In this subsection, the definition of the binary extended theta operation together with its properties are provided with detailed proofs.

Definition 3.1.1 Let and be BSSs over, such that ) for each and ) for each . The binary extended theta of and is the BSS , denoted by where and for each .

Here, .

For more on theta ( operation of sets, we refer to [62]. By Definition 3.1.1, if =, then if =, then and if , then .

Example 3.1.2 Let , , , and the BSSs and over , be defined as follows:

Let , where . Since , , and ,

In this example, the fact that some components become empty sets is related to parameter differences and the combinatorial structure of the theta operation. For example, in the operation , since the union of and is the universal set, the result of the operation is the empty set (∅) for . This behavior reflects the self-negation and parameter-based union properties of the theta operation and is not observed in classical union/intersection operations. Therefore, the example clearly demonstrates the algebraic logic of the theta operation and the effect of parameter differences on the result.

Example 3.1.3 (A Real-Life Quality Control Problem) Quality control of products manufactured in a factory is performed by two different inspectors. There are two separate product groups in the factory, each from a different production lines and tested independently. Products from the first production line form the first product group, while products from the second production line form the second product group. These two product groups undergo various quality tests using different testing equipment.

Inspector A is responsible for mechanical and electrical tests, while Inspector B is responsible for electrical and safety tests. Therefore, some tests are evaluated by only one inspector, while others are evaluated by both inspectors. The goal is to evaluate the reports of these two inspectors together to reliably identify both successful products for the two production lines and, in particular, defective products that must be eliminated.

Products on the first production line:

Products on the second production line:

Test types (set of parameters):

Here : Mechanical test, : Electrical test, : Safety test.

Tests evaluated by Inspector A:

,

Tests evaluated by Inspector B:

and the BSSs and over , be defined as follows:

According to Inspector A, during the mechanical test, products numbered and from the first production line and product numbered from the second production line were found to be successful. In the safety test, products numbered and from the first production line and product numbered from the second production line were found to be successful.

According to Inspector B, during the electrical test, products numbered and from the first production line and products numbered and from the second production line were found to be successful. In the safety test, product numbered from the first production line and product numbered from the second production line were found to be successful.

Let , where . Since , , and , this means that the mechanical test is evaluated only by Inspector A, the safety test is evaluated only by Inspector B, and the electrical test is evaluated by both inspectors.

This result indicates that, according to both inspectors in the electrical test, product numbered from the first production line; products numbered and from the second production line has definitely failed and should be safely discarded.

Additionally, since the mechanical test ​ is evaluated only by Inspector A,

and since security test ​ is evaluated only by Inspector B,

is obtained. Interpretation: The binary extended theta operation preserves existing information in tests reviewed by a single expert, but in tests reviewed jointly by two experts, it produces a common negative agreement rather than a positive agreement, thereby reliably identifying products that will definitely be rejected in multi-criteria quality control systems as it is presented in Table 1.

Table 1. Binary soft set operations: Focus and interpretation

Note 3.1.4 Binary restricted and binary extended intersection operations coincide in . That is, =

Proposition 3.1.5 is closed in .

Proof: It is evident that is a binary operation in . That is,

Namely, if and are BSSs over then so is. Similarly, let be a fixed subset of E, then

Namely, if and are BSSs over then so is.

Proposition 3.1.6 Let and be BSSs over . Then,, where .

Proof: Let ) for each , ) for each ) for each and , where

for each . Let, where

for each .

Assume that, where

for each and where

for each . Thus, , where .

From a different point of view, holds in the unique non-degenerate meaningful case when and , namely, .

That is, is associative in under certain conditions (UCC).

Note 3.1.7 The conditions necessary to achieve associativity are natural from a mathematical perspective and can be found in both theoretical models and practical applications. In particular, such equality can occur when certain evaluation criteria are equivalent or when different experts make similar classifications based on the same parameters. Therefore, equality is not unique to abstract or artificial scenarios but can arise meaningfully in specific application contexts.

Remark 3.1.8 Let and be BSSs over . Then,

Proof: The proof follows from Proposition 3.1.7. That is, is not associative in .

Proposition 3.1.9 Let and be BSSs over . Then,.

Proof: Let ) for each , ) for each , and , where

for each . Let , where

for each . Thereby, . That is, is commutative in . It is also obvious that , namely, is commutative in as well.

Theorem 3.1.10 Let be a BSS over . Then, .

Proof: Let ) for each and, where

Then, for each , implying that . Namely, is not idempotent in.

Note 3.1.11 This equality demonstrates that the proposed binary extended theta operation naturally generalizes the complement operation. In particular, when applied to identical inputs, the theta operation behaves like a unary negation operator. This property demonstrates the internal consistency of the proposed operation and confirms that the theta operation captures a type of self-negation under common rejection, which is not observed in existing binary soft operations such as extended union or extended intersection. This makes the operation more than a standard binary operation as it is a multi-layered algebraic structure that incorporates both binary and unary information. In particular, due to the single operator (complement) it contains, it systematically reveals the potential for self-negation not present in classical binary soft operations. This feature reinforces the flexibility and internal consistency of the theta operation, strengthening the value and general applicability of the proposed algebraic structure.

Proposition 3.1.12 Let be a BSS over . Then,

Proof: Let for each , and . Assume that , where

for each Thus, , implying that .

Note 3.1.13 According to Proposition 3.1.12, the identity element of is the BSS in . Here, note that for , there does not exist such that , because this situation requires , that is and , that is, this requires . From this, we conclude that in the set , there is no inverse element for any element other than the BSS with respect to Naturally, the inverse of the identity element of , is itself, i.e.

Theorem 3.1.14 is a commutative, not idempotent monoid, whose identiy is UCCs.

Proof: Using Proposition 3.1.5, Proposition 3.1.6, Proposition 3.1.9, Theorem 3.1.10, Proposition 3.1.12 and Note 3.1.13, the proof is clear.

Note 3.1.15 The theta operation in the classical set is not associative [62], and therefore does not form a monoid. In the context of binary soft sets, we can impose certain parameter conditions that make the operation associative—these conditions are not overly restrictive. Under these conditions, the binary extended theta operation is not only associative, but also enables the creation of meaningful binary algebraic structures in This demonstrates that the operation has an independent algebraic value beyond classical set theory.

Proposition 3.1.16 Let be a BSS over . Then, .

Proof: Let and for each . Then, for each . Assume that , where

for each Hence,

(,

implying that .

Proposition 3.1.17 Let be a BSS over . Then, .

Proof: Let and for each . Then, for each . Assume that , where

for each Hence,

,

implying that .

Proposition 3.1.18 Let be a BSS over . Then,

Proof: Let for each and . Then, . Assume that , where

for each . Thereby,

for each Thus, .

Proposition 3.1.19 Let and be BSSs over . Then, if and only if and .

Proof: Let ) and where

for each . Since , for each . Thus, for each , . Thus, )= . Therby, and .

Conversely, let and . Then, for each , and ) for all . Let where

for each . Thus,

implying that .

Proposition 3.1.20 Let and be BSSs over . Then, , , and .

Proof: Since the empty set is a subset of every set, and the universal set is a superset of every set, the proof is straightforward.

Theorem 3.1.21 Let , and be BSSs over . Then,

Proof: (i) Let ), for each and , where

Thus,

for each . Hence, . Moreover, since

is obtained.

(ii) Let ), for each and . Then, , for each . Let , where

Thus,

for each , implying that .

Conversely, let , and . Thus,

for each . Since, then , implying that and . That is, for each . Hence, .

(iii) Let), )) for each , and . Then, for each . Let , where

Let , where

for each Since and for each Thus,

(iv) Let ,, and , and be BSSs over be defined as follows:

Let , where

for each Let where

for each Here, , however is not satisfied.

(v) Let ), )) for each , and . Then, and for each . Let , where

Let , where

Since for each , is obtained.

3.2 Distributions of binary extended theta operation over other types of binary soft set operations

In this subsection, the distributions of binary extended theta operation over other types of BSS operations are provided, and the algebraic structures formed by the binary extended theta operation of BSSs in are thoroughly examined.

Theorem 3.2.1 Let, and be BSSs over . Then, the following distributions hold, where ;

1)

2)

Proof: (1) Let ) for each , ) for each ) for each Let , where

for each and where

for each . Assume that , where

for al and , where

for each . Let, where for each ,

Thus,

Here and are piecewise functions that form both sides of the equality. When examining the parameters line by line, the equality can be achieved by selecting the corresponding parameter sets as empty in the mismatched lines. This situation indicates that certain parameter separation conditions are necessary for the distribution laws to be valid.

Thereby, , where . It is obvious that means that .

Example 3.2.2 Let , , , and the BSSs , and over , be defined as follows:

One can easily show that

Here note that . Thus, the distribution holds. However, we have the following example:

Example 3.2.3 Let , , , and the BSSs , and over , be defined as follows:

One can easily show that

and

It is observed that

Here note that and . Thus, the distribution does not hold.

Theorem 3.2.4 Let, and be BSSs over . Then, the following distributions hold, where ;

1) (

2) (

Proof: (1) Let ) for each , ) for each ) for each and , where ,

Let where ,

Let's consider . Let where ,

Assume that , where ,

Let, where for each ,

Thus,

Hence, , where . It is obvious that means that .

Theorem 3.2.5 (, and (, are multiplicatively commutative and additively idempotent semirings with the identity and without zero UCCs.

Proof.Soylu (2026) showed that (, and (, are commutative, idempotent monoids with the identity, that is, a bounded semilattice (hence a semigroup). (, is a commutative monoid (hence a semigroup) whose identity is under the condition Ꝟ∩Ꝡ∩Ꝙ=∅, where , and are BSSs over . Moreover, distributes over and under the conditions . Consequently, UCCs, (, and (, are additive idempotent multiplicatively commutative semiring without zero, but with unity.

Theorem 3.2.6 Let, and be BSSs over . Then, the following distributions hold, where ;

1)

2)

Proof: (1) Let ) for each , ) for each ) for each Let , where

for each , and where

for each . Assume that , where

for all and , where

for each . Let, where for each ,

Thus,

Here, let's consider in , since =, if an element is in the complement of then it is either in or (. Thus, if , then either or . By taking into account this fact, , where . It is obvious that means that .

Theorem 3.2.7 Let, and be BSSs over . Then, the following distributions hold, where ;

1) (

2) (

Proof: (1) Let ) for each , ) for each ) for each and , where ,

Let where ,

Let's consider . Let where ,

Assume that , where ,

Let, where for each ,

Thus,

Here, let's consider in , since =, if an element is in the complement of then it is either in or (. Thus, if , then either or . By taking into account this fact, , where . It is obvious that means that .

Theorem 3.2.8 (, and (, are multiplicatively commutative and additively idempotent semirings with the identity and without zero UCCs.

Proof.Soylu (2026) showed that (, and (, are non-commutative, idempotent monoids with the identity, that is, a band (hence a semigroup). (, is a commutative monoid (hence a semigroup) whose identity is under the condition Ꝟ∩Ꝡ∩Ꝙ=∅, where , and are BSSs over . Moreover, distributes over and under the conditions . Consequently, UCCs, (, and (, are additive idempotent multiplicatively commutative semirings without zero, but with unity.

This work introduces a novel binary extended soft set operation for binary soft sets (BSSs). We aim to advance the BSS theory by proposing the concept of the "binary extended theta operation" of BSSs and closely analyzing the algebraic structures associated with this operation. The binary extended theta operation proposed in this study, instead of producing positive consensus, reveals common negative information by taking the intersection of the complements of binary soft sets on common parameters. In this respect, the theta operation provides a new algebraic perspective based on negative information for binary soft set theory and conceptually differs from existing binary extended operations in the literature. We also show that the collection of BSSs over the universe combined with binary extended theta operations and certain types of BSS operation forms various important algebraic structures under certain conditions. To expand this body of knowledge, future studies could examine different types of binary extended soft set operations as well as the corresponding distributions and their properties.

Authors' contribution

Aslıhan Sezgin: Conceptualization, validation, formal analysis, data curation, supervision, Orhan Karamustafaoğlu: Methodology, software, validation, investigation, resources, data curation, writing—original draft preparation, writing—review and editing, visualization, supervision. All authors have read and agreed to the published version of the manuscript.

Acknowledgement

The authors would like to thank the reviewers for their valuable suggestions for improving this article.

Funding

No funding was received for this work.

Data a vailability s tatement

The data sets generated and/or analysed during the current study are included in this article. If more information is needed, it is available from the corresponding author upon reasonable request.

Conflicts of i nterest

The authors declare no conflicts of interest.

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