and artificial intelligence.

Keywords: Q-fuzzy soft subset; Q-fuzzy soft subring; fuzzy soft algebra; aggregation operator; triangular norm (T-norm); homomorphism; inverse image; isomorphism; lattice structure; algebraic invariance.

1. INTRODUCTION

   Fuzzy algebraic structures provide an effective mathematical framework for modelling uncertainty, vagueness, and incomplete information in algebraic systems. Since the introduction of fuzzy sets by Zadeh [12], the concept has been successfully incorporated into various branches of mathematics, leading to the development of fuzzy groups, fuzzy rings, fuzzy ideals, and many other generalized algebraic structures. Among these, fuzzy subrings have attracted considerable attention due to their ability to extend classical ring theory to uncertain environments.

   The notion of Q-fuzzy sets further generalizes fuzzy sets by allowing membership values to be taken from the interval [0, ), thereby providing a richer and more flexible representation of uncertainty. This extension has proved useful in situations where traditional membership scales are insufficient for describing varying degrees of belongingness. Consequently, Q-fuzzy algebraic structures have become an important area of investigation in modern fuzzy mathematics.

   Meanwhile, Molodtsov’s soft set theory [13] introduced a parameterized approach to uncertainty modeling and has since become a powerful tool in decision-making, information processing, and algebraic investigations. The integration of fuzzy sets and soft sets led to the emergence of fuzzy soft structures, which combine the advantages of both theories. Significant contributions by Maji et al., Pei

   et al., Akta and Çaman [1], Acar et al. [4], and others established the foundations of soft algebraic systems and their applications.

   Researchers have also investigated fuzzy soft subrings and related algebraic structures under various operational frameworks. Aygünolu and Aygün [2] employed triangular norms (T-norms) in the study of fuzzy soft algebraic systems, providing an effective mechanism for combining fuzzy information. Furthermore, Solairaju and Geethalakshmi [10] examined fuzzy soft subrings and associated lattice- theoretic properties, illustrating the usefulness of T-norm-based operations in algebraic investigations. Despite these developments, a comprehensive treatment of Q-fuzzy soft subrings from the perspectives of algebraic foundations, morphisms, and aggregation operators remains limited. In particular, the structural behaviour of Q-fuzzy soft subrings under homomorphic mappings and aggregation mechanisms has not been sufficiently explored. Such investigations are important because they provide deeper insight into the preservation and construction of algebraic structures within uncertain environments.

   Motivated by this observation, the present paper develops a unified framework for the study of Q-fuzzy soft subrings. We establish their fundamental algebraic properties and characterize their behaviour under various operations. Special attention is devoted to morphisms, including homomorphisms, inverse images, direct images, and isomorphisms, with the aim of identifying algebraic properties that remain invariant under these mappings. In addition, aggregation operators derived from T-norms and related fuzzy operators are introduced and analysed as tools for combining and constructing Q-fuzzy soft subrings. Their effects on closure properties, intersections, and algebraic stability are investigated in detail.

   The paper also establishes several structural results concerning intersections, lattice-theoretic properties, and preservation theorems. Through illustrative examples, the proposed framework demonstrates how Q-fuzzy soft subrings can serve as a natural extension of existing fuzzy and soft algebraic models while providing greater flexibility for uncertainty representation.

   The organization of the paper is as follows. Section 2 recalls the necessary preliminary concepts related to fuzzy sets, soft sets, Q-fuzzy sets, and Q-fuzzy soft subsets. Section 3 introduces Q-fuzzy soft subrings and develops their basic algebraic properties. Section 4 investigates aggregation operators, intersections, and related structural characteristics. Section 5 studies morphisms, including homomorphisms, inverse images, direct images, and isomorphisms with examples, applications, and concluding structural results. Finally, the paper highlights potential directions for future research in fuzzy algebra, information processing, and uncertainty-based mathematical systems.

2. DEFINITIONS AND PRELIMINARIES

   To guarantee completeness, we present the basic definitions and notations for fuzzy soft subsets introduced by Ahmad et al. [17] [2009] and Maji et al. [8] [2003] in this section. In this study, let be a subring over which all subsets are taken. Let be the initial universe subset of discourse.

   Definition 2.1[20] The function known as a t-norm, represented by , is a mapping

   : [0,1] × [0,1] [0,1].

   The following conditions are satisfied for all 1, 2, 3, 4 [0,1]

   1. (1, 2) = (2, 1)

   2. (1, (2, 3)) = ((1, 2), 3)

   3. (1, 1) = (1, 1) = 1

   4. If 1 3 and 2 4, then (1, 2) (3, 4).

   Remark 2.2 A commonly applied t-norm is the minimum t-norm, defined by

   (1, 2) = min {1, 2}.

   Definition 2.3 [12] Let be a non-empty subset. A fuzzy subset K on is characterized by its membership function : [0,1], where for each , the value () represents the degree of membership of in .The complement of the fuzzy subset is defined by

   () = 1 (), . The cnstant mappings () = 0 and () = 1 represent the null fuzzy subset and the universal fuzzy subset, respectively. These are denoted by 0n and 1w. Definition 2.4 [4] Let be a universal subset and R be a subset of parameters. A pair (, ) is called a soft subset over if : ( ), where ( ) denotes the power subset of .

   Definition 2.5 [13] Let be a collection of parameters and be the universal subset of discourse. Let ( ) denote the power subset of . An ordered pair(() , ), where () :

   ( ), is called a soft subset over . Thus, a soft subset over represents a parameterized family of subsets of .That is, {() () } ( ), where each () () corresponds to the subset of elements of associated with the parameter r.

   Remark 2.6: A soft subset (() , ) over can be viewed as a mapping that takes to each parameter a subset of , thereby forming a parameterized structure which is suitable for uncertainty modelling.

   Definition 2.7 [17] Let be the universal subset of discourse. Let be a collection of parameters and let . A fuzzy soft subset on is defined by a pair (() , ), where () :

   ( ), where = [0,1] represents the unit interval, and ( ) represents the collection of all fuzzy

   subsets of . That is, for each , (() )(): [0,1] is a fuzzy subset of . Hence,

   () ()() gives the degree of membership values of with respect to the parameter r. The subset follow () is called a fuzzy soft subset over .

   Remark 2.8 A fuzzy soft subset () can be viewed as a parameterized family of fuzzy subsets of the universe of discourse , where each parameter corresponds to a fuzzy subset () (). Remark 2.9 For each , define () () = () ()(), , which simplifies notation in the following sections.

   Remark 2.10, for a given element , a classical soft subset () can be regarded as a fuzzy soft subset () , where the characteristic function associated with () at is defined by the mapping

   1, if (() )(),

   (() )() = {(() )()}() = {0, otherwise. (2.1)

   Thus, every classical soft subset can be interpreted as a special case of a fuzzy soft subset with respect to its characteristic function.

   Example 2.11 Let = {1, 2, 3}, = {1, 2}

   Define a classical soft subset () is given as follows (1) = {1, 3}, (2) = {2}

   1

   Now, the corresponding given fuzzy soft subset representation is given as

   For 1 , ((

   )

   ) () = 1, {1, 3},

   1

   {

   0, otherwise

   i.e., ((

   ) ) = {(1, 1), (2, 0), (3, 1)}

   For 2 , ((

   ) )

   2

   () = {1, {2},

   0, otherwise

   i.e., ((

   ) ) = {(1, 0), (2, 1), (3, 0)}

   2

   Definition 2.12 Assume that the subgroup has a non-null fuzzy soft subset() . If and only if

   s fuzzy subgroup () (a) is true for all a in is then () is assumed to be a s soft subgroup.

   Definition 2.13 [11] A fuzzy soft non-empty subset over the subgroup is represented by ()

   If and only if, then () is considered a fuzzy soft subgroup of . Then

   1. (() )( , ) { (() ) (), (() )()} (2.2)

   2. (() )( 1) (() ) () is true for each , . (2.3) where all in P satisfy the fuzzy subgroup () () = (() ) of .

      Definition 2.14 [4] A soft non-empty fuzzy subset over a subring (, +,.) is the one is known as () if and only if s sub subring (() )(a) is true for all a in is then () is usually referred to as a soft subring of .

      Definition 2.15 [2] Let us construct a fuzzy soft subset () over a non-null subring . Then, and only then, () is it said that is a subset over .

      (1) (() )( + ) {(() ) (), (() )()}		(2.4)
      (2) (() ) () (() ) ()		(2.5)
      (3) (() ) ( . ) T{(() ) (), (() )()}		(2.6)
      is true for each , and s fuzzy sub subring (() )() for all in Z.	= (() ) is true

      Example 2.16 A soft subring can be thought of as a fuzzy soft subring, because every soft subset is a fuzzy soft subset. A fuzzy subring is defined by the characteristic function associated with each subring. As a result, each soft subring can be thought of as a variant of a fuzzy soft subring.

      Example 2.17 Let be the collection of all natural numbers, and define a mapping

      : ( ) by () () = (() ): , for every , where

      1 , if = 2, for some ,

      (() )() = {() ()}() = {

      0, otherwise.

      Here, denotes the subset of all integers.

      Therefore, the subset (() constitutes a fuzzy soft subset. Moreover, since each (() defines a fuzzy subring structure on , the fuzzy soft subset (() ) forms a fuzzy soft subring.

      Example 2.18 Let = , = {1,2}

      Define ((

      ) ) () = {1, is even

      and ((

      ) )

      () = {0.5, is multiple of 3

      1

      0, otherwise

      2

      0, otherwise

      Then each ()is a fuzzy subring of . Hence, (() ) forms a fuzzy soft subring.

      Definition 2.19 [21] Let () be a fuzzy soft subset that is non-empty over , the universal subset of discourse. Then the soft subset ( )= {(( )) / each a in } is introduced to as a – level soft subset for all [0, 1].

      Definition 2.20 [1] Let (() ) be a non-empty fuzzy soft subring in (, +, .) and :

      be the mapping. The definition of (() ) is as follows (() ) : [0, 1] by

      (() ) () = (() )( ) (by using 2.1 definition)

3. Q~FUZZY SOFT SUB SUBRINGS AND ITS PROPERTIES

Definition 3.1 Let (1, 1, ) and (2, 2, ) be two fuzzy soft subsets.

Assume that 1 2 . Define = 1 2 Then the intersection of (1, 1, ) and (2, 2, )

is the fuzzy soft subset (, , ) where for all , (, , ) = (()(, ) =

((1)(, ), (2)(, )) for all , . In this case, we write (1, 1, ) (2, 2, ) = (, , )

Definition 3.2 Let {(, , )} be a family of fuzzy soft subsets. Assume that

Define = . Then the intersection of the family {(, , )} is the fuzzy soft subset

(, , ) where for every , ()(, ) = (()(, )) where denotes the iterated application of the T-norm for all , . In this case, we write , , ) = ( , , ) If = min , then Definitions 3.1 and 3.2 reduce to the classical intersection of fuzzy soft subsets.

Definition 3.3 Let (, +,)be a fuzzy soft subring. Let (, , ): × [0,1], (, , ) 0

Then (, , )is called a -fuzzy soft subring over if

(1QFSR) (, , )( + , ) ((, , )(, ), (, , )(, ))

(2QFSR) (, , )(, ) (, , )(, )

(3QFSR) (, , )( , ) ((, , )(, ), (, , )(, ))

for all , and . Also, for every , (, , ) defines a fuzzy subring of .

Theorem 3.4 Let (, , )be a -fuzzy soft subring over .

1. ( , , )(, ) (, , )(0, ),

   ( )

2. Define

= { (, , )

(, ) = (, , )

(0, }

( )

Then

is a -fuzzy soft subring of .

Proof. Let . ()(0, ) = ()( + (), ) (()(, ), ()(, ))

Using (2QFSR) ()(, ) ()(, ).Thus ()(0, ) (()(, ), ()(, ))

Since (, ) , we obtain ()(, ) ()(0, )

Part (i) proved

( )

Now let ,

then ()(, ) = ()(0, )

()(, ) = ()(0, )

()( + (), ) (()(, ), ()(, ))

= (()(0, ), ()(0, )) = ()(0, )

( )

Hence,

Similarly ()(, ) = ()(0, )

Part (ii) proved

Corollary 3.5 Let be finite. Let (, , )be a -fuzzy soft subring. Define: = { : (, , )(, ) = (, , )(0, )}

Then is called a -fuzzy soft crisp subring of .

Theorem 3.6 Let (1, 1, ) , (2, 2, ) be two -fuzzy soft subrings. Define: (, , ) = (1, 1, ) (2, 2, ) and = 1 2

(, ) = ((1)(, ), (2)(, ))

Then (, , ) is a -fuzzy soft subring of .

Proof. Straightforward

Remark 3.7 We define the notion of a T-power -fuzzy soft subring as follows. Let (1, , ) = (1, (2, , )), Then, extending Definition 3.3, we have ()(1 + + , ) (()(1, ), , ()(, )).

Theorem 3.8 Let (, , ) and (, , ) be two –fuzzy soft subrings. Define: (, , ) = (, , ) (, , )

= , For all (, ) × , (, ) = ((, ),

(, ))

Then (, , ) is a –fuzzy Q-fuzzy soft subring.

Proof Let , , ( + , ) = (( + , ),

( + , ))

(((, ), (, )), ( (, ), (, )))

(((, ),

(, )), ((, ),

(, ))) = ((, ), (, ))

First condition proved

(, ) = ((, ),

(, )) ((, ),

(, )) = (, )

Second condition proved

(, ) = ((, ),

(, ))

(((, ), (, )), (

(, ),

(, ))) ((, ), (, ))

Third condition proved Hence proved.

Theorem 3.9 Let {(, , ) }1 be a family of –fuzzy soft subring. Define: (, , ) = ( , , ) and =

1 1

(, ) = 1, (, ) , Then (, , ) is a –fuzzy Q-fuzzy soft subring.

Example 3.10 Let = , = {}

(, ) = {0.8, even

and

(, ) = {0.7, multiple of 3

0.3, otherwise

0.2, otherwise

Then (, ) = ((, ),

(, ))

This forms a -fuzzy Q-fuzzy soft subring.

Theorem 3.11 Let (, , ) be a –fuzzy Q-fuzzy soft subring.

Define the complement: (, , ) (, ) = 1 (, ),

then (, , ) is a –fuzzy soft subring (with respect to dual t-conorm ).

Proof Let , , then (, , ) ( + , ) = 1

( + , )

Since (, , )is a fuzzy soft subring, ( + , ) ((, ), (, ))

Thus, 1 ( + , ) 1 ((, ), (, ))

Using duality, 1 (, ) (1 , 1 )

Hence, ()( + , ) (()(, ), ()(, ))

First condition proved.

Also ()(, ) = 1 (, )

Since, (, ) (, ).Then, 1 (, ) 1 (, )

()(, ) ()(, )

Second condition proved.

Also ()(, ) = 1 (, )

Since, (, ) ((, ), (, ))

Then, 1 (, ) 1 ((, ), (, )) Using duality, (()(, ), ()(, )) Third condition proved

Thus(, , ) satisfies all conditions of a –fuzzy soft subring.

Hence proved.

Theorem 3.12 Let (, , ) be a conceivable (imaginable) fuzzy soft structure satisfying,

,( + , ) (,(, ), ,(, )) (3.6)

,(, ) ,(, ) (3.7)

,(, ) (,(, ), ,(, )) (3.8) then (, , ) is a –fuzzy soft subring.

Proof Let , , then the reduction of is (, ) = (, (, , (, )))

then, (, ) (, )

Thus, ,( + , ) (,(, ), ,(, )) (,(, ), ,(, ))

From (3.7), ,(, ) ,(, )

From (3.8), ,(, ) (,(, ), ,(, )). Using same argument,

(,(, ), ,(, ))

All three defining properties of a –fuzzy soft subring are satisfied. Hence proved.

SECTION 4: GENERALIZED OPERATIONS AND EXTENSIONS OF QFUZZY SOFT SUBRINGS

Definition 4.1 Let (, , ) and (, , )be two Qfuzzy soft subsets over . Define their union

(, , ) by: = and for all (, ) × ,

()(, ) = {

()(, ), ,

()(, ), ,

(()(, ), ()(, )), ,

where is a t-conorm. We write: (, , ) = (, , ) (, , )

()

Definition 4.2 Let {(, , ): } be a family of Qfuzzy soft subsets.

Define: = and ()(, ) = (()(, )). Thus, (, , ) = ( , , )

()

Theorem 4.3 Let (, , ) and (, , ) be two -Q-fuzzy soft subrings. Then their union

(, , ) = (, , ) (, , ), = is also an -Q-fuzzy soft subring.

Proof. Let , , then

( + , ) = (( + , ), ( + , ))

(((, ), (, )), ((, ), (, )))

(((, ), (, )), ((, ), (, )))

Thus, ( + , ) ((, ), (, ))

and (, ) = ((, ), (, )) ((, ), (, )) = (, )

Also (, ) = ((, ), (, ))

(((, ), (, )), ((, ), (, )))

(((, ), (, )), ((, ), (, )))

Hence proved.

Theorem 4.4 (, , ) is a Qfuzzy soft subring = { : (, ) }

is a Q-fuzzy soft subring for all Im().

Proof. Let , , i.e., (, ) , (, )

( + , ) ((, ), (, )) (, )

Thus + belongs to the = { : (, ) }

And (, ) (, )

(, ) ((, ), (, )) (, )

Thus belongs to the = { : (, ) }

Hence is a Q-fuzzy soft subring for all Im(). Theorem 4.5 Define +(, ) = (, ) + 1 (0, ). Then (+, , ) is a normal S containing (, , ).

Proof. Let , , then

+( + , ) = ( + , ) + 1 (0, )

((, ), (, )) + 1 (0, )

((, ) + 1 (0, ), (, ) + 1 (0, ))

= (+(, ), +(, )).

and

+(, ) = (, ) + 1 (0, )

(, ) + 1 (0, )(since (, ) (, ))

= +(, ).

Thus, +(, ) +(, ).

Also

+(, ) = (, ) + 1 (0, )

((, ), (, )) + 1 (0, )

((, ) + 1 (0, ), (, ) + 1 (0, ))

= (+(, ), +(, )).

and the normality condition is satisfied

+(0, ) = (0, ) + 1 (0, )

= 1.

Thus, (+, , ) is a normal Q-fuzzy soft subring containing (, , ).

Theorem 4.6 Let (, , ) and (, , ) be two -Q-fuzzy Q-fuzzy soft subrings. Define: (, ) = ((, ), (, )), , .

Then (, × , ) is also a Q-fuzzy soft subring of .

Proof. Let , , .

( + , ) = (( + , ), ( + , ))

(((, ), (, )), ((, ), (, )))

(((, ), (, )), ((, ), (, )))

= ((, ), (, )).

and

(, ) = ((, ), (, ))

((, ), (, )) (since , are subrings)

= (, ).

Hence, (, ) (, ).

Then

(, ) = ((, ), (, ))

(((, ), (, )), ((, ), (, )))

(((, ), (, )), ((, ), (, )))

= ((, ), (, )).

Thus (, × , ) is a Q-fuzzy soft subring of .

Theorem 4.7 Let (, , )be Qfuzzy soft subrings over . Define: (, ) = ( , )

Then,

( , , ) is a Qfuzzy soft subring over .

=1

=1

Proof. Let = (1, , ), = (1, , ) and . Then

( + , ) = ( + , ) ( (, ), (, )) ((, ), (, ))

and (, ) = (, ) (, ) = (, ). Also (, ) = (, ) ((, ), (, )) Hence proved.

Lemma 4.8 Let , be Qfuzzy Q-fuzzy soft subring s. If , then (, ) (, )

Proof. Straightforward

Corollary 4.9 Let {( , , ) : = 1,2, , } be a finite family of Q-fuzzy soft subrings of .

=1

Define the iterated product by (, , ) = (1, 2, , ), = , and for all (, ) × ,

(, ) = (1(, ), 2(, ), , (, )). Then (, , ) is a Qfuzzy Q-fuzzy soft subring of .

Proof Let , and .

We prove the three defining conditions using the t-norm .

( + , ) = (1( + , ), 2( + , ), , ( + , ))

Since each (, , ) is a Qfuzzy Q-fuzzy soft subring,

( + , ) ( (, ), (, )),

Thus, ( + , ) ((1(, ), 1(, )), , ((, ), (, )))

((1(, ), , (, )), (1(, ), , (, )))

Hence, ( + , ) ((, ), (, ))

and (, ) = (1(, ), 2(, ), , (, ))

Since each satisfies: (, ) (, )

Therefore, (, ) (1(, ), 2(, ), , (, )) = (, )

Also (, ) = (1(, ), 2(, ), , (, ))

Since each is a Q-fuzzy soft subring ,

(, ) ( (, ), (, ))

Thus, (, ) ((1(, ), 1(, )), , ((, ), (, )))

((1(, ), , (, )), (1(, ), , (, )))

Hence, (, ) ((, ), (, ))

All three defining conditions of a – soft subring are satisfied.

(1, 2, , ) is a -Q-fuzzy soft subring . Hence, (, , ) is a – Q-fuzzy soft subring.

Example 4.10 Let = , = {}. Define three Q-fuzzy soft subrings

(, ) = {0.9, even , (, ) = {0.8, 0 (mod 3) and (, ) = {0.7, > 0

1 0.5, otherwise 2

0.4, otherwise

3 0.6, 0

Define (, ) = (1(, ), 2(, ), 3(, )), = min

Compute: = 6: (6, ) = min (0.9,0.8,0.7) = 0.7,

= 3: (3, ) = min (0.5,0.8,0.7) = 0.5 and = 2: (2, ) = min (0.9,0.4,0.6) = 0.4

We have ( + , ) ((, ), (, )), (, ) (, ), (, ) ((, ), (, ))

Thus, is a Qfuzzy soft subring and proving Corollary 4.9.

SECTION 5 QFUZZY SOFT SUBRING HOMOMORPHISMS

Definition 5.1 Let and be two Qfuzzy soft subrings and let be a subset of parameters. Let (, , ) and (, , ) be Qfuzzy soft subsets over and , respectively. A pair , is called a Qfuzzy soft function from to if : is a mapping between subrings and : is a mapping between parameter subsets.

Definition 5.2 A Qfuzzy soft function , : is called a Qfuzzy soft homomorphism if for all

, , then ( + ) = () + () , () = ()() and () . Definition 5.3 A Qfuzzy soft homomorphism , is called a Qfuzzy soft isomorphism if is a Qfuzzy soft subring isomorphism and is bijective.

Definition 5.4 Let (, , ) be a Qfuzzy soft subset over .The image under , is

, (, , ) = ((), (), ) where ()(, ) = sup

()=, ()=

(, )

Definition 5.5 Let (, , )be a Qfuzzy soft subset over . The pre-image is

, 1(, , ) = (1(), 1(), ) where 1()(, ) = ()((), )

Theorem 5.6 Let (, , ) be a Qfuzzy soft subring over . Let , : be a Qfuzzy soft homomorphism. Then , (, , ) is a Qfuzzy soft subring over .

Proof Let 1, 2 , . Take 1, 2 such that (1) = 1, (2) = 2

()(1 + 2, ) = sup (1 + 2, ) , Since is Qfuzzy soft subring,

(1 + 2, ) ((1, ), (2, ))

Thus, ()(1 + 2, ) ((1, ), (2, ))

()(1 + 2, ) (()(1, ), ()(2, )) and

()(1, ) = sup (1, ) sup (1, )

Thus, ()(1, ) ()(1, )

Then ()(12, ) = sup (12, ) ((1, ), (2, ))

Hence ()(12, ) (()(1, ), ()(2, ))

Therefore, , (, , ) is a Qfuzzy soft subring over .

Theorem 5.7 Let (, , ) be a Qfuzzy soft subring over . Let , : be a Qfuzzy soft

homomorphism. Then , 1(, , ) is a Qfuzzy soft subring over .

Proof Let 1, 2 , .

1()(1 + 2, ) = ()((1 + 2), ) = ()((1) + (2), )

Since is Qfuzzy soft subring,

(()((1), ), ()((2), )). Thus,

(1()(1, ), 1()(2, ))

1()(1, ) = ()((1), ) = ()((1), ) ()((1), )

Thus, 1()(1, ) 1()(1, )

1()(12, ) = ()((12), ) = ()((1)(2), )

(()((1), ), ()((2), )). Thus,

(1()(1, ), 1()(2, ))

Hence proved.

Theorem 5.8 Let , be a Qfuzzy soft isomorphism. Then, (, , ) is Qfuzzy soft subring over T , (, , ) is Qfuzzy soft subring over . Proof Assume that , , ) is a Qfuzzy soft subring over . We prove that its image ((), (), ) is a Qfuzzy soft subring over .

Let 1, 2 , .

Since is surjective, there exist 1, 2 such that (1) = 1, (2) = 2.

Then ()(1 + 2, ) = sup

()=1+2

(, ).

Since is a homomorphism, (1 + 2) = 1 + 2.

Thus ()(1 + 2, ) (1 + 2, ).

Since is a Qfuzzy soft subring, (1 + 2, ) ((1, ), (2, )).

Hence ()(1 + 2, ) ((1, ), (2, )).

Taking supremum over all such 1, 2

()(1 + 2, ) (()(1, ), ()(2, )).

And let . Take such that () = . Then ()(, ) = sup (, ).

Since is Qfuzzy soft subring, (, ) (, ).

Thus ()(, ) ()(, ).

Then, ()(12, ) = sup (12, ).

Since (12, ) ((1, ), (2, )),

We obtain ()(12, ) (()(1, ), ()(2, )).

Hence , (, , ) is a Qfuzzy soft subring over .

Conversely, assume that , (, , ) is a Qfuzzy soft subring over . Since , is an isomorphism, is bijective and is bijective

Hence inverse exists, , 1.

We write (, , ) = , 1(, (, , )).

Since , (, , ) is a Qfuzzy soft subring, its inverse image under , 1 is also a Qfuzzy soft subring. Thus (, , ) is a Qfuzzy soft subring over . (, , ) is Qfuzzy soft subring , (, , ) is Qfuzzy soft subring. Example 5.9 Let = = , = {}.

Define

(, ) = {0.8, even

0.3, otherwise

and define () = 2, = identity

Then ()(, ) satisfies all Qfuzzy soft subring conditions. Hence the image is a Qfuzzy soft subring.

6. CONCLUSION WITH FUTURE RESEARCH

In this paper, we developed a unified and systematic framework for Qfuzzy soft algebraic structures by introducing and studying Qfuzzy soft subrings, Qfuzzy soft subrings, and their extension to Q fuzzy soft modules under a general T-norm. We established fundamental properties, including intersection, complement, homomorphic image and pre-image, isomorphism invariance, and generated Qfuzzy soft submodules, supported by rigorous theorems, proofs, and illustrative examples. The integration of module theory with Qfuzzy soft structures provide a natural and significant generalization of classical and fuzzy algebraic systems. This work opens several promising directions for future research, such as Qfuzzy soft quotient modules, homological aspects (exact sequences and projectivity), categorical formulations, extensions to intuitionistic and interval-valued Qfuzzy soft modules, and potential applications in decision-making systems, artificial intelligence, and uncertainty modelling.

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