S-approximation spaces extension model based on item-polytomous perspective

S-approximation spaces extension model based on item-polytomous perspective | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article S-approximation spaces extension model based on item-polytomous perspective Xiaojie Xie, Shujiao Liao, Jinjin Li This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4447331/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 5 You are reading this latest preprint version Abstract S-approximation spaces are important extension models of two-universe. However, existing S-approximation spaces are unable to represent and process item-polytomous knowledge, which limits the application of this model to a certain extent. Therefore, it is necessary to address these problems with other models.For this purpose, this paper proposes a new extension model named polytomous S-approximation spaces. First, the concepts in polytomous S-approximation spaces are introduced and the basic properties of the model under different operations and assumptions are discussed. Second, this paper outlines a relationship between knowledge space theory and polytomous S-approximation spaces. The upper and lower approximations of family of sets in polytomous S-approximation spaces are researched.A new method is established, based on this, for constructing a special type of polytomous knowledge structures.Third, the upper and lower approximation numbers in polytomous S-approximation spaces are proposed. Moreover, the approximation numbers capture interesting properties.Finally, based on the approximation numbers, it constructs two kinds of matroidal structures.In addition, the operations in the obtained matroidal structures are investigated.Compared with other models, the proposed polytomous S-approximation spaces further enrich the S-approximation spaces theory and the polytomous generalization of knowledge space theory. S-approximation spaces upper approximation lower approximation polytomous knowledge structure matroidal structure Full Text Cite Share Download PDF Status: Under Review Version 1 posted Editor assigned by journal 21 May, 2025 Reviewers agreed at journal 18 Mar, 2025 Reviewers invited by journal 17 Nov, 2024 Editor invited by journal 22 May, 2024 First submitted to journal 19 May, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4447331","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":379034180,"identity":"77954156-f8b2-452a-9742-4105a840307f","order_by":0,"name":"Xiaojie Xie","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/0lEQVRIie2PsUoDQRRFZ3kw2zyZdhZEf+FJwCgs8Vd2CKSyEGxSzjKydqlj5TfYiOUsKbZZzAekmcUf2DRiIehsmSKTlIJz4BUP7uFyGYtE/iLgTxJjnCXaFvMchdBHK1A6185Os6U9ui81F93DKiddhHPUQOeu7nK1SMtKKr1GYjbpt7f7lczwEUmajTjWXnnb4Bg0ZE+v+xUB7FJKWp1xqbzSbvBaWw4nAYVD+umVH+TnnVeqdyRbhBUBOLRY35IYUpU9rGQG7/2Wqd+iSle0U8yWtQluoXXz8iG/J+r5sXH113xyI4Sp+21AGQC5+yc6nB8i/cFIJBKJ/Gt+AV96TSX6bIBjAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0009-0001-1271-5447","institution":"Minnan Normal University","correspondingAuthor":true,"prefix":"","firstName":"Xiaojie","middleName":"","lastName":"Xie","suffix":""},{"id":379034181,"identity":"9659589a-7dc8-483c-92e2-ef3289aa7f6a","order_by":1,"name":"Shujiao Liao","email":"","orcid":"https://orcid.org/0000-0003-3640-3648","institution":"Minnan Normal University","correspondingAuthor":false,"prefix":"","firstName":"Shujiao","middleName":"","lastName":"Liao","suffix":""},{"id":379034182,"identity":"d2346685-9718-4bc0-9b50-b0de3e88bb4c","order_by":2,"name":"Jinjin Li","email":"","orcid":"","institution":"Minnan Normal University","correspondingAuthor":false,"prefix":"","firstName":"Jinjin","middleName":"","lastName":"Li","suffix":""}],"badges":[],"createdAt":"2024-05-20 07:26:40","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4447331/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4447331/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":71595840,"identity":"9f8bf83b-80f0-439e-b399-f1caf4c2fd13","added_by":"auto","created_at":"2024-12-17 04:11:18","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":320558,"visible":true,"origin":"","legend":"","description":"","filename":"snarticle.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4447331/v1_covered_446de938-c8cb-4902-87ba-01bf91966bd3.pdf"}],"financialInterests":"","formattedTitle":"S-approximation spaces extension model based on item-polytomous perspective","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"soft-computing","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"soco","sideBox":"Learn more about [Soft Computing](https://www.springer.com/journal/500)","snPcode":"500","submissionUrl":"https://submission.nature.com/new-submission/500/3","title":"Soft Computing","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"S-approximation spaces, upper approximation, lower approximation, polytomous knowledge structure, matroidal structure","lastPublishedDoi":"10.21203/rs.3.rs-4447331/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4447331/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eS-approximation spaces are important extension models of two-universe. 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