A wide variety of detailed numerical systems have been proposed to represent and analyze frameworks affected by uncertainty, ambiguity, and indeterminacy - essential traits found in areas such as engineering, economics, social sciences, and medical diagnostics. Foundational concepts, such as fuzzy set theory introduced by Zadeh [1] and traditional probabilistic models, offer initial tools for tackling these challenges; however, they often encounter epistemological and mathematical constraints. For instance, fuzzy set theory relies on the subjective assignment of membership grades. In contrast, probabilistic approaches require well-defined distributions and repeatable experiments - conditions that are often lacking in real-world, ambiguous scenarios. To overcome these challenges, Molodtsov [2] introduced soft set theory, a parameter-driven and logarithmically adaptable framework that does not rely on probabilistic or fuzzy assumptions. Initial operational definitions by Maji et al. [3] were later refined by Pei and Miao [4] from an information-theoretic perspective, thereby enhancing their relevance in social and multivalued systems. This groundwork was further advanced by Ali et al. [5], who introduced limited and extended operations, greatly enhancing the representational and mathematical versatility of soft sets. Numerous subsequent studies [6–26] have progressed the field by clarifying definitions, proposing new binary operations, and formalizing generalized soft equivalences. In recent years, the mathematical framework of soft set theory has undergone significant expansion, as evidenced by various initiatives aimed at developing coherent and extensible parallel operations. These developments have concentrated on generalizing the core concepts of soft subsethood and equivalence. Significant contributions from Qin and Hong [27] laid the foundation for broader theoretical models, which were later refined by Jun and Yang [28] and Liu et al. [29] through the creation of J-soft and L-soft equivalence relations. Feng and Li [30] further advanced the theory by classifying soft subsets under L-equality and demonstrating that specific remainder structures fulfill essential semigroup properties, such as associativity, commutativity, and distributivity. More recent generalizations (such as g-soft, gf-soft, and T-soft equivalences) have been introduced by Abbas et al. [31], [32] and Al-shami [33], and Al-shami and El-Shafei [34], incorporating congruence-based and lattice-theoretic approaches into the mathematical study of soft sets. A breakthrough was achieved by Çağman and Enginoğlu [35], whose foundational modification addressed key inconsistencies and established a coherent and mathematically robust basis for further theoretical developments. This strengthened theoretical foundation has enabled the systematic incorporation of soft set operations into classical mathematical frameworks. The concept of the soft intersection product was first introduced for rings [36], semigroups [37], and groups [38], forming the basis for the development of soft union ring, semigroup, and group theories. Similarly, the soft intersection product was defined for groups [39], semigroups [40], and rings [41], with corresponding algebraic theories subsequently developed. Due to inherent differences among these algebraic structures, the definitions and properties of these products exhibit structural variations. In particular, the presence of a unit element and inverses in groups imparts unique characteristics to the group-based definitions.

Building on this groundwork, the research presents a novel binary operation called the soft symmetric difference complement-union product, which is defined over soft sets with parameter domains influenced by group-theoretic concepts. This operation is formulated within a strictly axiomatic framework and undergoes extensive algebraic examination. Key properties, such as closure, associativity, commutativity, and idempotency, are rigorously proven, along with an investigation into their relationships with identity elements, null and absolute soft sets, and absorbing elements. Additionally, the operation is shown to be entirely compatible with generalized concepts of soft subsethood and soft equality, integrating smoothly with the existing algebraic framework of soft set theory. To assess its theoretical significance and structural role, a comprehensive comparative analysis is conducted against established soft set operations, concentrating on its behavior within layered inclusion hierarchies. It is obtained that the proposed product is a noncommutative semigroup in the collection of soft sets with a fixed parameter set. By abstracting group-theoretic axioms into parameter-dependent soft structures, this operation establishes a foundation for a generalized soft group theory. In this algebraic system, soft sets indexed by group-structured parameters mimic classical group behavior through formally defined operations. Consequently, this work not only introduces a notable algebraic advancement but also creates a solid theoretical basis for applying soft set theory in areas that require formal management of uncertainty, abstract algebraic representation, and multi-criteria decision-making frameworks. Manuscript structure: Section 2 outlines the essential algebraic foundations and formal definitions that support the theoretical framework. Section 3 presents the soft symmetric difference complement-union product and thoroughly explores its related algebraic theory. Finally, Section 4 summarizes the key findings and suggests possible avenues for further development of soft algebra in systems that tackle uncertainty.

Molodtsov's early work on soft set theory introduced a parameter-based method for modeling uncertainty, but it lacked the required algebraic rigor for integration into abstract algebraic systems. Çağman and Enginoğlu [35] addressed this limitation through an axiomatic reconstruction that provided a solid and logically coherent foundation. The present research relies entirely on this improved framework, which underpins all definitions, operations, and algebraic structures discussed in this study.

Definition 1 ([35]). Let E be a parameter set, U be a universal set, P(U) be the power set of U, and ℋ ⊆ E. Then, the soft set ℑℋ 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 ([35]). 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 ([35]). 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 ([35]). 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 ([42]). 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 6 ([42]). 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 7 ([42]). 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 [43-59].

This section presents a comprehensive algebraic analysis of the soft symmetric difference complement–union product, a newly introduced binary operation on soft sets derived from group-theoretically organized parameter domains. The study is conducted within a rigorous axiomatic framework, highlighting the essential characteristics of the operation (closure, associativity, commutativity, and idempotency) which confirm its role as an internal operation in soft set algebra. Furthermore, the operation is linked to generalized soft subsethood and soft equality, which are vital for defining morphisms and structuring algebraic substructures. The discussion particularly emphasizes the operation's position within stratified inclusion lattices, ensuring its structural soundness and seamless integration into the broader algebraic framework of soft set theory. It is obtained that the proposed product is a noncommutative semigroup in the collection of soft sets with a fixed parameter set.

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, and 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 8. Let ℑG and ℘G be two s. Then, the soft symmetric difference complement-union product ℑG⨂ ′ ℘G is defined by

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

G s /u G G G

for all x ∈ G.

Note 1: the soft symmetric difference complement-union 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) ∪ ℘ ( )) = ∐( (y) ∪ ( )) = ( ⨂ ′ )(x).

G s /u G G G G G G s /u G x=y x=y

Hence,

ℑG⨂ ′ ℘G = G⨂ ′ G.

s /u s /u

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

∙ ℴ ρ

ℴ ℴ ρ

ρ ρ ℴ

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

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

ρℴ, (ℑ ⨂ ′ ℘ )(ρ) = ((ℑ (ℴ) ∪ ℘ (ρ))∆(ℑ (ρ) ∪ ℘ (ℴ))) = {e, x, yx} is obtained. Hence,

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

G s /u G

Proposition 1. The set SG(U) is closed under the soft symmetric difference complement-union 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-union product is a binary operation in SG(U). Thereby, SG(U) is closed under the soft symmetric difference complement-union product.

Proposition 2. The soft symmetric difference complement-union product is associative in SG(U).

Proof: let ℑG, ℘G, and ℷG be three s. Then, for all x ∈ G, ℑ ⨂ ′ (℘ ⨂ ′ ℷ )(x) = ∐ (ℑ (y) ∪ (℘ ⨂ ′ ℷ )( ))

G s /u G s /u G G G s /u G

= ∐ (ℑG (y) ∪ ( ∐ (℘G(m) ∪ ℷG(n))))

x=y =mn

= ∐ (ℑG (y) ∪ (℘G(m) ∪ ℷG(n)))

x=y(mn)

= ∐ ((ℑG (y) ∪ ℘G(m)) ∪ ℷG(n))

x=(ym)n

= ∐ (( ∐ (ℑG (y) ∪ ℘G(m))) ∪ ℷG(n))

x=an a=ym

= ∐ ((ℑ ⨂ ′ ℘ )(a) ∪ ℷ (n))

G s /u G G x=an

= (ℑ ⨂ ′ ℘ )⨂ ′ ℷ (x).

G s /u G s /u G

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

s /u s /u s /u s /u

Example 2. 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 ℷ = {(ℴ, {e}), (ρ, {x, y})}. Since ℑ ⨂ ′ ℘ = {(ℴ, {e}), (ρ, {e, x, yx})}, then

G G s /u G

(ℑG⨂ ′ ℘G)⨂ ′ ℷG = {(ℴ, {e}), (ρ, {e, x})}. Moreover, since ℘G⨂ ′ ℷG = {(ℴ, {e, y}), (ρ, {e, x})}, then

s /u s /u s /u

ℑG⨂ ′ (℘G⨂ ′ ℷG) = {(ℴ, {e}), (ρ, {e, x})}. Thereby, (ℑG⨂ ′ ℘G)⨂ ′ ℷG = ℑG⨂ ′ (℘G⨂ ′ ℷG).

s /u s /u s /u s /u s /u s /u

Proposition 3. The soft symmetric difference complement-union product is not commutative in SG(U). However, if G is an abelian group, then the soft symmetric difference complement-union product is commutative in SG(U).

Proof: let ℑG and ℘G be two s and G be an abelian group. Then, for all x ∈ G,

(ℑ ⨂ ′ ℘ )(x) = ∐(ℑ (y) ∪ ℘ ( )) = ∐(℘ ( ) ∪ ℑ (y)) = (℘ ⨂ ′ ℑ )(x),

G s /u G G G G G G s /u G x=y x= y

implying that ℑG⨂ ′ ℘G = ℘G⨂ ′ ℑG.

s /u s /u

Proposition 4. The soft symmetric difference complement-union product is not idempotent in SG(U).

Proof: consider the ℑG in Example 1.

Then, ℑ ⨂ ′ ℑ = {(ℴ, {y}), (ρ, U)}, implying that ℑ ⨂ ′ ℑ ≠ ℑ .

Theorem 1. (SG(U), ⨂ ′ ) is a noncommutative semigroup. If G is abelian, then (SG(U), ⨂ ′ ) is a

s /u s /u

commutative semigroup.

Proof: the proof is followed by Propositions 1-4.

Proposition 5. Let ℑG be a constant . Then, I. ℑ ⨂ ′ ℑ = ℑ , where |G| = and is a positive odd integer.

G s /u G G

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

G s /u 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 ( )) = (A ⏟∆A∆ … ∆A) = A = ℑ (x).

G s /u G G G G x=y times A,where is even

Thereby, ℑG⨂ ′ ℑG = ℑG .

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

(ℑ ⨂ ′ ℑ )(x) = ∐(ℑ (y) ∪ ℑ ( )) = (A ⏟∆A∆ … ∆A) = ∅ = U (x).

G s /u G G G G x=y times A,where is even

Thereby, ℑG⨂ ′ ℑG = UG.

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

Proposition 6. Let ℑG be an . Then, I. ℑ ⨂ ′ U = U ⨂ ′ ℑ = ∅ , where |G| = and is a positive odd integer.

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

Proof: let ℑG be an .

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

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

x=y x=y

Thus, ℑG⨂ ′ UG = ∅G. Similarly, for all x ∈ G,

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

x=y x=y

Thus, UG⨂ ′ ℑG = ∅G.

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

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

x=y x=y

Thus, ℑG⨂ ′ UG = UG. Similarly, for all x ∈ G,

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

x=y x=y

Thus, UG⨂ ′ ℑG = UG.

Remark 2. UG is the absorbing element of the soft symmetric difference complement-union product in SG(U), where |G| = and is a positive even integer.

Proposition 7. Let ℑG be a constant . Then, I. ∅ ⨂ ′ ℑ = ℑ ⨂ ′ ∅ = ℑ , where |G| = and is a positive odd integer.

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

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

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

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

x=y x=y

Thereby, ∅G⨂ ′ ℑG = ℑG . Similarly, for all x ∈ G,

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

x=y x=y

Thereby, ℑG⨂ ′ ∅G = ℑG .

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

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

x=y x=y

Thereby, ∅G⨂ ′ ℑG = UG. Similarly, for all x ∈ G,

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

x=y x=y

Thereby, ℑG⨂ ′ ∅G = UG.

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

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

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 ( )) = ∅ (x).

G s /u G G G G

Thereby, ℑG ⨂ ′ ℑG = ∅G. Similarly, for all x ∈ G,

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

G s /u G G G G

Thereby, ℑG⨂ ′ ℑG = ∅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 /u G G G G

Thereby, ℑG ⨂ ′ ℑG = UG. Similarly, for all x ∈ G,

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

G s /u G G G G

Thereby, ℑG⨂ ′ ℑG = UG.

Proposition 9. Let ℑ and ℘ be two s such that ℑ ⊆ ̃ ℘ . Then,

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

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

Proof: let ℑ and ℘ be two s and ℑ ⊆ ̃ ℘ . Then, for all x ∈ G, ℑ (x) = A, ℘ (x) = B, where A and B

G G G S G G G

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) ∪ ℘ ( )) = ℘ (x).

G s /u G G G G

Thereby, ℑG⨂ ′ ℘G = ℘G . Similarly, for all x ∈ G,

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

G s /u G G G G

Thereby, ℘G⨂ ′ ℑG = ℘G .

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

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

G s /u G G G G

Thereby, ℑG⨂ ′ ℘G = UG. Similarly, for all x ∈ G,

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

G s /u G G G G

Thereby, ℘G⨂ ′ ℑG = UG.

Proposition 10. Let ℑ and ℘ be two s such that ℑ ⊆ ̃ ℘ . Then,

G G G A G

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

G s /u G G

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

G s /u G G

Proof: let ℑ and ℘ be two s and ℑ ⊆ ̃ ℘ . Then, ℑ (y) ⊆ ℘ ( ), for each y, ∈ G.

G G G A G G G

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

(ℑ ⨂ ′ ℘ )(x) = ∐(ℑ (y) ∪ ℘ ( )) = ∅ (x).

G s /u G G G G

Thereby, ℑG⨂ ′ ℘G = ∅G.

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

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

G s /u G G G G

′ ′ ′ ′

Thereby, ℑ ⨂ ′ ℘ = U . Here, note that if A ⊆ B, then A ∩ B = (A ∪ B) = ∅.

G s /u G G

Remark 3. Let ℑ and ℘ be two s such that ℑ ⊆ ̃ ℘ . Then,

G G G S G

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

G s /u G G

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

G s /u G G

Proof: the proof is similar to the proof of Proposition 10.

Proposition 11. Let ℑG and ℘G be two s. Then,(ℑG⨂ ′ ℘G) = ℑG⨂s/u℘G.

Proof: let ℑG and ℘G be two s. Then, for all x ∈ G,

(ℑ ⨂ ′ ℘ ) (x) = (∐(ℑ (y) ∪ ℘ ( ))) = △ (ℑ (y) ∪ ℘ ( ))

= △ (ℑG(y) ∪ ℘G( )) = (ℑG⨂s/u℘G)(x).

Thereby, (ℑG⨂ ′ ℘G) = ℑG⨂s/u℘G. Here note that, in classical set theory, A ∆B = A∆B, where A and B are fixed sets.

This research introduces the soft symmetric difference complement-union product, a novel binary operation on soft sets that is influenced by group-theoretic structures. The operation is examined within a comprehensive algebraic framework, emphasizing its connection to generalized soft equality and its role across various levels of soft subsethood. A comparative analysis highlights the operation's expressive potential and algebraic consistency in relation to existing soft set operations. The study also explores its relationship with key concepts such as null and absolute soft sets, as well as its coherence within group-parameterized binary operations. Key algebraic properties (like closure, associativity, commutativity, and idempotency) are thoroughly validated, including conditions concerning identity, inverse, and absorbing elements. It is obtained that the proposed product is a noncommutative semigroup in the collection of soft sets with a fixed parameter set. The resulting structure demonstrates strong internal consistency, extending traditional algebraic concepts into the domain of soft sets. Ultimately, this operation lays the groundwork for a generalized soft group theory, where soft sets indexed by group-structured parameters exhibit group-like behavior through abstract soft operations. As a result, this work not only strengthens the theoretical basis of soft set theory but also broadens its potential applications in areas such as algebraic modeling and decision-making under uncertainty.

Zeynep Ay: Investigation, Visualization, Conceptualization, Writing-Review, Validation.

Aslıhan Sezgin: Supervision, Visualization, Conceptualization, Validation, Review.

The authors have given consent for the publication of this manuscript.

This study does not involve any research conducted on human participants or animals.

No external funding was received for this research.

N/A.

The authors declare that they have no conflicts of interest regarding the publication of this article.