Infinite Atomized Semilattices

Infinite Atomized Semilattices | 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 Infinite Atomized Semilattices Fernando Martin Maroto, Antonio Ricciardo, David Méndez, Gonzalo G de Polavieja This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8392855/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 Atomized semilattices are subdirect product representations that facilitate theconstruction and manipulation of semilattices in practical contexts. This paper developsa careful axiomatization of atomized semilattices that extends to the infinite setting. Weextend the Full Crossing operator, which yields the quotient via principal congruences,to infinite structures. The notions of atom redundancy are generalized to the infinitecase, revealing a divergence between redundancy and weak redundancy that is absent infinite models. Finally, we prove that for every finitely generated semilattice there existsan atomization consisting exclusively of non-redundant atoms. semilattices congruences subdirect products Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 07 Feb, 2026 Reviewers invited by journal 14 Jan, 2026 Editor assigned by journal 13 Jan, 2026 Submission checks completed at journal 08 Jan, 2026 First submitted to journal 18 Dec, 2025 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-8392855","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":574490758,"identity":"d91882ad-d728-4468-91e3-c7413b005500","order_by":0,"name":"Fernando Martin Maroto","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAxElEQVRIiWNgGAWjYBAC9gYogx9EJBQQoYXnAJgyYJAE6U0wIEWLwQEITYQWsdOJnysq/sgZn1+d+OGBAYM8v9gBAlqkczdLnjljYGx24+1mCaDDDGfOTsCvxV46d4NkY5tB4rYbZzeAtCQY3CagBWTLT5CWzTPObv5BrJZtYFs28PduI9qWbZYNZ4yNJW7wbrNIMJAg7BeQw242VMjJ8fef3XzzR4WNPL80AS0IIAFWKUGschDgP0CK6lEwCkbBKBhJAACD9EMTcvQtwgAAAABJRU5ErkJggg==","orcid":"","institution":"Champalimaud Foundation","correspondingAuthor":true,"prefix":"","firstName":"Fernando","middleName":"Martin","lastName":"Maroto","suffix":""},{"id":574490760,"identity":"571a7c98-6b34-4752-b0d3-b6e38444968b","order_by":1,"name":"Antonio Ricciardo","email":"","orcid":"","institution":"Algebraic AI","correspondingAuthor":false,"prefix":"","firstName":"Antonio","middleName":"","lastName":"Ricciardo","suffix":""},{"id":574490761,"identity":"ee0c96f0-286d-4dc3-8c7a-d91047fef679","order_by":2,"name":"David Méndez","email":"","orcid":"","institution":"Champalimaud Foundation","correspondingAuthor":false,"prefix":"","firstName":"David","middleName":"","lastName":"Méndez","suffix":""},{"id":574490762,"identity":"fe42dd42-a903-46e0-a919-c8e382a30047","order_by":3,"name":"Gonzalo G de Polavieja","email":"","orcid":"","institution":"Champalimaud Foundation","correspondingAuthor":false,"prefix":"","firstName":"Gonzalo","middleName":"G","lastName":"de Polavieja","suffix":""}],"badges":[],"createdAt":"2025-12-18 08:23:41","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8392855/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8392855/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":100537339,"identity":"a72e820d-68c4-4c1e-bc23-22982cacf7b0","added_by":"auto","created_at":"2026-01-19 04:18:13","extension":"json","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":4895,"visible":true,"origin":"","legend":"","description":"","filename":"ed825181d4bd4e0a8a003eada5e7482b.json","url":"https://assets-eu.researchsquare.com/files/rs-8392855/v1/b17bed806a77049044396609.json"},{"id":100537340,"identity":"fe2ac506-9d59-4c77-9223-97cd5506a541","added_by":"auto","created_at":"2026-01-19 04:18:17","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":296807,"visible":true,"origin":"","legend":"","description":"","filename":"iasIII.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8392855/v1_covered_ca35fd1c-cc1f-4965-a441-80e3b64cdeeb.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Infinite Atomized Semilattices","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":"order","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"orde","sideBox":"Learn more about [Order](http://link.springer.com/journal/11083)","snPcode":"11083","submissionUrl":"https://submission.nature.com/new-submission/11083/3","title":"Order","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"semilattices, congruences, subdirect products","lastPublishedDoi":"10.21203/rs.3.rs-8392855/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8392855/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Atomized semilattices are subdirect product representations that facilitate theconstruction and manipulation of semilattices in practical contexts. 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