Soft Symmetric Difference Complement-Difference Product of Groups

1. Introduction

A broad range of comprehensive numerical systems has been suggested to illustrate and analyze frameworks influenced by uncertainty, ambiguity, and indeterminacy—key characteristics prevalent in fields like engineering, economics, social sciences, and medical diagnostics. Foundational theories such as fuzzy set theory—first introduced by Zadeh [1]—and traditional probabilistic models provide initial tools for addressing these issues, but they often face epistemological and mathematical limitations. Specifically, fuzzy set theory relies on the subjective assignment of membership grades. At the same time, probabilistic methods necessitate access to well-defined distributions and repeatable experiments—conditions that are frequently absent in real-world ambiguous situations. To address these limitations, Molodtsov [2] proposed soft set theory as a parameter-driven, logarithmically adaptable framework that eliminates the need for probabilistic or fuzzy assumptions. Initial operational definitions by Maji et al. [3] were later refined by Pei and Miao [4] through an information-theoretic perspective, enhancing their applicability in social and multivalued systems. This foundation was further developed by Ali et al. [5], [6], who introduced limited and extended operations, significantly improving the representational and mathematical flexibility of soft sets. Numerous subsequent studies have advanced the field by clarifying definitions, introducing new binary operations, and formalizing generalized soft equivalences.

In recent years, the mathematical landscape of soft set theory has significantly expanded, as evidenced by various efforts aimed at developing coherent and extensible parallel operations. These advancements have focused on generalizing the fundamental concepts of soft subsethood and equivalence. Key contributions by Pei and Miao [4], Feng et al. [7], [8], and Qin and Hong [9] established the groundwork for broader theoretical models, which were later enhanced by Jun and Yang [10] and Liu et al. [11] through the development of J-soft and L-soft equivalence relations. Feng and Li [12] further advanced the theory by categorizing soft subsets under L-equality and demonstrating that specific remainder structures satisfy essential semigroup properties, including associativity, commutativity, and distributivity. More recent generalizations—such as g-soft, gf-soft, and T-soft equivalences—have been introduced by Abbas et al. [13], [14] and Alshami and El-Shafei [15], integrating congruence-based and lattice-theoretic approaches into the mathematical analysis of soft sets. A significant breakthrough came from Çağman and Enginoğlu [16], whose foundational modification resolved key inconsistencies and established a coherent and mathematically sound basis for ongoing theoretical advancements. This enhanced theoretical foundation has facilitated the systematic integration of soft set operations into classical mathematical structures. The subtle distinction between the union and difference operations has been modified to fit within the contexts of ring theory [17], semigroup theory [18], and group theory [19]. Conversely, the dual operation—the soft union-difference product—has been applied in group-theoretic [20], semigroup-theoretic [21], and ring-theoretic [22] frameworks. The resulting algebraic behaviors are fundamentally linked to the underlying structures.

Building on this foundation, this research introduces a new binary operation known as the soft symmetric difference complement-difference product, which is defined over soft sets with parameter domains shaped by group-theoretic principles. This operation is developed within a strictly axiomatic framework and is subjected to comprehensive algebraic analysis. Important properties such as closure, associativity, commutativity, and idempotency are rigorously established, along with an exploration of their connections to identity elements, null and absolute soft sets, and absorbing elements. Furthermore, the operation is demonstrated to be fully compatible with generalized notions of soft subsethood and soft equality, seamlessly integrating with the existing algebraic structure of soft set theory. To evaluate its theoretical importance and structural role, a detailed comparative analysis is performed against established soft set operations, focusing on its behavior within layered inclusion hierarchies. By abstracting group-theoretic axioms into parameter-dependent soft structures, this operation lays the groundwork for a generalized soft group theory. In this algebraic system, soft sets indexed by group-structured parameters emulate classical group behavior through formally defined operations. As a result, this work not only introduces a significant algebraic advancement but also establishes a robust theoretical foundation for applying soft set theory in fields that require formal management of uncertainty, abstract algebraic representation, and multi-criteria decision-making frameworks. Manuscript Structure: Section 2 provides the fundamental algebraic foundations and formal definitions that underpin the theoretical framework. Section 3 introduces the soft symmetric difference complement-difference product and thoroughly investigates its associated algebraic theory. Finally, Section 4 summarizes the main conclusions and proposes potential directions for further development of soft algebra in systems that address uncertainty.

2. Preliminaries

Molodtsov's [2] initial development of soft set theory established a parameter-driven approach to represent uncertainty, but it did not possess the necessary algebraic precision for incorporation into abstract algebraic systems. This shortcoming was successfully remedied by the axiomatic reconstruction by Çağman and Enginoğlu [16], which offered a logically sound and structurally strong basis. The current research is entirely based on this enhanced framework, which supports all definitions, operations, and algebraic structures presented in this work.

Definition 1 ([16]). Let E be a parameter set, U be a universal set, and P(U) be the power set of U, and ℋ ⊆ E. Then, the soft set ʆΥ over U is a function such that ℑℋ: E → P(U), where for all w ∉ ℋ, ℑℋ(w) = ∅. That is,

ℑℋ = {(w, ℑℋ(w)): w ∈ E}.

From now on, the soft set over U is abbreviated by .

Definition 2 ([16]). Let ℑℋ be an . If ℑℋ(w) = ∅ for all w ∈ E, then ℑℋ is called a null ЅЅ and indicated by ∅E, and if ℑℋ(w) = U, for all w ∈ E, then ℑℋ is called an absolute ЅЅ and indicated by UE.

Definition 3 ([16]). Let ℑℋ and ℘ℵ be two s. If ℑℋ(w) ⊆ ℘ℵ(w), for all w ∈ E, then ℑℋ is a soft subset of ℘ and indicated by ℑ ⊆ ̃ ℘ . If ℑ (w) = ℘ (w), for all w ∈ E, then ℑ is called soft equal to ℘ , and

ℵ ℋ ℵ ℋ ℵ ℋ ℵ

denoted by ℑℋ = ℘ℵ.

Definition 4 ([16]). Let fℋ be an . Then, the complement of fℋ denoted by fℋ , is defined by the soft set

c c ′

fℋ : E → P(U) such that fℋ (e) = U\fℋ(e) = (fℋ(e)) , for all e ∈ E.

Definition 5 ([16]). Let ℑℋ and ℘ℵ be two s. Then, the symmetric difference of ℑℋ and ℘ℵ is the ℑ ∆ ̃℘ , where (ℑ ∆ ̃℘ )(w) = ℑ (w)∆℘ (w), for all w ∈ E.

ℋ ℵ ℋ ℵ ℋ ℵ

Definition 6 ([23]). Let ℑ and ℘ be two s. Then, ℑ is called a soft S-subset of ℘ , denoted by ℑ ⊆ ̃ ℘

K ℵ K ℵ K S ℵ

if for all w ∈ E, ℑK(w) = ℳ and ℘ℵ(w) = , where ℳ and are two fixed sets and ℳ ⊆ . Moreover, two ЅЅs ℑ and ℘ are said to be soft S-equal, denoted by ℑ = ℘ , if ℑ ⊆ ̃ ℘ and ℘ ⊆ ̃ ℑ .

K ℵ K S ℵ K S ℵ ℵ S K

It is obvious that if ℑK =S ℘ℵ, then ℑK and ℘ℵ are the same constant functions, that is, for all w ∈ E, ℑK(w)= ℘ℵ(w) = ℳ, where ℳ is a fixed set.

Definition 7 ([23]). Let ℑK and ℘ℵ be two s. Then, ℑK is called a soft A-subset of ℘ℵ, denoted by ℑ ⊆ ̃ ℘ , if, for each , ∈ E, ℑ ( ) ⊆ ℘ ( ).

K A ℵ K ℵ

Definition 8 ([23]). Let ℑK and ℘ℵ be two s. Then, ℑK is called a soft S-complement of ℘ℵ, denoted by ℑK =S (℘ℵ) , if, for all w ∈ E, ℑK(w) = ℳ and ℘ℵ(w) = , where ℳ and are two fixed sets and ℳ = ′. Here, = U\ .

For additional information on ЅЅs, we refer to [24]-[50].

From now on, let G be a group, and SG(U) denotes the collection of all s over U, whose parameter sets are G; that is, each element of SG(U) is an parameterized by G. Moreover, let ∆ represent the classical symmetric difference operation, which is commutative and associative, in classical set theory. Then, the symmetric difference of the family = {Ci: i ∈ I} such that I is an index set, is denoted by

△ =△ Ci = C1∆C2∆ … ∆Cn.

Definition 9. Let G and ℊG be two s. Then, the soft symmetric difference-lambda product G⨂s/lℊG is defined by

( G⨂s/lℊG)(x) = △ ( G (y)λℊG( )) = △ ( G (y) ∪ (ℊG( )) ) , y, ∈ G,

x=y x=y for all x ∈ G.

3. Soft Symmetric Difference Complement-Difference Product of Group

This section provides an in-depth algebraic examination of the soft symmetric difference complement– difference product, a newly defined binary operation on soft sets that are based on group-theoretically structured parameter domains. The analysis is conducted within a strict axiomatic framework, demonstrating the operation's key properties—closure, associativity, commutativity, and idempotency—thus establishing its function as an internal operation in soft set algebra. Additionally, the operation is shown to correspond with generalized soft subsethood and soft equality, which are crucial for defining morphisms and organizing algebraic substructures. The discussion particularly highlights the operation's placement within stratified inclusion lattices, ensuring its structural integrity and smooth integration into the broader algebraic framework of soft set theory.

From now on, the symmetric difference complement of the family = {Ci: i ∈ I} such that I is an index set, is denoted by

∐ = ∐ Ci = (C1∆C2∆ … ∆Cn) .

Definition 1. Let ℑG and ℘G be two s. Then, the soft symmetric difference complement-difference product ℑG⨂ ′ ℘G is defined by

(ℑ ⨂ ′ ℘ )(x) = ∐(ℑ (y) ∖ ℘ ( )), y, ∈ G,

G s /d G G G

for all x ∈ G.

Note here that since G is a group, there always exists y, ∈ G such that = y , for all x ∈ G. Let the order of the group G be n, that is, |G| = n. Then, it is obvious that there exist n different combinations of writing styles for each x ∈ G such that x = y , where y, ∈ G.

Note 1: The soft symmetric difference complement-difference product is well-defined in SG(U). In fact, let ℑG, ℘G, G, G ∈ SG(U) such that (ℑG, ℘G) = ( G, G). Then, ℑG = G and ℘G = G, implying that ℑG(x) = G(x) and ℘G(x) = G(x), for all x ∈ G. Thereby, for all x ∈ G,

(ℑ ⨂ ′ ℘ )(x) = ∐(ℑ (y) ∖ ℘ ( )),

G s /d G G G

∐( G(y) ∖ G( ))

= ( ⨂ ′ )(x).

G s /d G

Hence, ℑG⨂ ′ ℘G = G⨂ ′ G.

s /d s /d

Example 1. Consider the group G = {ℴ, ρ} with the following operation:

Let ℑG and ℘G be two s over U = D2 = {< x, y >: x = y = e, xy = yx} = {e, x, y, yx} as follows:

ℑG = {(ℴ, {x, yx}), (ρ, {e, x})} and ℘G = {(ℴ, {e, y}), (ρ, {e})}.

Since ℴ = ℴℴ = ρρ, (ℑG⨂ ′ ℘G)(ℴ) = ((ℑG (ℴ) ∖ ℘G(ℴ))∆(ℑG (ρ) ∖ ℘G(ρ))) = {e, x, y}, and since ρ =

ℴρ = ρℴ, (ℑG⨂ ′ ℘G)(ρ) = ((ℑG (ℴ) ∖ ℘G(ρ))∆(ℑG (ρ) ∖ ℘G(ℴ))) = {e, x, y} is obtained. Hence,

ℑ ⨂ ′ ℘ = {(ℴ, {e, x, y}), (ρ, {e, x, y})}.

G s /d G

Proposition 1. The set SG(U) is closed under the soft symmetric difference complement-difference product. That is, if ℑG and ℘G are two s, then so is ℑG⨂ ′ ℘G.

Proof: It is obvious that the soft symmetric difference complement-difference product is a binary operation in SG(U). Thereby, SG(U) is closed under the soft symmetric difference complement-difference product. Proposition 2. The soft symmetric difference complement-difference product is not associative in SG(U). Proof: Consider the group G and the s ℑG and ℘G in Example 1, let ℷG be an over U = {e, x, y, yx} such that ℷG = {(ℴ, {x, y}), (ρ, {y})}.

Since ℑG⨂ ′ ℘G = {(ℴ, {e, x, y}), (ρ, {e, x, y})}, then

(ℑ ⨂ ′ ℘ )⨂ ′ ℷ = {(ℴ, {e, y, yx}), (ρ, {e, y, yx})}.

G s /d G s /d G

Moreover, since ℘G⨂ ′ ℷG = {(ℴ, U), (ρ, U)}, then

ℑ ⨂ ′ (℘ ⨂ ′ ℷ ) = {(ℴ, U), (ρ, U)}.

G s /d G s /d G

Thereby, (ℑG⨂ ′ ℘G)⨂ ′ ℷG ≠ ℑG⨂ ′ (℘G⨂ ′ ℷG).

s /d s /d s /d s /d

Proposition 3. The soft symmetric difference complement-difference product is not commutative in SG(U). Proof: Consider the s ℑG and ℘G over U = {e, x, y, yx} in Example 1, then,

(ℑG⨂ ′ ℘G)(x) = {(ℴ, {e, x, y}), (ρ, {e, x, y})} and(℘G⨂ ′ ℑG)(x) = {(ℴ, {x, yx}) , (ρ, {x, yx})}

s /d s /d

implying that ℑG⨂ ′ ℘G ≠ ℘G⨂ ′ ℑG.

s /d s /d

Proposition 4. The soft symmetric difference complement-difference product is not idempotent in SG(U). Proof: Consider the ℑG in Example 1, then,

ℑ ⨂ ′ ℑ = {(ℴ, U), (ρ, {x, y})}.

G s /d G

Implying that ℑG⨂ ′ ℑG ≠ ℑG.

Proposition 5. Let ℑG be a constant . Then, ℑG⨂ ′ ℑG = UG.

Proof: Let ℑG be a constant that, for all x ∈ G, ℑG (x) = A, where A is a fixed set. Then, for all x ∈ G,

(ℑ ⨂ ′ ℑ )(x) = ∐(ℑ (y) ∖ ℑ ( )) = U (x).

G s /d G G G G

Thereby, ℑG⨂ ′ ℑG = UG.

Remark 1. Let SG (U) be the collection of all constant . Then, the soft symmetric difference complement-difference product is not idempotent in SG (U) either.

Proposition 6. Let ℑG be an . Then, ℑG⨂ ′ UG = UG.

Proof: Let ℑG be an . Then, for all x ∈ G, (ℑ ⨂ ′ U )(x) = ∐(ℑ (y) ∖ U ( )) = ∐(ℑ (y) ∖ U) = U (x).

G s /d G G G G G x=y x=y

Thus, ℑG⨂ ′ UG = UG.

Remark 2. UG is the right absorbing element of the soft symmetric difference complement-difference product in SG(U).

Proposition 7. UG is not the left absorbing element of the soft symmetric difference complement-difference product in SG(U).

Proof: Consider the ℑG in Example 1, then,

U ⨂ ′ ℑ = {(ℴ, {x, y}), (ρ, {x, y})}.

G s /d G

Implying that UG⨂ ′ ℑG ≠ ℑG.

Remark 3. UG is not the absorbing element of the soft symmetric difference complement-difference product in SG(U).

Proposition 8. Let ℑG be a constant . Then, I. U ⨂ ′ ℑ = ℑ , where |G| = and is a positive odd integer.

G s /d G G

II. U ⨂ ′ ℑ = U , where |G| = and is a positive even integer.

G s /d G G

Proof: Let ℑG be a constant such that, for all x ∈ G, ℑG (x) = A, where A is a fixed set.

I. Let |G| = , where is a positive odd integer. Then, for all x ∈ G,

c c c

(U ⨂ ′ ℑ )(x) = ∐(U (y) ∖ ℑ ( )) = ( ℑ ⏟ ∆ℑ … ∆ℑ ) = ℑ (x).

G s /d G G G G G G G x=y times ℑG ,where is odd

Thereby, ⨂ ′ ℑ = ℑ .

II. Let |G| = , where is a positive even integer. Then, for all x ∈ G,

c c c

(U ⨂ ′ ℑ )(x) = ∐(U (y) ∖ ℑ ( )) = ( ℑ ⏟ ∆ℑ … ∆ℑ ) = U (x).

G s /d G G G G G G G x=y times ℑG ,where is even

Thereby, UG ⨂ ′ ℑG = UG.

Remark 4. UG is the absorbing element of the soft symmetric difference complement-difference product in SG (U), where |G| = and is a positive even integer, by Proposition 6 and Proposition 8(II). Besides, UG is the left identity element of the soft symmetric difference complement-difference product in SG (U), where |G| = and is a positive odd integer, by Proposition 8(I).

Proposition 9. Let ℑG be an . Then, ∅G⨂ ′ ℑG = UG.

Proof: Let ℑG be an . Then, for all x ∈ G,

(∅ ⨂ ′ ℑ )(x) = ∐(∅ (y) ∖ ℑ ( )) = ∐(∅ ∖ ℑ ( )) = U (x).

G s /d G G G G G x=y x=y

Thereby, ∅G⨂ ′ ℑG = UG.

Proposition 9. Let ℑG be a constant . Then,

I. ℑ ⨂ ′ ∅ = ℑ , where |G| = and is a positive odd integer.

G s /d G G II. ℑ ⨂ ′ ∅ = U , where |G| = and is a positive even integer.

G s /d G G

Proof: Let ℑG be a constant such that, for all x ∈ G, ℑG (x) = A, where A is a fixed set.

I. Let |G| = , where is a positive odd integer. Then, for all x ∈ G,

(ℑ ⨂ ′ ∅ )(x) = ∐(ℑ (y) ∖ ∅ ( )) = ∐(ℑ (y) ∖ ∅) = ℑ (x).

G s /d G G G G G x=y x=y

Thereby, ℑG⨂ ′ ∅G = ℑG

II. Let |G| = , where is a positive even integer. Then, for all x ∈ G,

(ℑ ⨂ ′ ∅ )(x) = ∐(ℑ (y) ∖ ∅ ( )) = ∐(ℑ (y) ∖ ∅) = U (x).

G s /d G G G G G x=y x=y

Thereby, ℑG⨂ ′ ∅G = UG. ◻

Proposition 10. Let ℑG be a constant . Then,

I. ℑ ⨂ ′ ℑ = ℑ , where |G| = and is a positive odd integer.

G s /d G G

II. ℑ ⨂ ′ ℑ = U , where |G| = and is a positive even integer.

G s /d G G

Proof: Let ℑG be a constant such that, for all x ∈ G, ℑG (x) = A, where A is a fixed set.

I. Let |G| = , where is a positive odd integer. Then, for all x ∈ G,

(ℑ ⨂ ′ ℑ )(x) = ∐(ℑ (y) ∖ ℑ ( )) = ℑ (x).

G s /d G G G G

Thereby, ℑ ⨂ ′ ℑ = ℑ .

G s /d G G

II. Let |G| = , where is a positive even integer. Then, for all x ∈ G,

(ℑ ⨂ ′ ℑ )(x) = ∐(ℑ (y) ∖ ℑ ( )) = U (x).

G s /d G G G G

Thereby, ℑ ⨂ ′ ℑ = U .

G s /d G G

Proposition 11. Let ℑG be a constant . Then,

I. ℑ ⨂ ′ ℑ = ℑ , where |G| = and is a positive odd integer.

G s /d G G

II. ℑ ⨂ ′ ℑ = U , where |G| = and is a positive even integer.

G s /d G G

Proof: Let ℑG be a constant such that, for all x ∈ G, ℑG (x) = A, where A is a fixed set.

I. Let |G| = , where is a positive odd integer. Then, for all x ∈ G,

c c c

(ℑ ⨂ ′ ℑ )(x) = ∐(ℑ (y) ∖ ℑ ( )) = ℑ (x).

G s /d G G G G

Thereby, ℑ ⨂ ′ ℑ = ℑ .

G s /d G G

II. Let | | = , where is a positive even integer. Then, for all x ∈ G,

(ℑ ⨂ ′ ℑ )(x) = ∐(ℑ (y) ∖ ℑ ( )) = U (x).

G s /d G G G G

Thereby, ℑ ⨂ ′ ℑ = U .

G s /d G G

Proposition 12. Let ℑ and ℘ be two s such that ℑ ⊆ ̃ ℘ . Then, ℑ ⨂ ′ ℘ = U .

G G G A G G s /d G G Proof: Let ℑ and ℘ be two s and ℑ ⊆ ̃ ℘ . Then, ℑ (y) ⊆ ℘ ( ), for each y, ∈ G. Then, for all x ∈ G,

G G G A G G G

(ℑ ⨂ ′ ℘ )(x) = ∐(ℑ (y) ∖ ℘ ( )) = U (x).

G s /d G G G G

Thereby, ℑG⨂ ′ ℘G = UG.

Proposition 13. Let ℑ and ℘ be two s such that ℑ ⊆ ̃ ℘ . Then, ℑ ⨂ ′ ℘ = U .

G G G S G G s /d G G

Proof: The proof is similar to the proof of Proposition 12.

Proposition 14. Let ℑG and ℘G be two s such that ℑG ⊆S (℘G) . Then,

I. ℑ ⨂ ′ ℘ = ℑ , where |G| = and is a positive odd integer.

G s /d G G

II. ℑ ⨂ ′ ℘ = U , where |G| = and is a positive even integer.

G s /d G G

Proof: Let ℑG and ℘G be two s and ℑG ⊆S (℘G) . Then, for all x ∈ G, ℑG(x) = A, ℘G(x) = B, where A and B are two fixed sets and A ⊆ B′.

I. Let |G| = , where is a positive odd integer. Then, for all x ∈ G,

(ℑ ⨂ ′ ℘ )(x) = ∐(ℑ (y) ∖ ℘ ( )) = ℑ (x).

G s /d G G G G

Thereby, ℑ ⨂ ′ ℘ = ℑ .

G s /d G G

II. Let |G| = , where is a positive even integer. Then, for all x ∈ G,

(ℑ ⨂ ′ ℘ )(x) = ∐(ℑ (y) ∖ ℘ ( )) = U (x).

G s /d G G G G

Thereby, ℑ ⨂ ′ ℘ = U .

G s /d G G

Proposition 15. Let ℑG and ℘G be two s such that (℘G) ⊆S ℑG. Then, I. ℑ ⨂ ′ ℘ = ℘ , where |G| = and is a positive odd integer.

G s /d G G

II. ℑ ⨂ ′ ℘ = U , where |G| = and is a positive even integer.

G s /d G G

Proof: Let ℑG and ℘G be two s and (℘G) ⊆S ℑG. Then, for all x ∈ G, ℑG(x) = A, ℘G(x) = B, where A and B are two fixed sets and B′ ⊆ A.

I. Let |G| = , where is a positive odd integer. Then, for all x ∈ G,

(ℑ ⨂ ′ ℘ )(x) = ∐(ℑ (y) ∖ ℘ ( )) = ℘ (x).

G s /d G G G G

Thereby, ℑ ⨂ ′ ℘ = ℘ . Here note that ℑ (y) ∖ ℘ ( ) = ℘ ( ) ∩ ℑ (y), for all y, ∈ G,

G s /d G G G G G G

II. Let | | = , where is a positive even integer. Then, for all x ∈ G,

(ℑ ⨂ ′ ℘ )(x) = ∐(ℑ (y) ∖ ℘ ( )) = U (x).

G s /d G G G G

Thereby, ℑ ⨂ ′ ℘ = U .

G s /d G G

Proposition 16. Let ℑG and ℘G be two s. Then, (ℑG⨂ ′ ℘G) = ℑG⨂s/p℘G.

Proof: Let ℑG and ℘G be two s. Then, for all x ∈ G, (ℑ ⨂ ′ ℘ ) (x) = (∐(ℑ (y) ∖ ℘ ( ))) ,

G s /d G G G

= △ (ℑG(y) ∖ ℘G( )) ,

= △ (ℑG(y) ∩ ℘G ( )),

= △ (ℑG (y) ∪ ℘G( )).

Thereby, (ℑG⨂ ′ ℘G) = ℑG⨂s/p℘G.

4. Conclusion

This research presents the soft symmetric difference complement–difference product, a new binary operation on soft sets that are influenced by group-theoretic structures. The operation is analyzed within a detailed algebraic framework, focusing on its relationship with generalized soft equality and its function across different levels of soft subsethood. A comparative analysis showcases the operation's expressive capabilities and algebraic coherence in relation to existing soft set operations. The study also investigates its alignment with fundamental concepts like null and absolute soft sets, as well as its consistency within group-parameterized binary operations. Essential algebraic properties—such as closure, associativity, commutativity, and idempotency—are thoroughly confirmed, including conditions related to identity, inverse, and absorbing elements. The resulting structure exhibits strong internal consistency, extending traditional algebraic concepts into the realm of soft sets. Fundamentally, this operation establishes a basis for a generalized soft group theory, where soft sets indexed by group-structured parameters mimic group-like behavior through abstract soft operations. Consequently, this work not only enhances the theoretical foundation of soft set theory but also expands its potential applications in fields like algebraic modeling and decision-making under uncertainty.

Author Contributions

İbrahim Durak: Investigation, visualization, conceptualization, writing-review, validation Aslıhan Sezgin: Supervision, visualization, conceptualization, validation, review

Conflicts of Interest

The authors stated that there are no conflicts of interest regarding the publication of this article.