Binary extended theta operation of binary soft sets

Aslıhan Sezgin; Orhan Karamustafaoğlu

doi:10.55976/dma.42026152210-23

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Binary extended theta operation of binary soft sets

Aslıhan Sezgin ¹ , Orhan Karamustafaoğlu ² ^*

1. Department of Mathematics and Science Education, Faculty of Education, Amasya University, P.O. Box 05100, Amasya, Turkey
2. Department of Mathematics and Science Education, Faculty of Education, Amasya University, P.O. Box 05100, Amasya, Turkey

Correspondence: orhan.karamustafaoglu@amasya.edu.tr

DOI: https://doi.org/10.55976/dma.42026152210-23

Received

19 January 2026
Revised

13 February 2026
Accepted

13 March 2026
Published

27 April 2026

Binary soft sets Binary soft set operations Binary extended theta operation Semiring

Abstract

<p>Binary soft set theory, first introduced by Açıkgöz and Taş in 2016, has become widely accepted as a technique for addressing and modeling uncertainty. Numerous theoretical and practical problems have been solved using this approach. Scholars have shown sustained interest in the theory's core concepts and operations since its inception. In this study, we propose the binary extended theta operation, a special binary soft set operation, and provide a thorough analysis of its basic algebraic features. We also study the distribution of this operation over certain types of binary soft set operations. By considering its algebraic properties and distribution rules, we show that, when combined with specific binary soft set operations, the binary extended theta operation forms many important algebraic structures within the collection of binary soft sets over the universe under certain conditions. The fundamental conceptual difference between the proposed binary extended theta operation and existing binary extended operations in the literature is that unlike approaches based on positive information aggregation, the theta operation systematically extracts negative information through common parameters and offers a unique and complementary tool, particularly for decision problems requiring reliable elimination, risk exclusion, and error detection. Further applications, including cryptology and decision-making, rely on operations of binary soft sets, making this theoretical subject essential from both theoretical and practical perspectives.</p>

<h5>Introduction</h5>
<p>Soft set theory has become a powerful mathematical tool for handling uncertainty since its introduction by Molodtsov <a href="#reference_list_0">[1]</a> in 1999. Maji et al. <a href="#reference_list_1">[2]</a>, Pei and Miao <a href="#reference_list_2">[3]</a>, and Ali et al. <a href="#reference_list_3">[4,5]</a> are the pioneers in soft set operations and developed the algebraic basis of soft set theory. The uni-int decision-making method for soft sets was proposed by Çağman and Enginoğlu <a href="#reference_list_5">[6]</a>.</p>
<p>Soft int-groups were introduced by Çağman et al. <a href="#reference_list_6">[7]</a>, while Sezer et al. <a href="#reference_list_7">[8]</a> introduced soft N-groups to further generalize group theory within the framework of soft sets. Later, Sezer et al. <a href="#reference_list_8">[9]</a> expanded on this by introducing the notion of N-group SI-actions. Atagün and Sezer <a href="#reference_list_9">[10]</a> explored soft semimodules, establishing connections between module theory and soft sets. A notable shift towards applications began with Khan et al. <a href="#reference_list_10">[11]</a>, who characterized Abel Grassmann's groupoids via soft ideals. Atagün and Sezgin <a href="#reference_list_11">[12]</a> analyzed soft substructures of groups and semirings. Around the same time, soft sets were applied to near-ring theory <a href="#reference_list_12">[13]</a> and ideal theory <a href="#reference_list_13">[14]</a>, further expanding the theory's domain. Atagün et al. <a href="#reference_list_14">[15]</a> explored P-properties in near-rings, while additional work on prime, maximal, and principal soft ideals of rings and soft covered ideals of semigroups were discussed in <a href="#reference_list_15">[16,17]</a>. In addition, the social relevance of soft set theory has been highlighted in emerging fields such as social networks, as explored by Riaz et al. <a href="#reference_list_17">[18]</a>. Soft intersection almost subsemigroups and ideals of semigroups were addressed in <a href="#reference_list_18">[19,20]</a>. Finally, the most recent contribution presents a complete study of the star-product and-product of soft sets <a href="#reference_list_20">[21,22]</a>.</p>
<p>Since its introduction by Açıkgöz and Taş <a href="#reference_list_22">[23]</a>, the concept of the binary soft set (BSS) has attracted considerable interest among decision scientists and mathematicians. By linking binary relations to components of the parameter set, this novel theory extends Molodtsov's soft set theory and offers a more adaptable framework for handling uncertainty in complicated settings.</p>
<p>Binary soft topological spaces [24–27], soft subspaces <a href="#reference_list_27">[28]</a>, vague BSSs <a href="#reference_list_28">[29]</a>, separation axioms, [30–32], weak binary soft structures <a href="#reference_list_32">[33]</a>, bipolar and fuzzy binary soft models [34–36], and binary soft functions <a href="#reference_list_36">[37]</a> are foundational studies.</p>
<p>The model's potential in expert systems, artificial intelligence, and advanced decision theory has been demonstrated by various recent contributions [38–40]. In fields like healthcare <a href="#reference_list_37">[38]</a> and multi-agent systems <a href="#reference_list_40">[41]</a>, soft mappings and BSS-based decision processes have shown promise. Besides, extensions into neutrosophic structures and hypersoft sets showed how the theory can be used to complex logical systems [39,42–44]. Integration of BSSs with fuzzy and intuitionistic fuzzy frameworks <a href="#reference_list_44">[45]</a>, neutrosophic systems <a href="#reference_list_41">[42,46]</a>, and plithogenic models <a href="#reference_list_40">[41]</a> have become a significant trend. Additionally, several studies explore binary soft α-open sets <a href="#reference_list_46">[47]</a>, soft θ-open and ω-open sets <a href="#reference_list_47">[48]</a>, soft ωβ-open sets <a href="#reference_list_48">[49]</a>, and soft homogeneous components <a href="#reference_list_49">[50]</a>, enhancing the structural analysis of binary soft topologies.</p>
<p>Contributions have also focused on extending classical mathematical results to BSSs, such as selection principles <a href="#reference_list_50">[51]</a>, object interaction sets <a href="#reference_list_51">[52]</a>, and applications in game theory <a href="#reference_list_52">[53]</a>. Meanwhile, binary soft closure spaces and limit points have been reconsidered in recent studies to refine the notions of continuity and compactness <a href="#reference_list_30">[31,54]</a>. The ongoing development of BSS theory is further evidenced by continued work on structural properties, mappings <a href="#reference_list_29">[30,32]</a>, topological functions, fuzzy BSSs and their applications, fuzzy parameterized fuzzy BSSs and their operations, binary hypersoft sets, and fuzzy binary soft topological spaces <a href="#reference_list_54">[55-59]</a>. Dalkılıç <a href="#reference_list_59">[60]</a> contributed by introducing and thoroughly developing the concepts of relations on BSSs, blocks, partitions, compositions, and binary soft functions.</p>
<p>The concept of BSSs was first introduced by Açıkgöz and Taş <a href="#reference_list_22">[23]</a>. Soylu <a href="#reference_list_60">[61]</a> revisited the foundational concepts of BSS theory, and updated existing definitions as necessary. Additionally, many new concepts regarding the BSS such as empty BSS, absolute BSS, and relative null/whole BSS with respect to a fixed parameter set, were introduced. Subsequently, different types of operations on BSSs, such as binary restricted operations and binary extended operations were presented.  </p>
<p>The vital role that BSSs plays in contemporary mathematics and decision analysis is confirmed by this extensive body of literature. By investigating the binary extended theta operation of BSSs, the current study builds on this foundation and attempts to advance the theory of BSSs by offering both theoretical insights and practical applications aligned with current research. The primary motivation for this study is that existing binary extended operations defined in the binary soft set literature are largely focused on positive information integration. In particular, operations such as extended union and extended intersection aim to combine acceptable or positive information from different sources. In contrast, a critical element in many real-world applications is determining which elements are universally rejected by all sources, i.e., identifying elements that must be definitively eliminated. In this context, the binary extended theta operation proposed in this study, rather than producing positive consensus, reveals common negative information by taking the intersection of the complements of binary soft sets on common parameters. Thus, the theta operation conceptually differs from existing binary extended operations in the literature and provides a new algebraic perspective based on negative information for binary soft set theory. By incorporating a real-life micro-example into the study, it has been clearly demonstrated that the theta process plays a practical and meaningful role, particularly in scenarios involving quality control, risk analysis, and reliable elimination, in contrast to classical binary extended processes. We demonstrate that the binary extended theta operation produces numerous significant algebraic structures within the collection of soft sets over the universe under certain conditions, by considering its algebraic properties and distribution principles. The structure of this study is as follows: The fundamental concepts of BSSs are reviewed in Section 2. In section 3, the binary extended theta operation, is defined and its algebraic properties are thoroughly examined together with its distributions over other BSS operations. A thorough examination of the algebraic structures produced in the set of BSSs over the universe by these operations is provided. In the conclusion section, we discuss the significance of the study's findings and their potential applications.</p>
<p><em>To read more details, please download the <a href="https://ojs.luminescience.cn/DMA/article/view/522/440">PDF</a> to see the full text.</em></p>
<h5>Conclusion</h5>
<p>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.</p>
<h5>Authors' contribution</h5>
<p>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.</p>
<h5>Acknowledgement</h5>
<p>The authors would like to thank the reviewers for their valuable suggestions for improving this article.</p>
<h5>Funding</h5>
<p>No funding was received for this work.</p>
<h5>Data availability statement</h5>
<p>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.</p>
<h5>Conflicts of interest</h5>
<p>The authors declare no conflicts of interest.</p>

References

[1]References

[2]Molodtsov D. Soft set theory-first results. Computers & Mathematics with Applications.1999;42(1): 19–31. doi: 10.1016/S0898-1221(99)00056-5.

[3]Maji PK, Biswas R, Roy AR. Soft set theory. Computers & Mathematics with Applications.2003;45(1): 555–562. doi: 10.1016/S0898-1221(02)00216-X.

[4]Pei D, Miao D. From soft sets to information systems. In: Proceedings of Granular Computing. 2005 IEEE International Conference on Granular Computing. 2005;2: 617–621. doi: 10.1109/GRC.2005.1547365.

[5]Ali MI, Feng F, Liu X, Min WK, Shabir M. On some new operations in soft set theory. Computers & Mathematics with Applications.2009;57(9): 1547–1553. doi: 10.1016/j.camwa.2008.11.009.

[6]Ali MI, Shabir M, Naz M. Algebraic structures of soft sets associated with new operations. Computers & Mathematics with Applications.2011;61: 2647–2654. doi: 10.1016/j.camwa.2011.03.011.

[7]Çağman N, Enginoğlu S. Soft set theory and uni-int decision making. European Journal of Operational Research.2010;7(2): 848–855. doi: 10.1016/j.ejor.2010.05.004.

[8]Çağman N, Çitak F, Aktaş H. Soft int-group and its applications to group theory. Neural Computing and Applications.2012;2: 151–158. doi: 10.1007/s00521-011-0752-x.

[9]Sezer AS, Atagün AO, Çağman N. A new view to N-group theory: soft N-groups. Fasciculi Mathematici.2013;51: 123–140.

[10]Sezer AS, Atagün AO, Çağman N. N-group SI-action and its applications to N-group theory. Fasciculi Mathematici.2014;54: 139–153.

[11]Atagün AO, Sezer AS. Soft sets, soft semimodules and soft substructures of semimodules. Mathematical Sciences Letters.2015;4(3): 235. doi: 10.12785/msl/040303.

[12]Khan A, Izhar M, Sezgin A. Characterizations of Abel Grassmann's groupoids by the properties of their double-framed soft ideals. International Journal of Analysis and Applications.2017;15(1): 62–74. https://etamaths.com/index.php/ijaa/article/view/1328.

[13]Atagün AO, Sezgin A. Int-soft substructures of groups and semirings with applications. Applied Mathematics & Information Sciences.2017;11(1): 105–113. doi: 10.18576/amis/110113.

[14]Atagün AO, Sezgin A. A new view to near-ring theory: Soft near-rings. South East Asian Journal of Mathematics & Mathematical Sciences.2018;14(3).

[15]Jana C, Pal M, Karaaslan F, Sezgin A. (α, β)-Soft intersectional rings and ideals with their applications. New Mathematics and Natural Computation.2019;15(02): 333–350. doi: 10.18576/amis/110113.

[16]Atagün AO, Kamaci H, Tastekin I, Sezgin A. P-properties in near-rings. Journal of Mathematical and Fundamental Sciences.2019;51(2): 152–167. doi: 10.5614/j.math.fund.sci.2019.51.2.5.

[17]Atagün AO, Sezgin A. More on prime, maximal and principal soft ideals of soft rings. New Mathematics and Natural Computation.2022;18(01): 195–207. doi: 10.1142/S1793005722500119.

[18]Sezgin A, Orbay M. Analysis of semigroups with soft intersection ideals. Acta Universitatis Sapientiae, Mathematica.2022;14(1): 166-210. doi: 10.2478/ausm-2022-0012.

[19]Riaz M, Hashmi MR, Karaaslan F, Sezgin A, Shamiri MMAA, Khalaf MM. Emerging trends in social networking systems and generation gap with neutrosophic crisp soft mapping. CMES–Computer Modeling in Engineering and Sciences.2023;136(2): 1759–1783. doi: 10.32604/cmes.2023.023327.

[20]Sezgin A, İlgin A. Soft intersection almost ideals of semigroups. Journal of Innovative Engineering and Natural Science.2024;4(2): 466–481. doi: 10.61112/jiens.1464344.

[21]Sezgin A, İlgin A. Soft intersection almost subsemigroups of semigroups. International Journal of Mathematics and Physics.2024;15(1): 13–20. doi:10.26577/ijmph.2024v15i1a2.

[22]Sezgin A, Şenyiğit E. A new product for soft sets with its decision-making: Soft star-product. Big Data and Computing Visions.2025;5(1): 52–73. doi: 10.22105/bdcv.2024.492834.1221.

[23]Sezgin A, Atagün AO, Çağman N. A complete study on and-product of soft sets. Sigma Journal of Engineering and Natural Sciences.2025;43(1): 1–14. doi: 10.14744/sigma.2025.00002.

[24]Açıkgöz A, Taş N. Binary soft set theory. European Journal of Pure and Applied Mathematics.2016;30(2): 1113–1121.

[25]Benchalli SS, Patil PG, Dodamani AS, Pradeepkumar J. On binary soft topological spaces. International Journal of Applied Mathematics.2017;30(6): 437–453. doi: doi: 10.12732/ijam.v30i6.1

[26]Hussain S. Binary soft connected spaces and an application of binary soft sets in decision making problem. Fuzzy Information and Engineering.2019;11(4): 506–521. doi: 10.1080/16168658.2020.1773600.

[27]Hussain S. On some structures of binary soft topological spaces. Hacettepe Journal of Mathematics and Statistics.2019;48(3): 644–656. doi: 10.15672/HJMS.2017.536

[28]Patil PG, Bhat NN. New separation axioms in binary soft topological spaces. Italian Journal of Pure and Applied Mathematics.2020;(44): 775–783.

[29]Khattak AM, Ullah Z, Amin F, Abdullah S, Jabeen S, Khattak NA, Khattak ZA. Soft sub spaces and soft b-separation axioms in binary soft topological spaces. Journal of New Theory.2018;(23): 48–62.

[30]Remya PB, Shalini AF. Vague binary soft sets and their properties. International Journal of Engineering, Science and Mathematics.2018;7(11): 56–73.

[31]Patil PG, Bhat NN. Binary soft locally closed sets. Malaya Journal of Matematik.2021;9(3): 83–90. https://doi.org/10.26637/MJM0903/003.

[32]Subhashini J, Metilda PG. A note on fuzzy binary soft separation axioms. In: AIP Conference Proceedings.2022;2385: 130012. doi: 10.1063/5.0070744.

[33]Patil PG, Adaki AG. Results on binary soft topological spaces. Ratio Mathematica.2023;48. doi: 10.23755/rm.v48i0.1136.

[34]Khattak AM, Haq ZU, Burqi MZ, Abdullah S. Weak soft binary structures. Journal of New Theory.2019;(26): 64–72.

[35]Al-Shami TM. Bipolar soft sets: relations between them and ordinary points and their applications. Complexity.2021;2021: 6621854. doi: 10.1155/2021/6621854.

[36]Metilda PG, Subhashini J. Fuzzy binary soft separation axioms and its properties. In: Journal of Physics: Conference Series.2021;1947(1): 012020. doi: 10.1088/1742-6596/1947/1/012020.

[37]Metilda PG, Subhashini J. Remarks on fuzzy binary soft limit points. In: AIP Conference Proceedings.2022;2385: 130035. doi: 10.1063/5.0070723.

[38]Patil PG, Bhat NN. Some properties and characterizations of binary soft functions. Southeast Asian Bulletin of Mathematics.2023;47(3).

[39]Hussain S, Alkhalifah M. An application of binary soft mappings to the problem in medical expert systems. Journal of Applied Mathematics & Informatics.2020;38(5–6): 533–545. doi: 10.14317/jami.2020.533.

[40]Kamacı H, Saqlain M. n-ary fuzzy hypersoft expert sets. Neutrosophic Sets and Systems.2021;43(1): 15.

[41]Saleh HY, Salih AA, Asaad BA, Mohammed RA. Binary bipolar soft points and topology on binary bipolar soft sets with their symmetric properties. Symmetry.2023;16(1): 23. doi: 10.3390/sym16010023.

[42]Basumatary B, Wary N, Saeed M, Saqlain M. On some properties of plithogenic neutrosophic hypersoft almost topological group. Neutrosophic Sets and Systems.2021;43(1).

[43]Mohamed SY, Tamilselvi AS, Srividhya G. Neutrosophic bipolar vague binary topological space. Ratio Mathematica.2023;50. doi:10.23755/rm.v50i0.1544. ISSN: 1592-7415.

[44]Nagomi R, Shalini AF. Neutrosophic vague binary soft topological spaces. International Journal of Humanities and Sciences.2024;1(2): 45–55. doi: 10.34256/ijohs126.

[45]Nawaz M, Liu Y, Ramaswamy S, Saeed MM, Khattak AM. Analysis of some structures in ternary soft topological spaces. European Journal of Pure and Applied Mathematics.2025;18(1): 5567–5567. doi: 10.29020/nybg.ejpam.v18i1.5567.

[46]Sivasankari H, Subhashini J. Intuitionistic fuzzy binary soft sets and its properties. Indian Journal of Science and Technology.2024;17(35): 3636–3642. doi: 10.17485/IJST/v17i35.1161.

[47]Manoharan R, Peter MJ. Exploring quantum and binary soft τ–open sets: practical applications through soft topologies. ResearchSquare Preprint.2024. doi: 10.21203/rs.3.rs-3952730/v1.

[48]Parvathy CR, Sofia A. Binary soft simply* alpha open sets and continuous function. Italian Journal of Pure and Applied Mathematics. 2024; 51: 398–410.

[49]Al Ghour S. Between soft θ-openness and soft ω₀-openness. Axioms.2023;12(3): 311. doi: 10.3390/axioms12030311.

[50]Al-Ghour S. Soft ωβ-open sets and their generated soft topology. Computational and Applied Mathematics.2024;43(4): 209. doi: 10.1007/s40314-024-02731-5.

[51]Al-Ghour S. Soft homogeneous components and soft products. Fuzzy Information and Engineering.2024;16(1): 24–32. doi: 10.26599/FIE.2023.9270029.

[52]Kočinac LD, Al-Shami TM, Çetkin V. Selection principles in the context of soft sets: Menger spaces. Soft Computing.2021;25(20): 12693–12702. doi: 10.1007/s00500-021-06069-6.

[53]Dalkılıç O, Cangul IN. Determining interactions between objects from different universes: (inverse) object interaction set for binary soft sets. Soft Computing.2024;28(21): 12869–12877. doi: 10.1007/s00500-024-10318-9.

[54]Kollias I, Leventides J, Papavassiliou VG. On the solution of games with arbitrary payoffs: an application to an over-the-counter financial market. International Journal of Finance & Economics.2024;29(2): 1877–1895. doi: 10.1002/ijfe.2758.

[55]Majeed RN. Binary τech soft closure spaces. TWMS Journal of Applied and Engineering Mathematics.2024;14(2): 683–695.

[56]Metilda G, Subhashini J. Some results on fuzzy binary soft point. Space.2021;1: 589–592.

[57]Metilda PG, Subhashini J. An application of fuzzy binary soft set in decision making problems. Webology.2021;18(6): 3672–3680.

[58]Patil PG, Teli R, PN A. Fuzzy parameterized fuzzy binary soft sets and their application in decision making. TWMS Journal of Applied and Engineering Mathematics.2025;15(8): 2041–2049.

[59]Omar ZA, Asaad BA. Binary hypersoft sets. Filomat.2025;39(27): 9449–9472. doi: 10.2298/FIL2527449O.

[60]Patil PG, Reddy CJS, Teli R, Elluru V. New structures in fuzzy binary soft topological spaces. International Journal of Mathematics Trends and Technology.2025;71(4): 55-60. doi: 10.14445/22315373/IJMTT-V71I4P106.

[61]Dalkılıç O. Unifying relationships in uncertain environments: examining relations in binary soft sets for expressing inter-object correspondence. The Journal of Supercomputing.2025;81(16): 1–29. doi: 10.1007/s11227-025-08036-6.

[62]Soylu MS. Binary soft set with its basic concepts and operations. MSc Thesis, Amasya University, Amasya, Türkiye.2026.

[63]Sezgin A, Çağman N, Atagün AO, Aybek FN. Complemental binary operations of sets and their application to group theory. Matrix Science Mathematic.2023;7(2): 114–121. doi: 10.26480/msmk.02.2023.114.121.

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Sezgin, A., & Karamustafaoğlu, O. (2026). Binary extended theta operation of binary soft sets. Decision Making and Analysis, 4(1), 10–23. https://doi.org/10.55976/dma.42026152210-23

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