Numerous researchers have proposed a variety of mathematical frameworks aimed at modeling and addressing complex problems characterized by uncertainty, vagueness, and ambiguity in domains such as engineering, economics, social sciences, and healthcare. Molodtsov [1] identified inherent limitations in existing frameworks. For instance, fuzzy set theory [2] often encounters challenges in the appropriate specification of membership functions, while probability theory relies on extensive trials to establish the existence of a mean value.

To overcome these limitations, Molodtsov [1] introduced soft set theory as a novel mathematical paradigm, demonstrating its potential applicability in diverse areas such as probability theory, game theory, and operations research. In contrast to classical approaches, soft set theory provides a more adaptable framework by eliminating rigid requirements related to approximate descriptions. The seminal work by Maji et al. [3], which applied soft sets to decision-making, laid the groundwork for subsequent advancements. Building upon this foundation, numerous studies [4–10] introduced enhanced decision-making methodologies grounded in soft set theory. Notably, Çağman and Enginoğlu [11] proposed a soft set-based decision-making model, and further introduced the concept of soft matrices [12], formulating decision procedures based on their AND, OR, AND-NOT, and OR-NOT operations. These formulations proved effective in resolving real-world problems under uncertainty.

The adaptability and utility of soft set theory have led to its widespread adoption in decision-making contexts [13-24], where significant developments have emerged. These include bijective and exclusive disjunctive soft sets, generalized uni-int frameworks, soft approximations, operator-based decision processes, reduced and cardinality-inverse soft matrices, soft semantics, and a range of mean and generalized operators on fuzzy soft matrices, as well as soft set-valued mappings.

In recent years, scholarly interest has increasingly turned to the theoretical underpinnings of soft set theory. Maji et al. [25] conducted a foundational analysis involving concepts such as soft subsets and supersets, soft set equality, and basic operations including union, intersection, AND-product, and OR-product. Pei and Miao [26] refined these notions by exploring connections between soft sets and information systems and redefining intersection and subset relations. Further contributions by Ali et al. [27] introduced new operations such as restricted union, restricted intersection, restricted difference, and extended intersection. Subsequent investigations [28-40] have focused on the algebraic structure of soft set operations, correcting earlier conceptual inconsistencies and proposing new methodologies.

Significant progress has been made in the formalization of soft set operations, as evidenced by a wide array of newly defined and rigorously analyzed operations [41–48]. Central to the theory are soft equal relations and soft subsets. Maji et al. [25] initiated the formal definition of soft subsets, which was later extended by Pei and Miao [26] and Feng et al. [29]. Qin and Hong [49] introduced new notions of soft congruence and equality. To generalize Maji’s distributive laws, Jun and Yang [50] incorporated broader soft subset classes and proposed J-soft equal relations. Inspired by their work, Liu et al. [51] explored soft L-subsets and soft L-equal relations, revealing that distributive laws do not universally apply across all soft equalities.

Building on this foundation, Feng et al. [52] investigated classifications of soft subsets and the properties of soft product operations introduced in [24], such as the AND- and OR-products, within the framework of soft L-subsets. Their comprehensive analysis addressed commutativity, associativity, and distributivity, resolving previously incomplete findings and demonstrating that soft L-equal relations constitute congruence relations in free soft algebras, where resulting quotient structures form commutative semigroups. For further developments in the theory of soft equalities (including generalized soft equality, soft lattices, relaxed parameter constraints, g-soft and gf-soft equality, and T-soft equality) see [53–57].

Çağman and Enginoğlu [11] revisited the definition of soft set by Maji et al. [25] and its operations to enhance practical applicability. They introduced four distinct product operations: AND-, OR-, AND-NOT-, and OR-NOT-products, alongside the uni-int decision function. These innovations were integrated into a unified decision-making framework, demonstrated through practical applications involving uncertainty. Sezgin et al. [58] subsequently analyzed the AND-product, a pivotal operation in soft set-based decision-making, within various equality frameworks, such as soft L-equality and J-equality. Their work systematically examined its algebraic properties, including idempotency, commutativity, and associativity, comparing these with results involving soft F-subsets, M-equality, L-equality, and J-equality.

The concept of the soft union product was first introduced for rings [59], semigroups [60], and groups [61], forming the basis for the development of soft union ring, semigroup, and group theories. Similarly, the soft intersection product was defined for groups [62], semigroups [63], and rings [64], 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.

In this study, we propose a novel product for soft sets whose parameter sets form a group structure, referred to as the soft intersection-difference product, constructed within the definitional framework of Çağman and Enginoğlu [11]. A detailed analysis of its algebraic properties is undertaken, considering various soft subsets and equality relations, with the aim of inspiring the development of a new soft group theory rooted in this construct. The remainder of the paper is organized as follows: Section 2 revisits essential concepts in soft set theory; Section 3 introduces the soft intersection-difference product and presents a comprehensive algebraic analysis in relation to different types of soft subsets and equalities. The concluding section offers a summary of the results and outlines directions for future research.

This section is devoted to revisiting a selection of foundational definitions and structural properties that serve as a theoretical basis for the developments presented in the subsequent section. Although the notion of the soft set was initially proposed by Molodtsov [1], the conceptual framework, including key definitions and operational structures, was later substantially revised by Çağman and Enginoğlu [11] to enhance both theoretical rigor and applicability. Accordingly, the present study is grounded in this refined formulation, which will be employed consistently throughout the paper.

Definition 1 ([11]). Let E be a parameter set, U be a universal set, P(U) be the power set of U, and Υ ⊆ E. Then, the soft set ʆΥ over U is a function such that ʆΥ: E → P(U), where for all ƙ ∉ Υ, ʆΥ(ƙ) = ∅. That is, ʆΥ = {(ƙ, ʆΥ(ƙ)): ƙ ∈ E}.

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

Definition 2 ([11]). Let ʆℋ be an ЅЅ. If ʆℋ(ƙ) = ∅, for all ƙ ∈ E, then ʆℋ is called a null ЅЅ and indicated by ∅E, and if ʆℋ(ƙ) = U, for all ƙ ∈ E, then ʆℋ is called an absolute ЅЅ and indicated by UE.

Definition 3 ([11]). Let ʆℋ, ʆℵ be two ЅЅs. If ʆℋ(ƙ) ⊆ ʆℵ(ƙ), for all ƙ ∈ E, then ʆℋ is a soft subset of ʆℵ and indicated by ʆ ⊆ ̃ ʆ . If ʆ (ƙ) = ʆ (ƙ), for all ƙ ∈ E, then ʆ is called soft equal to ʆ , and denoted by ʆ = ʆ .

ℋ ℵ ℋ ℵ ℋ ℵ ℋ ℵ

Definition 4 ([11]). Let ʆ , ʆ be two ЅЅs. The union of ʆ and ʆ is the ЅЅ ʆ ∪ ̃ ʆ , where (ʆ ∪ ̃ ʆ )(w) =

ℋ ℵ ℋ ℵ ℋ ℵ ℋ ℵ

ʆℋ(w) ∪ ʆℵ(w), for all w ∈ E.

Definition 5 ([11]). Let ʆℋ be an ЅЅ. Then, the complement of ʆℋ, denoted by (ʆℋ) , is defined by the ЅЅ

c c ′

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

Definition 6 ([65]). Let ʆ and ʠ be two ЅЅs. Then, ʆ is called a soft S-subset of ʠ , denoted by ʆ ⊆ ̃ ʠ , if

K L K L K S L

for all e ∈ E, ʆK(e) = Ā and ʠK(e) = В, where Ā and В are two fixed sets and Ā ⊆ В. Moreover, two ЅЅs ʆK and ʠ are said to be soft S-equal, denoted by ʆ = ʠ , if ʆ ⊆ ̃ ʠ and ʠ ⊆ ̃ ʆ .

L K S L K S L L S K

It is obvious that if ʆK =S ʠL, then ʆK and ʠL are the same constant functions, that is, for all e ∈ E, ʆK(e)= ʠK(e) = Ā, where Ā is a fixed set.

Definition 7 (65]). Let ʆ and ʠ be two ЅЅs. Then, ʆ is called a soft A-subset of ʠ , denoted by ʆ ⊆ ̃ ʠ ,

K L K L K A L

if, for each ℊ, h ∈ E, ʆK(ℊ) ⊆ ʠL(h).

Definition 8 ([65]). Let ʆK and ʠL be two ЅЅs. Then, ʆK is called a soft S-complement of ʠL, denoted by ʆK =S (ʠL) , if, for all e ∈ E, ʆK(e) = Ā and ʠL(e) = В, where Ā and B are two fixed sets and Ā = В′. Here, B = U\B.

For additional information on ЅЅs, we refer to [66-82].

Definition 9 ([61]). Let ʆG and ʠG be two ЅЅs, where G is a group. Then, the soft intersection-union product ʆG⨂i/uʠG is defined by (ʆG⨂i/uʠG)(ƙ) = ⋂ (ʆG (ƚ) ∪ ʠG(ɯ)), ƚ, ɯ ∈ G,

for all ƙ ∈ G.

Definition 10 ([83]). Let ʆ and ʠ be two ЅЅs where G is a group. Then, the soft intersection-symmetric difference product ʆ ⨂ / ʠ is defined by (ʆG⨂i/sʠG)(ƙ) = ⋂ (ʆG (ƚ)∆ʠG(ɯ)), ƚ, ɯ ∈ G,

for all ƙ ∈ G.

Definition 11 ([83]). Let ʆG and ʠG be two ЅЅs, where G is a group. Then, the soft union-difference product ʆG⨂u/dʠG is defined by (ʆG⨂u/dʠG)(ƙ) = ⋃ (ʆG (ƚ) ∖ ʠG(ɯ)) , ƚ, ɯ ∈ G,

for all ƙ ∈ G.

In this section, we introduce a novel binary operation for soft sets, referred to as the soft intersection-difference product, defined in the context of soft sets whose parameter set is a group structure. A rigorous algebraic analysis of this product is carried out, with particular attention devoted to its characteristics under diverse notions of soft equalities and various classifications of soft subsets. Theoretical findings are further elucidated through representative examples that illustrate the structural characteristics of the proposed operation.

From now on, G denotes a group, SG(U) is the collections of ЅЅs, whose parameter set is G, and all the ЅЅs are the elements of SG(U).

Definition 12. Let ʆG and ʠG be two ЅЅs. Then, the soft intersection-difference product ʆG⨂i/dʠG is defined by (ʆG⨂i/dʠG)(ƙ) = ⋂ (ʆG(ƚ) ∖ ʠG(ɯ)), ƚ, ɯ ∈ G,

for all ƙ ∈ G.

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

Note 1: The soft intersection-difference product is well-defined in SG(U). In fact, let ʆG, ʠG, ϦG, ζG ∈ SG(U) such that (ʆG, ʠG) = (ϦG, ζG). Then, ʆG = ϦG and ʠG = ζG, implying that ʆG(ƙ) = ϦG(ƙ) and ʠG(ƙ) = ζG(ƙ), for all ƙ ∈ G. Thereby, for all ƙ ∈ G,

(ʆG⨂i/dʠG)(ƙ) = ⋂ƙ=ƚɯ(ʆG (ƚ) ∖ ʠG(ɯ)) = ⋂ƙ=ƚɯ(ϦG (ƚ) ∖ ζG(ɯ)) = (ϦG⨂i/dζG)(ƙ).

Hence, ʆG⨂i/dʠG = ϦG⨂i/dζG.

Example 1. Consider the group G = {Ձ, ɓ} with the following operation:

∙ Ձ ɓ Ձ Ձ ɓ ɓ ɓ Ձ

Let ʆG and ʠG be two ЅЅs over U= D2 = {< x, y >: x = y = e, xy = yx} = {e, x, y, yx} as follows:

ʆG = {(Ձ, {e, x, yx}), (ɓ, {x, yx})}, ʠG = {(Ձ, {e, y, yx}), (ɓ, {e, y})}.

Since Ձ = ՁՁ = ɓɓ, (ʆG⨂i/dʠG)(Ձ) = (ʆG(Ձ) ∖ ʠG(Ձ)) ∩ (ʆG(ɓ) ∖ ʠG(ɓ)) = {x}, and since ɓ = Ձɓ = ɓՁ, (ʆG⨂i/dʠG)(ɓ) = (ʆG(Ձ) ∖ ʠG(ɓ)) ∩ (ʆG(ɓ) ∖ ʠG(Ձ)) = {x} is obtained. Hence, ʆG⨂i/dʠG = {(Ձ, {x}), (ɓ, {x})}.

Proposition 1. The set SG(U) is closed under the soft intersection-difference product. That is, if ʆG and ʠG are two ЅЅs, then so is ʆG⨂i/dʠG.

Proof: It is obvious that the soft intersection-difference product is a binary operation in SG(U). Thereby, SG(U) is closed under the soft intersection-difference product.

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

Proof: Consider the group G and the ЅЅs ʆG and ʠG in Example 1. Let իG be an ЅЅ over U = {e, x, y, yx} such that իG = {(Ձ, {e, y}), (ɓ, {y, yx})}.

Since ʆG⨂i/dʠG = {(Ձ, {x}), (ɓ, {x})}, then (ʆG⨂i/dʠG)⨂i/dիG = {(Ձ, {x}), (ɓ, {x})}.

Moreover, since ʠG⨂i/dիG = {(Ձ, ∅), (ɓ, ∅)}, then ʆG⨂i/d(ʠG⨂i/dիG) = {(Ձ, {x, yx}), (ɓ, {x, yx})}.

Thereby, (ʆG⨂i/dʠG)⨂i/dիG ≠ ʆG⨂i/d(ʠG⨂i/dիG).

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

Proof: Consider the ЅЅs ʆG and ʠG in Example 1. Then, ʆG⨂i/dʠG = {(Ձ, {x}), (ɓ, {x})}, and ʠG⨂i/dʆG = {(Ձ, {y}), (ɓ, {y})}, implying that ʆG⨂i/dʠG ≠ ʠG⨂i/dʆG.

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

Proof: Consider the ЅЅ ʆG = {(Ձ, {e, x, yx}), (ɓ, {x, yx})} in Example 1. Then, ʆG⨂i/dʆG = {(Ձ, ∅), (ɓ, ∅)}, implying that ʆG⨂i/dʆG ≠ ʆG.

Proposition 5. Let ʆG be a constant ЅЅ. Then, ʆG⨂i/dʆG = ∅G.

Proof: Let ʆG be a constant ЅЅ such that, for all ƙ ∈ G, ʆG(ƙ) = Ā, where Ā is a fixed set. Then, for all ƙ ∈ G, ⋂ (ʆG (ƚ) ∖ ʆG(ɯ)) = ∅G(ƙ) (ʆG⨂i/dʆG)(ƙ) = .

Thereby, ʆG⨂i/dʆG = ∅G.

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

Proposition 6. ∅G is the left absorbing element of the soft intersection-difference product in SG(U).

Proof: Let x ∈ G. Then, (∅G⨂i/dʆG)(ƙ) = ⋂ (∅G(ƚ) ∖ ʆG(ɯ)) ⋂ (Ø ∖ ʆG(ɯ)) = ∅G(ƙ).

Thus, ∅G⨂i/dʆG = ∅G.

Proposition 7. ∅G is not the right absorbing element of the soft intersection-difference product in SG(U). Proof: Consider the ЅЅ ʆG = {(Ձ, {e, x, yx}), (ɓ, {x, yx})} in Example 1. Then, ʆG⨂i/d∅G = {(Ձ, {x, yx}), (ɓ, {x, yx})}, implying that ʆG⨂i/d∅G ≠ ∅G.

Remark 2. ∅G is not the absorbing element of soft intersection-difference product in SG(U).

Proposition 8. Let ʆG be a constant ЅЅ. Then, ʆG⨂i/d∅G = ʆG.

Proof: Let ʆG be a constant ЅЅ such that, for all ƙ ∈ G, ʆG(ƙ) = Ā, where Ā is a fixed set. Then, for all ƙ ∈ G, (ʆG⨂i/d∅G)(ƙ) = ⋂ (ʆG(ƚ) ∖ ∅G(ɯ)) = ʆG(ƙ).

Thereby, ʆG⨂i/d∅G = ʆG.

Remark 3. ∅G is not the absorbing element of soft intersection-difference product in SG (U) either. Moreover, ∅G is the right identity element of soft intersection-difference product in SG (U).

Proposition 9. Let ʆG be an ЅЅ. Then, ʆG⨂i/dUG = ∅G.

Proof: Let ʆG be an ЅЅ. Then, for all ƙ ∈ G, (ʆG⨂i/dUG)(ƙ) = ⋂ƙ=ƚɯ(ʆG(ƚ) ∖ UG(ɯ)) = ∅G(ƙ).

Thereby, ʆG⨂i/dUG = ∅G.

Proposition 10. Let ʆG be a constant ЅЅ. Then, UG⨂i/dʆG = ʆG .

Proof: Let ʆG be a constant ЅЅ such that, for all ƙ ∈ G, ʆG(ƙ) = Ā, where Ā is a fixed set. Then, for all ƙ ∈ G, (ʆG⨂i/dUG)(ƙ) = ⋂ (UG(ƚ) ∖ ʆG(ɯ)) = ʆG (ƙ).

Thereby, UG⨂i/dʆG = ʆG .

Proposition 11. Let ʆG be a constant ЅЅ. Then, ʆG ⨂i/dʆG = ʆG .

Proof: Let ʆG be a constant ЅЅ such that, for all ƙ ∈ G, ʆG(ƙ) = Ā, where Ā is a fixed set. Then, for all ƙ ∈ G,

c c c

(ʆG ⨂i/dʆG)(ƙ) = ⋂ (ʆG (ƚ) ∖ ʆG(ɯ)) = ʆG (ƙ).

Thereby, ʆG ⨂i/dʆG = ʆG .

Proposition 12. Let ʆG be a constant ЅЅ. Then, ʆG⨂i/dʆG = ʆG.

Proof: Let ʆG be a constant ЅЅ such that, for all ƙ ∈ G, ʆG(ƙ) = Ā, where Ā is a fixed set. Then, for all ƙ ∈ G, (ʆG⨂i/dʆG )(ƙ) = ⋂ (ʆG(ƚ) ∖ ʆG (ɯ)) = ʆG(ƙ).

Thereby, ʆG⨂i/dʆG = ʆG.

Proposition 13. Let ʆ , ʠ and ի be three ЅЅs. If ʆ ⊆ ̃ ʠ , then

G G G G G

ʆ ⨂ ի ⊆ ̃ ʠ ⨂ ի and ի ⨂ ʠ ⊆ ̃ ի ⨂ ʆ .

G i/d G G i/d G G i/d G G i/d G

Proof: Let ʆG, ʠG and իG be three ЅЅs such that ʆG ⊆ ʠG. Then, for all ƙ ∈ G, ʆG(ƙ) ⊆ ʠG(ƙ) and (ʠG(ƙ)) ⊆ (ʆG(ƙ)) . Thus, for all ƙ ∈ G,

(ʆG⨂i/dիG)(ƙ) = ⋂ (ʆG(ƚ) ∖ իG(ɯ)) ⊆ ⋂ (ʠG(ƚ) ∖ իG(ɯ)) = (ʠG⨂i/dիG)(ƙ),

for all ƙ ∈ G, implying that ʆ ⨂ ի ⊆ ̃ ʠ ⨂ ի . Similarly, for all ƙ ∈ G,

G i/d G G i/d G

(իG⨂i/dʠG)(ƙ) = ⋂ (իG(ƚ)\ ʠG(ɯ)) ⊆ ⋂ (իG(ƚ) ∖ ʆG(ɯ)) = (իG⨂i/dʆG)(ƙ),

for all ƙ ∈ G, implying that ի ⨂ ʠ ⊆ ̃ ի ⨂ ʆ .◻

G i/d G G i/d G

Proposition 14. Let ʆ , ʠ , Ϧ , and ζ be four ЅЅs. If Ϧ ⊆ ̃ ʆ and ζ ⊆ ̃ ʠ , then ζ ⨂ ʆ ⊆ ̃ ʠ ⨂ Ϧ and

G G G G G G G G G i/d G G i/d G

Ϧ ⨂ ʠ ⊆ ̃ ʆ ⨂ ζ .

G i/d G G i/d G

Proof: Let ʆ , ʠ , Ϧ and ζ be four ЅЅs such that Ϧ ⊆ ̃ ʆ and ζ ⊆ ̃ ʠ . Then, for all ƙ ∈ G, Ϧ (ƙ) ⊆ ʆ (ƙ) and

G G G G G G G G G G ′ ′ ′ ′

ζG(ƙ) ⊆ ʠG(ƙ). Thus, (ʆG(ƙ)) ⊆ (ϦG(ƙ)) and (ʠG(ƙ)) ⊆ (ζG(ƙ)) , for all ƙ ∈ G. Hence, for all ƙ ∈ G,

(ζG⨂i/dʆG)(ƙ) = ⋂ (ζG(ƚ)\ ʆG(ɯ)) ⊆ ⋂ (ʠG(ƚ) ∖ ϦG(ɯ)) = (ʠG⨂i/dϦG)(ƙ),

implying that ζ ⨂ Ϧ ⊆ ̃ ʠ ⨂ ʆ . Similarly, for all ƙ ∈ G,

G i/d G G i/d G

(ϦG⨂i/dʠG)(ƙ) = ⋂ (ϦG(ƚ) ∖ ʠG(ɯ)) ⊆ ⋂ (ʆG(ƚ)\ ζG(ɯ)) = (ʆG⨂i/dζG)(ƙ),

is obtained. Thereby, Ϧ ⨂ ʠ ⊆ ̃ ʆ ⨂ ζ .◻

G i/d G G i/d G

Theorem 1. Let ʆG and ʠG be two ЅЅs. Then, ʆG⨂i/dʠG = UG if and only if ʆG = UG and ʠG = ∅G.

Proof: Let ʆG and ʠG be two ЅЅs. Suppose that ʆG = UG and ʠG = ∅G. Hence, for all ƙ ∈ G, ʆG(ƙ) = UG(ƙ) = U andʠG(ƙ) = ∅G(ƙ) = ∅. Thus, for all ƙ ∈ G,

(ʆG⨂i/dʠG)(ƙ) = ⋂ƙ=ƚɯ(ʆG(ƚ) ∖ ʠG(ɯ)) =⋂ƙ=ƚɯ(UG(ƚ) ∖ ∅G(ɯ)) = UG(ƙ).

Thereby, ʆG⨂i/dʠG = UG.

Conversely, suppose that ʆG⨂i/dʠG = UG. Then, (ʆG⨂i/dʠG)(ƙ) = UG(ƙ) = U, for all ƙ ∈ G. Thus, for all ƙ ∈ G, UG(ƙ) = U = ʆG⨂i/dʠG(ƙ) = ⋂ (ʆG(ƚ) ∖ ʠG(ɯ)).

This implies that ʆG(ƚ) ∖ ʠG(ɯ) = U, for all ƚ, ɯ ∈ G. Thus, ʆG(ƚ) = U and ʠG(ɯ) = ∅, for all ƚ, ɯ ∈ G. Thereby, ʆG = UG and ʠG = ∅G.

Proposition 15. Let ʆ and ʠ be two ЅЅs. If ʆ ⊆ ̃ ʠ , then ʆ ⨂ ʠ = ∅ .

G G G A G G i/d G G

Proof: Let ʆ and ʠ be two ЅЅs and ʆ ⊆ ̃ ʠ . Thus, ʆ (Ձ) ⊆ ʠ (ɓ) for each Ձ, ɓ ∈ G. Hence, for all ƙ ∈ G,

G G G A G G G

ʆG⨂i/dʠG(ƙ) = ⋂ (ʆG(ƚ) ∖ ʠG(ɯ)) = ∅ = ∅G(ƙ).

Thus, ʆG⨂i/dʠG = ∅G is obtained.

Proposition 16. Let ʆG and ʠG be two ЅЅs. If ʆG ⊆S (ʠG) , then ʆG⨂i/dʠG = ʆG.

Proof: Let ʆG and ʠG be two ЅЅs and ʆG ⊆S (ʠG) . Hence, for all Ձ ∈ G, ʆG(Ձ) = Ā and ʠG(Ձ) = В, where Ā and В are two fixed sets and Ā ⊆ В . Thus, for all ƙ ∈ G, ʆG⨂i/dʠG(ƙ) = ⋂ (ʆG(ƚ) ∖ ʠG(ɯ)) = ⋂ (ʆG(ƚ) ∩ ʠG (ɯ)) = ʆG(ƙ).

Thus, ʆG⨂i/dʠG = ʆG.

Proposition 17. Let ʆ and ʠ be two ЅЅs. If (ʠ ) ⊆ ̃ ʆ , then ʆ ⨂ ʠ = ʠ .

G G G S G G i/d G G

Proof: Let ʆ and ʠ be two ЅЅs and ʠ ⊆ ̃ ʆ . Hence, for all Ձ ∈ G, ʆ (Ձ) = Ā and ʠ (Ձ) = В, where Ā and

G G G S G G G

В are two fixed sets and В ⊆ Ā. Thus, for all ƙ ∈ G, ʆG⨂i/dʠG(ƙ) = ⋂ (ʆG(ƚ) ∖ ʠG(ɯ)) = ⋂ (ʆG(ƚ) ∩ ʠG (ɯ)) = ʠG (ƙ).

Thus, ʆG⨂i/dʠG = ʠG .

Proposition 18. Let ʆG and ʠG be two ЅЅs. Then, (ʆG⨂i/dʠG) = ʆG⨂u/pʠG.

Proof: Let ʆG and ʠG be two ЅЅs. Then, for all ƙ ∈ G,

(ʆG⨂i/dʠG) (ƙ) = ( ⋂ (ʆG(ƚ) ∖ ʠG(ɯ))) = ⋃ (ʆG(ƚ) ∖ ʠG(ɯ))

= ⋃ (ʆG(ƚ) ∩ ʠG (ɯ)) = ⋃ (ʆG (ƚ) ∪ ʠG(ɯ)) = (ʆG⨂u/pʠG)(ƙ).

Thereby, (ʆG⨂i/dʠG) = ʆG⨂u/pʠG.

For more on soft union-plus product of groups, we refer to [85].

Proposition 19. Let ʆ and ʠ be two ЅЅs. Then, ʆ ⨂ ʠ ⊆ ̃ ʆ ⨂ ʠ .

G G G i/d G G i/u G

Proof: Let ʆG and ʠG be two ЅЅs. Then, for all ƙ ∈ G, (ʆG⨂i/dʠG)(ƙ) = ⋂ (ʆG(ƚ) ∖ ʠG(ɯ)) ⊆ ⋂ (ʆG(ƚ) ∪ ʠG(ɯ)) = (ʆG⨂i/uʠG)(ƙ).

Thus, ʆ ⨂ ʠ ⊆ ̃ ʆ ⨂ ʠ . ◻

G i/d G G i/u G Proposition 20. Let ʆG and ʠG be two ЅЅs. If one of the assertions following is satisfied, then ʆG⨂i/dʠG = ʆG⨂i/uʠG:

I. ʠG = ∅G

II. ʆG = ∅G and ʠG (ƙ) ∩ ʠG(ƚ) = ∅ for all ƙ, ƚ ∈ G Proof: Let ʆG and ʠG be two ЅЅs I. Let ʠG = ∅G.Then, for all ƙ ∈ G, ʠG(ƙ) = ∅G(ƙ) = ∅. Thus, for all ƙ ∈ G, (ʆG⨂i/dʠG)(ƙ) = ⋂ (ʆG (ƚ) ∖ ʠG(ɯ)) = ⋂ (ʆG (ƚ) ∖ ∅G(ɯ))

= ⋂ (ʆG (ƚ) ∪ ∅G(ɯ)) = ⋂ (ʆG (ƚ) ∪ ʠG(ɯ)) = (ʆG⨂i/uʠG)(ƙ).

Thus, ʆG⨂i/dʠG = ʆG⨂i/uʠG.

II. Let ʆG = ∅G and ʠG (ƙ) ∩ ʠG(ƚ) = ∅, for all ƙ, ƚ ∈ G. Then, for all ƙ ∈ G, ʆG(ƙ) = ∅G(ƙ) = ∅. Hence, for all ƙ ∈

(ʆG⨂i/dʠG)(ƙ) = ⋂ (ʆG (ƚ) ∖ ʠG(ɯ)) = ⋂ (∅G(ƚ) ∖ ʠG(ɯ))

= ⋂ (ʆG(ƚ) ∪ ʠG(ɯ)) = ∅ = (ʆG⨂i/uʠG)(ƙ).

Thus, ʆG⨂i/dʠG = ʆG⨂i/uʠG.

Proposition 21. Let ʆ and ʠ be two ЅЅs. Then, ʆ ⨂ ʠ ⊆ ̃ ʆ ⨂ ʠ .

G G G i/d G G i/s G

Proof: Let ʆG and ʠG be two ЅЅs. Then, for all ƙ ∈ G, (ʆG⨂i/dʠG)(ƙ) = ⋂ƙ=ƚɯ(ʆG (ƚ) ∖ ʠG(ɯ)) ⊆ ⋂ƙ=ƚɯ(ʆG (ƚ)∆ʠG(ɯ)) = (ʆG⨂i/sʠG)(ƙ).

Thus, ʆ ⨂ ʠ ⊆ ̃ ʆ ⨂ ʠ . ◻

G i/d G G i/s G

Proposition 22. Let ʆG and ʠG be two ЅЅs. If one of the assertions following is satisfied, then ʆG⨂i/dʠG = ʆG⨂i/sʠG:

I. ʠ ⊆ ̃ ʆ

G S G

II. ʆG =S ʠG

III. ʆG = ∅G and ʠG (ƙ) ∩ ʠG(ƚ) = ∅ for all ƙ, ƚ ∈ G

IV. ʠG = ∅G

Proof: Let ʆG and ʠG be two ЅЅs.

I. Suppose that ʠ ⊆ ̃ ʆ . Hence, for all Ձ ∈ G, ʠ (Ձ) = Ā and ʆ (Ձ) = В, where Ā and В are two fixed sets and

G S G G G

Ā ⊆ В. Thus, for all ƙ ∈ G, (ʆG⨂i/dʠG)(ƙ) = ⋂ (ʆG (ƚ) ∖ ʠG(ɯ)) = ⋂ (ʆG (ƚ)∆ʠG(ɯ)) = (ʆG⨂i/sʠG)(ƙ).

Thus, ʆG⨂i/dʠG = ʆG⨂i/sʠG.

II. It follows from Proposition 22 (I).

III. Let ʆG = ∅G and ʠG (ƙ) ∩ ʠG(ƚ) = ∅ for all ƙ, ƚ ∈ G. Then, for all ƙ ∈ G,ʆG(ƙ) = ∅G(ƙ) = ∅. Hence, for all ƙ ∈

(ʆG⨂i/dʠG)(ƙ) = ⋂ƙ=ƚɯ(ʆG (ƚ) ∖ ʠG(ɯ)) = ⋂ƙ=ƚɯ(∅G (ƚ) ∖ ʠG(ɯ)) = ∅G(ƙ).

Similarly, for all ƙ ∈ G, (ʆG⨂i/dʠG)(ƙ) = ⋂ (ʆG (ƚ)∆ʠG(ɯ)) = ⋂ (∅G (ƚ)∆ʠG(ɯ)) = ∅G(ƙ).

Hence, ʆG⨂i/dʠG = ʆG⨂i/sʠG.

IV. Let ʠG = ∅G. Then, for all ƙ ∈ G, ʠG(ƙ) = ∅G(ƙ) = ∅. Thus, for all ƙ ∈ G, (ʆG⨂i/dʠG)(ƙ) = ⋂ (ʆG (ƚ) ∖ ʠG(ɯ)) = ⋂ (ʆG (ƚ) ∖ ∅G(ɯ)) =

= ⋂ (ʆG (ƚ)∆ʠG(ɯ)) = (ʆG⨂i/sʠG)(ƙ).

Thereby, ʆG⨂i/dʠG = ʆG⨂i/sʠG.

Proposition 23. Let ʆ and ʠ be two ЅЅs. Then, ʆ ⨂ ʠ ⊆ ̃ ʆ ⨂ ʠ .

G G G i/d G G u/d G

Proof: Let ʆG and ʠG be two ЅЅs. Then, for all ƙ ∈ G, (ʆG⨂i/dʠG)(ƙ) = ⋂ (ʆG (ƚ) ∖ ʠ(ɯ)) ⊆ ⋃ (ʆG (ƚ) ∖ ʠ(ɯ)) = (ʆG⨂u/dʠG)(ƙ).

Thus, ʆ ⨂ ʠ ⊆ ̃ ʆ ⨂ ʠ . ◻

G i/d G G u/d G

Proposition 24. Let ʆG and ʠG be two ЅЅs. If one of the assertions following is satisfied, then ʆG⨂i/dʠG = ʆG⨂u/dʠG.

I. ʠ ⊆ ̃ ʆ

G S G

II. ʆ ⊆ ̃ ʠ

G A G

III. ʆG =S ʠG IV. ʆG =S (ʠG)

Proof: Let ʆG and ʠG be two ЅЅs.

I. Suppose that ʠ ⊆ ̃ ʆ . Hence, for all Ձ ∈ G, ʠ (Ձ) = Ā, ʆ (Ձ) = В, where Ā and В are two fixed sets and Ā ⊆

G S G G G

В. Thus, for all ƙ ∈ G, (ʆG⨂i/dʠG)(ƙ) = ⋂ (ʆG (ƚ) ∖ ʠG(ɯ)) = ⋃ (ʆG (ƚ) ∖ ʠG(ɯ)) = (ʆG⨂u/dʠG)(ƙ).

Hence, ʆG⨂i/dʠG = ʆG⨂u/dʠG.

II. Suppose that ʆ ⊆ ̃ ʠ . Then, ʆ (Ձ) ⊆ ʠ (ɓ), for each Ձ, ɓ ∈ G. Thus, for all ƙ ∈ G,

G A G G G

(ʆG⨂i/dʠG)(ƙ) = ⋂ (ʆG (ƚ) ∖ ʠG(ɯ)) = ∅ = ⋃ (ʆG (ƚ) ∖ ʠG(ɯ))

= (ʆG⨂u/dʠG)(ƙ).

Hence, ʆG⨂i/dʠG = ʆG⨂u/dʠG.

III. It follows from Proposition 24 (I).

IV. Let ʆG =S (ʠG) . Then, for all ƙ ∈ G, ʆG(ƙ) = Ā, ʠG(ƙ) = В, where Ā and В are two fixed sets and Ā = В .

Thus, for all ƙ ∈ G, (ʆG⨂i/dʠG)(ƙ) = ⋂ (ʆG (ƚ) ∖ ʠG(ɯ)) = ⋃ (ʆG (ƚ) ∖ ʠG(ɯ)) = (ʆG⨂u/dʠG)(ƙ).

Hence, ʆG⨂i/dʠG = ʆG⨂u/dʠG.

Proposition 25. The soft intersection-difference product distributes over soft intersection operation from the right side.

Proof: Let ʆG, ʠG and իG be three ЅЅs. Then, for all ƙ ∈ G, ((ʆ ∩ ̃ ʠ )⨂ ի )(ƙ) = ⋂ ((ʆ ∩ ̃ ʠ )(ƚ) ∖ ի (ɯ))

G G i/d G G G G

= ⋂ ((ʆG(ƚ) ∩ ʠG(ƚ)) ∖ իG(ɯ))

= ⋂ [(ʆG(ƚ) ∖ իG(ɯ))⋂(ʠG(ƚ) ∖ իG(ɯ))]

= [ ⋂ (ʆG(ƚ) ∖ իG(ɯ))] ∩ [ ⋂ (ʠG(ƚ) ∖ իG(ɯ))]

= (ʆG ⨂ իG) (ƙ) ∩ (ʠG ⨂ իG) (ƙ)

= ((ʆ ⨂ ի ) ∩ ̃ (ʠ ⨂ ի )) (ƙ).

G i G G i G

Thus, (ʆ ∩ ̃ ʠ )⨂ ի = (ʆ ⨂ ի ) ∩ ̃ (ʠ ⨂ ի ).

G G i/d G G i/d G G i/d G

Example 2. Consider the ЅЅs ʆG and ʠG in Example 1. Let իG be an ЅЅ as follows: իG = {(Ձ, {e, yx}), (ɓ, {x, y})}. Since ʆG⨂i/dիG = {(Ձ, ∅), (ɓ, ∅)} and ʠG⨂i/dիG = {(Ձ, ∅), (ɓ, ∅)}, then (ʆ ⨂ ի ) ∩ ̃ (ʠ ⨂ ի ) = {(Ձ, ∅), (ɓ, ∅)}.

G i/d G G i/d G

Moreover, since ʆ ∩ ̃ ʠ = {(Ձ, {e, yx}), (ɓ, ∅)}, (ʆ ∩ ̃ ʠ )⨂ ի = {(Ձ, ∅), (ɓ, ∅)}.

G G i/d G

Thus, (ʆ ∩ ̃ ʠ )⨂ ի = (ʆ ⨂ ի ) ∩ ̃ (ʠ ⨂ ի ).◻

G G i/d G G i/d G G i/d G

This study introduces a novel binary operation for soft sets whose parameter sets possess a group structure, termed the soft intersection-difference product. A comprehensive examination of its foundational algebraic properties is undertaken, with particular emphasis on its interaction with various classes of soft subsets and equality relations. The theoretical framework proposed herein not only addresses existing gaps but also offers a potential pathway for the development of a new branch of soft group theory grounded in this product. Prospective research directions may include the formulation of additional soft product operations and a deeper exploration of soft equality structures, both of which are expected to contribute significantly to the advancement of the theoretical and applied dimensions of soft set theory.

All authors contributed to the study's conception and design. AS supervised the study, performed material preparation. ZA performed data collection, and analysis, and wrote the first draft of the manuscript. All authors read and approved the final manuscript.

The author has approved the manuscript for publication.

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

No funding was received to conduct this study.

The datasets generated during and/or analyzed during the current study are available from the corresponding author (Aslıhan Sezgin, aslihan.sezgin@amasya.edu.tr) on reasonable request.

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