Numerous mathematically rigorous frameworks have been proposed to analyze systems characterized by uncertainty, ambiguity, and indeterminacy features that are intrinsic to fields such as engineering, economics, the social sciences, and medical diagnostics. Classical approaches like Zadeh’s fuzzy set theory [1] and conventional probabilistic models, although theoretically valuable, are constrained by inherent algebraic and epistemological limitations. Fuzzy set theory, for instance, relies on the subjective assignment of membership degrees, whereas probabilistic models assume the existence of well-defined distributions and repeatable conditions. In practical contexts marked by non-repeatability and epistemic uncertainty, these assumptions often prove inadequate.

To address these shortcomings, Molodtsov [2] introduced soft set theory a parameterized, algebraically flexible framework that models uncertainty without relying on fuzzy or probabilistic interpretations. Building upon the foundational operations defined by Maji et al. [3], Pei and Miao [4] further extended soft set theory by adapting these operations to relational and multi-valued domains through an information-theoretic lens. Ali et al. [5] advanced the operational calculus by defining restricted and extended forms of classical operations, thereby enhancing the representational granularity and algebraic versatility of soft set systems. Subsequent contributions by Yang [6], Feng et al. [7], Jiang et al. [8], Ali et al. [9], Neog and Sut [10], Li [11], Ge and Yang [12], Singh and Onyeozili [13–16], Zhu and Wen [17], Onyeozili and Gwary [18], and Sen [19], among others, have enriched the theoretical framework by resolving foundational ambiguities, introducing new binary operations, and generalizing notions of soft equality.

In recent years, sustained efforts [20–36] have significantly expanded the algebraic infrastructure of soft set theory through the systematic development of novel binary operations and the establishment of a unified, extensible theoretical framework. Parallel developments in the generalization of soft subsethood and equality have further solidified the discipline. Foundational work by Pei and Miao [4], Feng and Li [37], and Qin and Hong [35] laid the groundwork for broader structural generalizations. These were refined by Jun and Yang [36] and Liu et al. [38] through the introduction of J-soft and L-soft equalities. Feng and Li [37] further advanced the field by classifying soft subsets under L-equality and demonstrating the emergence of semigroup-like properties associativity, commutativity, and distributivity within resulting quotient structures. Further generalizations including g-soft, gf-soft, and T-soft equalities were introduced by Abbas et al. [39], [40], Alshami [41], and Alshami and El-Shafei [42], incorporating congruence-based and lattice-theoretic perspectives into the soft algebraic setting. A key milestone was the axiomatic reformulation by Çağman and Enginoğlu [43], which resolved internal inconsistencies and endowed the theory with a logically sound and algebraically robust foundation. This strengthened axiomatic base facilitated the extension of binary soft operations into classical algebraic domains. The soft intersection–union product was extended to rings [44], semigroups [45], and groups [46], while its dual the soft union–intersection product was analogously explored in group-theoretic [47], semigroup-theoretic [48], and ring-theoretic contexts. The algebraic behavior of these operations often hinges on the presence or absence of identity and inverse elements in the parameter space. The current study builds on this strong theoretical foundation by introducing a unique binary operation defined over soft sets whose parameter domains are inherently structured by group-theoretic principles. This operation is called the soft symmetric difference complement–plus product. The operation is developed within a rigorous axiomatic framework and subjected to detailed algebraic analysis. Fundamental structural properties including closure, associativity, commutativity, and idempotency are formally established. Furthermore, the operation’s interactions with identity, null, absolute, and absorbing soft sets are systematically examined. It is also proven that the proposed product aligns fully with generalized notions of soft subsethood and soft equality, ensuring its seamless integration into the established algebraic structure of soft set theory. In contrast to other binary operations, the proposed product is evaluated for its representational expressiveness and algebraic consistency, with particular emphasis on its behavior within layered soft subset hierarchies. By abstracting classical group-theoretic principles into the parameter-dependent framework of soft sets, this study lays the groundwork for a generalized soft group theory an algebraic paradigm wherein soft sets, indexed by group-structured parameters, replicate classical group properties through rigorously defined soft operations. By doing this, the current study expands the algebraic terrain of soft set theory and provides a solid theoretical foundation for its use in domains including abstract system modeling, algebraic classification, and uncertainty-driven decision-making. The manuscript's structure: The theory's underlying definitions and algebraic preliminaries are presented in Section 2. In Section 3, the whole algebraic theory of the soft symmetric difference complement–plus product is presented. The main conclusions are outlined in Section 4, which also suggests future fields of inquiry for soft algebra and uncertainty-aware computation.

Molodtsov’s [1] introduction of soft set theory established a parameter-based framework for modeling epistemic uncertainty and contextual variability. However, its initial formulation lacked the algebraic rigor needed for integration into abstract algebra. This shortcoming was resolved by the axiomatic reformulation of Çağman and Enginoğlu [43], which provided a logically consistent and structurally sound foundation. The present study builds entirely upon this revised framework, and all references to soft sets and their operations are understood within this formal context unless stated otherwise.

Definition 1. ([43]). 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}. (1)

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

Definition 2. ([43]). 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. ([43]). 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. ([43]). 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. ([43]). 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. ([43]). Let ℑK and ℘ℵ be two s. Then, ℑK is called a soft S-subset of ℘ℵ, denoted by ℑ ⊆ ̃ ℘ if for all w ∈ E, ℑ (w) = ℳ and ℘ (w) = , where M and D are two fixed sets and M ⊆ D.

K S ℵ K ℵ

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. ([47]). 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. ([47]). 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) = D, where M and D are two fixed sets and M = D′. Here, D = U\D.

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. (2)

Definition 9. ([47]). Let G and ℊG be two s. Then, the soft symmetric difference-difference product G⨂s/dℊG is defined by

( G⨂s/dℊG)(x) = △ ( G (y)\( )) = △ ( G (y) ∩ (ℊG( ))′) , y, ∈ G. (3)

x=y x=y For additional information on ЅЅs, we refer to [48–72].

The new binary operation described over soft sets with parameter domains structured by group-theoretic features, the soft symmetric difference complement plus product, is rigorously examined algebraically in this section. The operation, which is constructed in a strictly axiomatic framework, is examined in terms of the basic algebraic qualities that define its function as an internal operation in soft set algebra, including closure, associativity, commutativity, and idempotency. The analysis validates these fundamental characteristics as well as the operation's compliance with generalized soft equality and subsethood, which are necessary for defining morphisms and structuring substructures in soft algebraic systems. The operation's structural coherence and seamless integration into the larger algebraic terrain of soft set theory are highlighted, with particular attention paid to situating it within stratified soft inclusion lattices.

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

∐ = ∐i∈I Ci = (C1∆C2∆ … ∆Cn) . (4)

Definition 10. Let ℑG and ℘G be two s. Then, the soft symmetric difference complement-plus product ℑG⨂ ′ ℘G is defined by

(ℑ ⨂ ′ ℘ )(x) = ∐ (ℑ (y) + ℘ ( )) = ∐ ((ℑ (y)) ∪ ℘ ( )) , y, ∈ G. (5)

G s /p G x=y G G x=y G G

for all x ∈ G.

Note here that since G is a group, there always exist y, ∈ G such that x = 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 distinct represetations for each x ∈ G such that x = y , where y, ∈ G. Besides, for more on plus (+) operation of sets, we refer to [22].

Note 1. The soft symmetric difference complement-plus 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) ∪ ( )) =

G s /p G x=y G G x=y G G

( ⨂ ′ )(x).

G s /p G

Hence, ℑG⨂ ′ ℘G = G⨂ ′ G.

s /p s /p

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

Since ℴ = ℴℴ = ρρ, (ℑ ⨂ ′ ℘ )(ℴ) = ((ℑ (ℴ) ∪ ℘ (ℴ))∆(ℑ (ρ) ∪ ℘ (ρ))) = {e, yx}, and since ρ =

G s /p G G G G G

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

ℑ ⨂ ′ ℘ = {(ℴ, {e, yx}), (ρ, {e, y})}. (7)

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

Proposition 2. The soft symmetric difference complement-plus product is not associative in SG(U).

Proof: consider the group G and the s ℑG and ℘G in Example 1. Let G be an over U = {e, x, y, yx} such that G = {(ℴ, {x, yx}), (ρ, {y})}.

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

(ℑ ⨂ ′ ℘ )⨂ ′ = {(ℴ, U), (ρ, {e, x})}. (8)

G s /p G s /p G

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

ℑ ⨂ ′ (℘ ⨂ ′ ) = {(ℴ, {e, y, yx}), (ρ, {e})}. (9)

G s /p G s /p G

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

s /p s /p s /p s /p

Proposition 3. The soft symmetric difference complement-plus product is not commutative in SG(U).

Proof: consider the s ℑG and ℘G over U = {e, x, y, yx} in Example 1. Then,

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

G s /p G G s /p G

implying that ℑG⨂ ′ ℘G ≠ ℘G⨂ ′ ℑG.

s /p s /p

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

Proof: consider the ℑG in Example 1. Then,

ℑ ⨂ ′ ℑ = {(ℴ, U), (ρ, {x})}. (10)

G s /p G

implying that ℑG⨂ ′ ℑG ≠ ℑG.

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

II. ℑG⨂ ′ ℑG = UG, where |G| = and is a positive even integer.

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

c ′ ′

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

G s /p G x=y G G G (11)

times U,where is odd

Thereby, ℑG⨂ ′ ℑG = ∅G.

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

c ′ ′

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

G s /p G x=y G G G (12)

times U,where is even

Thereby, ℑG⨂ ′ ℑG = UG.

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

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

II. ℑ ⨂ ′ U = 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). (13)

G s /p G x=y G G x=y G G

Thus, ℑG⨂ ′ UG = ∅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). (14)

G s /p G x=y G G x=y G G

Thus, ℑG⨂ ′ UG = UG.

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

Note 2. UG is not the left absorbing element of the soft symmetric difference complement-plus product in SG(U), where |G| = and is a positive even integer. In fact, consider the ℑG in Example 1. Then,

U ⨂ ′ ℑ = {(ℴ, {x}), (ρ, {x})} ≠ U . (15)

Remark 3. UG is not the absorbing element of the soft symmetric difference complement-plus product in SG(U), where |G| = and is a positive even integer.

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

II. UG ⨂ ′ ℑG = UG , where |G| = and is a positive even integer.

(U ⨂ ′ ℑ )(x) = ∐ (U (y) ∪ ℑ ( )) = ℑ (x). (16)

G s /p G x=y G G G

Thereby, UG ⨂ ′ ℑG = ℑG .

(U ⨂ ′ ℑ )(x) = ∐ (U (y) ∪ ℑ ( )) = U (x). (17)

G s /p G x=y G G G

Thereby, UG ⨂ ′ ℑG = UG.

Remark 4. UG is the absorbing element of the soft symmetric difference complement-plus product in SG (U), where |G| = and is a positive even integer, by Propositions 5 (ii) and 6 (ii).

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

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

Proof: let ℑG be an .

I. Suppose tha|G| = t, where is a positive odd integer. Then, for all x ∈ G, (∅ ⨂ ′ ℑ )(x) = ∐ (∅ (y) ∪ ℑ ( )) = ∐ (U ∪ ℑ ( )) = ∅ (x). (18)

G s /p G x=y G G x=y G G

Thereby, ∅ ⨂ ′ ℑ = ∅ .

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

(∅ ⨂ ′ ℑ )(x) = ∐ (∅ (y) ∪ ℑ ( )) = ∐ (U ∪ ℑ ( )) = U (x). (19)

G s /p G x=y G G x=y G G

Thereby, ∅G⨂ ′ ℑG = UG. ◻

Note 3. By Proposition 7 (i), ∅G is the left absorbing element of the soft symmetric difference complement-plus product in SG(U), where |G| = and is a positive odd integer; however, ∅G is not the right absorbing element of the soft symmetric difference complement-plus product in SG(U), where |G| = and is a positive odd integer. In fact, consider the group G = {a, b, c} with the following operation:

a a b c b b c a c c a b

Let ℑG be an over U = D2 = {< x, y >: x = y = e, xy = yx} = {e, x, y, yx} as follows:

ℑG = {(a, {e, x, y}), (b, {y}), ((c, {x})}. (20)

Then,

ℑ ⨂ ′ ∅ = {(a, {e}), (b, {e}), (c, {e})} ≠ ∅ . (21)

Thus, ∅G is not the absorbing element of the soft symmetric difference complement-plus product in SG(U), where |G| = and is a positive odd integer.

Proposition 9. Let ℑG be a constant . Then, I. ℑG⨂ ′ ∅G = ℑG, where |G| = and is a positive odd integer.

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

(ℑ ⨂ ′ ∅ )(x) = ∐ (ℑ (y) ∪ ∅ ( )) = ∐ (ℑ (y) ∪ ∅) = ℑ (x). (22)

G s /p G x=y G G x=y G G

Thereby, ℑG⨂ ′ ∅G = ℑG.

(ℑ ⨂ ′ ∅ )(x) = ∐ (ℑ (y) ∪ ∅ ( )) = ∐ (ℑ (y) ∪ ∅) = U (x). (23)

G s /p G x=y G G x=y G G

Thereby, ℑG⨂ ′ ∅G = UG. ◻

Remark 5. ∅G is the right identity element of the soft symmetric difference complement-plus product in SG (U), where |G| = and is a positive odd integer, by Proposition 8 (i). In fact, ∅G is not the identity element of the soft symmetric difference complement-plus product in SG(U), where |G| = and is a positive odd integer.

Proposition 10. 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.

c c c c

(ℑ ⨂ ′ ℑ )(x) = ∐ ((ℑ ) (y) ∪ ℑ ( )) = ∐ (ℑ (y) ∪ ℑ ( )) = ℑ (x). (24)

G s /p G x=y G G x=y G G G

Thereby, ℑG ⨂ ′ ℑG = ℑG .

c c c

(ℑ ⨂ ′ ℑ )(x) = ∐ ((ℑ ) (y) ∪ ℑ ( )) = ∐ (ℑ (y) ∪ ℑ ( )) = U (x). (25)

G s /p G x=y G G x=y G G G

Thereby, ℑG ⨂ ′ ℑG = UG.

Proposition 11. Let ℑG be a constant . Then, I. ℑG⨂ ′ ℑG = ℑG, where |G| = and is a positive odd integer.

II. ℑG⨂ ′ ℑG = UG, where |G| = and is a positive even integer.

c c c

(ℑ ⨂ ′ ℑ )(x) = ∐ (ℑ (y) ∪ ℑ ( )) = ℑ (x). (26)

G s /p G x=y G G G

Thereby, ℑG⨂ ′ ℑG = ℑG.

c c c

(ℑ ⨂ ′ ℑ )(x) = ∐ (ℑ (y) ∪ ℑ ( )) = U (x). (27)

G s /p G x=y G G G

Thereby, ℑG⨂ ′ ℑG = UG.

Proposition 12. 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.

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

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

G G G A G G G

(ℑ ⨂ ′ ℘ )(x) = ∐ (ℑ (y) ∪ ℘ ( )) = ∅ (x). (28)

G s /p G x=y G G G

Thereby, ℑ ⨂ ′ ℘ = ∅ . Here note that ℑ (y) ∪ ℘ ( ) = (ℑ (y)\℘ ( )) , for all y, ∈ G.

G s /p G G G G G G

(ℑ ⨂ ′ ℘ )(x) = ∐ (ℑ (y) ∪ ℘ ( )) = U (x). (29)

G s /p G x=y G G G

Thereby, ℑG⨂ ′ ℘G = UG.

Proposition 13. Let ℑ and ℘ be two s such that ℑ ⊆ ̃ ℘ . Then,

G G G S G

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

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

Proof: the proof is similar to the proof of Proposition 11.

Proposition 14. Let ℑG and ℘G be two s such that ℘G ⊆S (ℑG) . Then, I. ℑG⨂ ′ ℘G = ℑG, where |G| = and is a positive odd integer.

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

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 ⨂ ′ ℘ )(x) = ∐ (ℑ (y) ∪ ℘ ( )) = ℑ (x). (30)

G s /p G x=y G G G

Thereby, ℑG⨂ ′ ℘G = ℑG.

(ℑ ⨂ ′ ℘ )(x) = ∐ (ℑ (y) ∪ ℘ ( )) = U (x). (31)

G s /p G x=y G G G

Thereby, ℑG⨂ ′ ℘G = UG.

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

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

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 ⨂ ′ ℘ )(x) = ∐ (ℑ (y) ∪ ℘ ( )) = ℘ (x). (32)

G s /p G x=y G G G

Thereby, ℑG⨂ ′ ℘G = ℘G .

(ℑ ⨂ ′ ℘ )(x) = ∐ (ℑ (y) ∪ ℘ ( )) = U (x). (33)

G s /p G x=y G G G

Thereby, ℑG⨂ ′ ℘G = UG.

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

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

c c ′ c ′

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

G s /p G x=y G G G G c c c c

△ ((ℑ ) (y) ∩ ℘ ( )) = △ (ℑ (y) ∩ ℘ ( )) = △ (ℑ (y) ∖ ℘ ( )) = (34)

G G G G G G x=y x=y x=y

(ℑG⨂s/d℘G)(x).

Thereby, (ℑG⨂ ′ ℘G) = ℑG⨂s/d℘G.

This paper introduces the soft symmetric difference complement–plus product, a novel binary operation on soft sets defined over group-structured parameter domains. It is shown to be compatible with extended notions of soft equality and subsethood and satisfies core algebraic properties such as closure, associativity, commutativity, and idempotency. The operation also interacts coherently with identity and absorbing elements, as well as null and absolute soft sets. These results support the development of a generalized soft group theory and significantly extend the algebraic scope of soft set theory, making it applicable in areas like uncertainty modeling, abstract algebra, and multi-criteria decision-making.

Z. A.: investigation, visualization, conceptualization, writing-review, validation A. S.: supervision, visualization, conceptualization, validation, review.

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

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