Elementary properties of free lattices

Authors:

16 May 2024

journal article
research article
Published by Walter de Gruyter GmbH in Forum Mathematicum

Vol. 37 (2) , 581-592
DOI:10.1515/forum-2023-0358

Abstract

We start a systematic analysis of the first-order model theory of free lattices. Firstly, we prove that the free lattices of finite rank are not positively indistinguishable, as there is a positive

\exists \forall

-sentence true in

Keywords FREE LATTICES PROFINITE COMPLETIONS MODEL THEORY POSITIVE FIRST-ORDER LOGICAll Related Versions Version 1, 2023-10-05, ArXiv (Unconfirmed version) Version 2, 2024-03-04, ArXiv (Unconfirmed version) Version 3, 2024-03-27, ArXiv (Unconfirmed version)References Publications with shared reference Related articles Cited This publication has 21 references indexed in Scilit: First order rigidity of non-uniform higher rank arithmetic groups Inventiones Mathematicae, 2019 Free and Finitely Presented Lattices 2016 DIOPHANTINE GEOMETRY OVER GROUPS VI: THE ELEMENTARY THEORY OF A FREE GROUP Geometric and Functional Analysis, 2006 Sublattices of complete lattices with continuity conditions Algebra universalis, 2005 Positive theories of free monoids Algebra and Logic, 1993 $L_{\infty _\omega }$ free algebras Algebra universalis, 1989 Categoricity results for L∞κ Annals of Pure and Applied Logic, 1988 Some varieties of semidistributive lattices 1985 Splitting lattices generate all lattices Algebra universalis, 1977 From a lattice to its ideal lattice Algebra universalis, 1974 Read more Read more Publications with shared reference occurs when two publications share one or more common references. This publication has 5055 Publications with shared reference indexed in Scilit, top 10 records with the most shared references are listed below: View publications in search results Cited by 3 articlesScroll to topHave questions? Email us at support@scilit.com © 2026 MDPI (Basel, Switzerland) unless otherwise stated. MDPI initiatives MDPI Journals MDPI Books Sciforum Proceedings Series Preprints.org SciProfiles Encyclopedia JAMS Legal Privacy Terms About Contact Disclaimer Follow us Join us Career © 2026 MDPI (Basel, Switzerland) unless otherwise stated.window.__NUXT__={};window.__NUXT__.config={public:{strapiUrl:"",strapiVersion:"",blog:"https://scilit.wordpress.sciforum.net",scilitBaseUrl:"https://api.scilit.com",mdpiBaseUrl:"https://www.mdpi.com",mdpiApiBaseUrl:"https://api.mdpi.com",mdpiExpandSearchBaseUrl:"https://nlpcn.mdpi.ai/scilit/",cookiebotId:"",googleGtmId:"GTM-KCMN9GD",baseUrl:"https://www.scilit.com",countryAnalysisEnabled:false,sentry:{dsn:"https://94d22edd757f30785802cf53863a1658@self-hosted-sentry.mdpi.com/7",environment:"prod"},strapi:{url:"http://localhost:1337",prefix:"/api",admin:"/admin",version:"v5",cookie:{},auth:{},cookieName:"strapi_jwt",devtools:false},piniaPluginPersistedstate:{},device:{defaultUserAgent:"Mozilla/5.0 (Macintosh; Intel Mac OS X 10_13_2) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/64.0.3282.39 Safari/537.36",enabled:true,refreshOnResize:false},"seo-utils":{canonicalQueryWhitelist:["page","sort","filter","search","q","category","tag"],canonicalLowercase:true},"nuxt-robots":{version:"5.7.1",isNuxtContentV2:false,debug:false,credits:false,groups:[{userAgent:["*"],disallow:["/publications?*","/publishers?*","/sources?*","/scholars?*","/organizations?*","/profile","/api"],allow:["/api/post","/api/posts","/api/most-popular"],contentUsage:[],contentSignal:[],_indexable:true,_rules:[{pattern:"/publications?*",allow:false},{pattern:"/publishers?*",allow:false},{pattern:"/sources?*",allow:false},{pattern:"/scholars?*",allow:false},{pattern:"/organizations?*",allow:false},{pattern:"/profile",allow:false},{pattern:"/api",allow:false},{pattern:"/api/post",allow:true},{pattern:"/api/posts",allow:true},{pattern:"/api/most-popular",allow:true}],_normalized:true},{userAgent:["Nuclei","WikiDo","Riddler","PetalBot","Zoominfobot","Go-http-client","Node/simplecrawler","CazoodleBot","dotbot/1.0","Gigabot","Barkrowler","BLEXBot","magpie-crawler"],comment:["Block non helpful bots"],disallow:["/"],allow:[],contentUsage:[],contentSignal:[],_indexable:false,_rules:[{pattern:"/",allow:false}],_normalized:true}],sitemap:["/sitemap.xml"],header:true,robotsEnabledValue:"index, follow, max-image-preview:large, max-snippet:-1, max-video-preview:-1",robotsDisabledValue:"noindex, nofollow",cacheControl:"max-age=14400, must-revalidate",botDetection:true,pageMetaRobots:{}}},app:{baseURL:"/",buildId:"518c2cec-b19c-4910-9995-4b747fff0c98",buildAssetsDir:"/_nuxt/",cdnURL:""}}{"@context":"https://schema.org","@graph":[{"@id":"https://www.scilit.com/#website","@type":"WebSite","inLanguage":"en","name":"Scilit","url":"https://www.scilit.com/","publisher":{"@id":"https://www.scilit.com/#organization"}},{"@id":"https://www.scilit.com/publications/6ce36eb08296bb8bbc318c8e6709880a#webpage","@type":"WebPage","description":"We start a systematic analysis of the first-order model theory of free lattices.","inLanguage":"English","name":"Elementary properties of free lattices","url":"https://www.scilit.com/publications/6ce36eb08296bb8bbc318c8e6709880a","about":{"@id":"https://www.scilit.com/#identity"},"breadcrumb":{"@id":"https://www.scilit.com/publications/6ce36eb08296bb8bbc318c8e6709880a#breadcrumb"},"isPartOf":{"@id":"https://www.scilit.com/#website"},"mainEntity":{"@id":"https://www.scilit.com/publications/6ce36eb08296bb8bbc318c8e6709880a#article"},"potentialAction":[{"@type":"ReadAction","target":["https://www.scilit.com/publications/6ce36eb08296bb8bbc318c8e6709880a"]}],"primaryImageOfPage":{"@id":"https://www.scilit.com/#logo"}},{"@id":"https://www.scilit.com/#identity","@type":"Organization","email":"mailto:support@scilit.com","name":"Scilit","telephone":"+41 61 683 77 34","url":"https://www.scilit.com","address":{"@type":"PostalAddress","addressLocality":"Basel","addressCountry":"CH","postalCode":"CH-4052","streetAddress":"Grosspeteranlage 5"}},{"@id":"https://www.scilit.com/publications/6ce36eb08296bb8bbc318c8e6709880a#article","@type":"ScholarlyArticle","abstract":"We start a systematic analysis of the first-order model theory of free lattices. Firstly, we prove that the free lattices of finite rank are not positively indistinguishable, as there is a positive \u003Cmath xmlns:env=\"http://degruyter.com/resources/metadata\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns:dgror=\"http://degruyter.com/resources/fetched-ror-id\" xmlns:m=\"http://degruyter.com/resources/metadata\" xmlns:ali=\"http://www.niso.org/schemas/ali/1.0/\" xmlns:tei=\"http://www.tei-c.org/ns/1.0\">                         \u003Cmrow>                            \u003Cmo>∃\u003C/mo>                            \u003Cmo>∀\u003C/mo>                         \u003C/mrow>                      \u003C/math> -sentence true in \u003Cmath xmlns:env=\"http://degruyter.com/resources/metadata\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns:dgror=\"http://degruyter.com/resources/fetched-ror-id\" xmlns:m=\"http://degruyter.com/resources/metadata\" xmlns:ali=\"http://www.niso.org/schemas/ali/1.0/\" xmlns:tei=\"http://www.tei-c.org/ns/1.0\">                         \u003Cmsub>                            \u003Cmi>","dateModified":"2024-05-16T00:00:00Z","datePublished":"2024-05-16T00:00:00Z","headline":"Elementary properties of free lattices","inLanguage":"English","sameAs":"https://doi.org/10.1515/forum-2023-0358","url":"https://www.scilit.com/publications/6ce36eb08296bb8bbc318c8e6709880a","author":[{"@id":"https://www.scilit.com/scholars/10799473#person"},{"@id":"https://www.scilit.com/scholars/13351095#person"}],"identifier":[{"@type":"PropertyValue","propertyID":"DOI","value":"10.1515/forum-2023-0358"}],"isPartOf":{"@type":"Periodical","name":"Forum Mathematicum"},"keywords":["Free lattices","profinite completions","model theory","positive first-order logic"],"mainEntityOfPage":{"@id":"https://www.scilit.com/publications/6ce36eb08296bb8bbc318c8e6709880a#webpage"},"publisher":{"@id":"https://www.scilit.com/organizations/25#publisher"}},{"@id":"https://www.scilit.com/scholars/13351095#person","@type":"Person","name":"Gianluca Paolini","url":"https://sciprofiles.com/profile/author/Kzh4RStiVjJMUmVzZ3E0SCtiVzFrdCtVNHgwWXh3SlRMVm9NVzVFeWtuaz0="},{"@id":"https://www.scilit.com/organizations/25#publisher","@type":"Organization","name":"Walter de Gruyter GmbH","url":"https://www.scilit.com/organizations/25"},{"@id":"https://www.scilit.com/publications/6ce36eb08296bb8bbc318c8e6709880a#breadcrumb","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://www.scilit.com/"},{"@type":"ListItem","position":2,"name":"Publications","item":"https://www.scilit.com/publications"},{"@type":"ListItem","position":3,"name":"Elementary properties of free lattices","item":"https://www.scilit.com/publications/6ce36eb08296bb8bbc318c8e6709880a"}]},{"@id":"https://www.scilit.com/#logo","@type":"ImageObject","caption":"Scilit","contentUrl":"https://www.scilit.com/scilit.svg","inLanguage":"en","url":"https://www.scilit.com/scilit.svg"},{"@id":"https://www.scilit.com/#organization","@type":"Organization","logo":"https://www.scilit.com/scilit.svg","name":"Scilit","url":"https://www.scilit.com","address":{"@type":"PostalAddress","addressLocality":"Basel","addressCountry":"CH","postalCode":"CH-4052","streetAddress":"Grosspeteranlage 5"}}]}[["ShallowReactive",1],{"data":2,"state":218,"once":232,"_errors":233,"serverRendered":40,"path":235,"pinia":236},["ShallowReactive",3],{"blog-config":4,"strapi-config":27,"fetch-publication-6ce36eb08296bb8bbc318c8e6709880a":28,"publication-editors-229524481":47,"shared-reference-count-229524481":122,"preprints-229524481-JOURNAL-ARTICLE":123,"funders-229524481":140,"references-10-90078324-4563728-43617515-43343524-43360654-14151142-45722729-45323288-120850442-35424366-55884942-35319196-35191675-93640273-106156264-3231073-13983626-14053476-17809749-99200338-23083865":141},{"postsPerPage":5,"isInfinite":6,"cacheTime":7,"categories":8},6,false,1,[9,14,19,23],{"id":10,"title":11,"slug":12,"icon":13},1405,"Research and Article Preparation","research-article-preparation","lock_open_right",{"id":15,"title":16,"slug":17,"icon":18},1408,"Publishing Landscape","publishing-landscape","forum",{"id":20,"title":21,"slug":22,"icon":13},1411,"Data and Metrics","data-metrics",{"id":24,"title":25,"slug":26,"icon":13},1400,"Scilit Features","features",{"cacheTime":7},{"identifier":29,"id":30,"publisher_group_identifier":31,"container_group_identifier":32,"unique_author_identifiers":33,"score":36,"type":37,"subtype":38,"title":39,"show_abstract":40,"abstract":41,"abstract_source":42,"abstract_display_right":43,"keywords":44,"pubdate":45,"pubyear":46,"pubdate_print":47,"volume":48,"issue":49,"fpage":50,"lpage":51,"is_oa":6,"doi":52,"pmid":53,"pmcid":53,"publisher":54,"publisherId":55,"workContainer":56,"workContainerId":57,"conference":53,"conferenceId":53,"authors":58,"authors_unique_ids":61,"authors_profile_claimed":64,"affiliations":65,"authors_links":68,"emails":74,"citations":76,"references":77,"providerId":99,"preprint_url":53,"pdf_pmc":53,"pdf_unpaywall":53,"pdf_core":53,"pdf_publisher":53,"has_relations":6,"has_relation_labels":101,"has_relation_ids":103,"is_relation_labels":105,"subjects":107,"institutes":110,"institute_ids":113,"grids":116,"language":118,"hcp_subjects":119,"chemical_substances":53,"mesh_terms":53,"collaboration":121,"sustainable_development_goals":53,"license":53},"6ce36eb08296bb8bbc318c8e6709880a","229524481","019e6262381a7109a1f4eadc12ef8927","019f25df081c70409c720702560d0f7e",[34,35],"019f26f4285a73f8a32a2c939cdc2b30","019f2749684a70d1ad3e20b4d6498fb6",8.110392,"JOURNAL-ARTICLE","RESEARCH-ARTICLE","Elementary properties of free lattices",true,"We start a systematic analysis of the first-order model theory of free lattices. Firstly, we prove that the free lattices of finite rank are not positively indistinguishable, as there is a positive \u003Cmath xmlns:env=\"http:\u002F\u002Fdegruyter.com\u002Fresources\u002Fmetadata\" xmlns:xlink=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxlink\" xmlns:mml=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" xmlns:dgror=\"http:\u002F\u002Fdegruyter.com\u002Fresources\u002Ffetched-ror-id\" xmlns:m=\"http:\u002F\u002Fdegruyter.com\u002Fresources\u002Fmetadata\" xmlns:ali=\"http:\u002F\u002Fwww.niso.org\u002Fschemas\u002Fali\u002F1.0\u002F\" xmlns:tei=\"http:\u002F\u002Fwww.tei-c.org\u002Fns\u002F1.0\">                         \u003Cmrow>                            \u003Cmo>∃\u003C\u002Fmo>                            \u003Cmo>∀\u003C\u002Fmo>                         \u003C\u002Fmrow>                      \u003C\u002Fmath> -sentence true in \u003Cmath xmlns:env=\"http:\u002F\u002Fdegruyter.com\u002Fresources\u002Fmetadata\" xmlns:xlink=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxlink\" xmlns:mml=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" xmlns:dgror=\"http:\u002F\u002Fdegruyter.com\u002Fresources\u002Ffetched-ror-id\" xmlns:m=\"http:\u002F\u002Fdegruyter.com\u002Fresources\u002Fmetadata\" xmlns:ali=\"http:\u002F\u002Fwww.niso.org\u002Fschemas\u002Fali\u002F1.0\u002F\" xmlns:tei=\"http:\u002F\u002Fwww.tei-c.org\u002Fns\u002F1.0\">                         \u003Cmsub>                            \u003Cmi>","PublisherWebsite",2,"Free lattices | profinite completions | model theory | positive first-order logic","2024-05-16T00:00:00Z",2024,null,"37","2","581","592","10.1515\u002Fforum-2023-0358","","Walter de Gruyter GmbH",25,"Forum Mathematicum",21369,[59,60],"J. B. Nation","Gianluca Paolini",[62,63],10799473,13351095,[6,6],[66,67],"Department of Mathematics , 3949 University of Hawaii , Honolulu , HI 96822 , USA","Department of Mathematics “Giuseppe Peano” , 241041 University of Torino , Via Carlo Alberto 10 , Torino 10123 , Italy",[69,72],{"name":59,"url":70,"avatar":71,"message":47},"https:\u002F\u002Fsciprofiles.com\u002Fprofile\u002Fauthor\u002FVWZXQytLY0hEY3ZtZnRFWVE4MXlXWndsZ0JGajZTRE5EYXhSQU9ibjNOOD0=","https:\u002F\u002Fadmin.scilit.com\u002Fbundles\u002Fmdpisciprofileslink\u002Fimg\u002Funknown-user.png",{"name":60,"url":73,"avatar":71,"message":47},"https:\u002F\u002Fsciprofiles.com\u002Fprofile\u002Fauthor\u002FKzh4RStiVjJMUmVzZ3E0SCtiVzFrdCtVNHgwWXh3SlRMVm9NVzVFeWtuaz0=",["Reactive",75],[],3,[78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98],90078324,4563728,43617515,43343524,43360654,14151142,45722729,45323288,120850442,35424366,55884942,35319196,35191675,93640273,106156264,3231073,13983626,14053476,17809749,99200338,23083865,["Reactive",100],[],["Reactive",102],[],["Reactive",104],[],["Reactive",106],[],[108,109],"Algebraic Topology","Mathematical Theory",[111,112],"University of Hawaii System","University of Turin",[114,115],"2006","471",["Reactive",117],[],"English",["Reactive",120],[],4,5055,{"article_id":30,"preprints":124,"published":139},[125,131,135],{"article_id":126,"date_published":127,"version":7,"provider":128,"confirmed":129,"identifier":130},221684382,"2023-10-05","ArXiv",0,"b7ae6f5d3ee3bb51c19db549e1f58d0a",{"article_id":132,"date_published":133,"version":43,"provider":128,"confirmed":129,"identifier":134},229316953,"2024-03-04","4caff7d118cdc2c6b15f922bcfecf029",{"article_id":136,"date_published":137,"version":76,"provider":128,"confirmed":129,"identifier":138},229316954,"2024-03-27","d2ed49cb796c12b50166444bce05d806",[],[],["Reactive",142],[143,159,164,172,180,188,193,203,208,213],{"identifier":144,"id":145,"publisher_group_identifier":146,"container_group_identifier":147,"type":53,"title":148,"abstract":53,"abstract_source":53,"abstract_display_right":53,"keywords":53,"pubdate":47,"pubyear":149,"pubdate_print":47,"volume":47,"issue":47,"fpage":47,"lpage":47,"is_oa":53,"doi":53,"publisher":53,"publisherId":150,"workContainer":151,"workContainerId":152,"conference":53,"conferenceId":53,"authors":153,"citations":129,"references":154,"providerId":155,"preprint_url":53,"pdf_pmc":53,"pdf_unpaywall":53,"pdf_core":53,"pdf_publisher":53,"has_relations":6,"has_relation_labels":156,"is_relation_labels":157,"language":47,"hcp_subjects":158},"7f1cc95490cf582ec880cf42e7b27f07","120850442","019e6262385473c2bad22eb65ff64d8f","019f25dea59473a19866fe06758eebf7","First order rigidity of non-uniform higher rank arithmetic groups",2019,49,"Inventiones Mathematicae",12580,[],[],[],[],[],[],{"identifier":160,"id":161,"publisher_group_identifier":146,"container_group_identifier":53,"type":53,"title":162,"abstract":53,"abstract_source":53,"abstract_display_right":53,"keywords":53,"pubdate":47,"pubyear":163,"pubdate_print":47,"volume":47,"issue":47,"fpage":47,"lpage":47,"is_oa":53,"doi":53,"publisher":53,"publisherId":150,"workContainer":53,"workContainerId":53,"conference":53,"conferenceId":53,"authors":153,"citations":129,"references":154,"providerId":155,"preprint_url":53,"pdf_pmc":53,"pdf_unpaywall":53,"pdf_core":53,"pdf_publisher":53,"has_relations":6,"has_relation_labels":156,"is_relation_labels":157,"language":47,"hcp_subjects":158},"87ae6b55f825bc277e43f25f51d5d622","90078324","Free and Finitely Presented Lattices",2016,{"identifier":165,"id":166,"publisher_group_identifier":146,"container_group_identifier":167,"type":53,"title":168,"abstract":53,"abstract_source":53,"abstract_display_right":53,"keywords":53,"pubdate":47,"pubyear":169,"pubdate_print":47,"volume":47,"issue":47,"fpage":47,"lpage":47,"is_oa":53,"doi":53,"publisher":53,"publisherId":150,"workContainer":170,"workContainerId":171,"conference":53,"conferenceId":53,"authors":153,"citations":129,"references":154,"providerId":155,"preprint_url":53,"pdf_pmc":53,"pdf_unpaywall":53,"pdf_core":53,"pdf_publisher":53,"has_relations":6,"has_relation_labels":156,"is_relation_labels":157,"language":47,"hcp_subjects":158},"d550a6242e1d6b5fd0d8d0610487ade0","45323288","019f25debb8573dd89a037bd44df82e5","DIOPHANTINE GEOMETRY OVER GROUPS VI: THE ELEMENTARY THEORY OF A FREE GROUP",2006,"Geometric and Functional Analysis",14401,{"identifier":173,"id":174,"publisher_group_identifier":146,"container_group_identifier":175,"type":53,"title":176,"abstract":53,"abstract_source":53,"abstract_display_right":53,"keywords":53,"pubdate":47,"pubyear":177,"pubdate_print":47,"volume":47,"issue":47,"fpage":47,"lpage":47,"is_oa":53,"doi":53,"publisher":53,"publisherId":150,"workContainer":178,"workContainerId":179,"conference":53,"conferenceId":53,"authors":153,"citations":129,"references":154,"providerId":155,"preprint_url":53,"pdf_pmc":53,"pdf_unpaywall":53,"pdf_core":53,"pdf_publisher":53,"has_relations":6,"has_relation_labels":156,"is_relation_labels":157,"language":47,"hcp_subjects":158},"44958bda9264b02f0942fc5032d9c15b","45722729","019f25dea53c729db64c5e730cc44571","Sublattices of complete lattices with continuity conditions",2005,"Algebra universalis",12549,{"identifier":181,"id":182,"publisher_group_identifier":146,"container_group_identifier":183,"type":53,"title":184,"abstract":53,"abstract_source":53,"abstract_display_right":53,"keywords":53,"pubdate":47,"pubyear":185,"pubdate_print":47,"volume":47,"issue":47,"fpage":47,"lpage":47,"is_oa":53,"doi":53,"publisher":53,"publisherId":150,"workContainer":186,"workContainerId":187,"conference":53,"conferenceId":53,"authors":153,"citations":129,"references":154,"providerId":155,"preprint_url":53,"pdf_pmc":53,"pdf_unpaywall":53,"pdf_core":53,"pdf_publisher":53,"has_relations":6,"has_relation_labels":156,"is_relation_labels":157,"language":47,"hcp_subjects":158},"a6eb4013e20a550fb74dfd18451ed0eb","43617515","019f25df509f70aca05ee2b37208c733","Positive theories of free monoids",1993,"Algebra and Logic",28914,{"identifier":189,"id":190,"publisher_group_identifier":146,"container_group_identifier":175,"type":53,"title":191,"abstract":53,"abstract_source":53,"abstract_display_right":53,"keywords":53,"pubdate":47,"pubyear":192,"pubdate_print":47,"volume":47,"issue":47,"fpage":47,"lpage":47,"is_oa":53,"doi":53,"publisher":53,"publisherId":150,"workContainer":178,"workContainerId":179,"conference":53,"conferenceId":53,"authors":153,"citations":129,"references":154,"providerId":155,"preprint_url":53,"pdf_pmc":53,"pdf_unpaywall":53,"pdf_core":53,"pdf_publisher":53,"has_relations":6,"has_relation_labels":156,"is_relation_labels":157,"language":47,"hcp_subjects":158},"4562406fabaf26c2f9210cde53685546","4563728","$$L_{\\infty _\\omega }$$ free algebras",1989,{"identifier":194,"id":195,"publisher_group_identifier":196,"container_group_identifier":197,"type":53,"title":198,"abstract":53,"abstract_source":53,"abstract_display_right":53,"keywords":53,"pubdate":47,"pubyear":199,"pubdate_print":47,"volume":47,"issue":47,"fpage":47,"lpage":47,"is_oa":53,"doi":53,"publisher":53,"publisherId":200,"workContainer":201,"workContainerId":202,"conference":53,"conferenceId":53,"authors":153,"citations":129,"references":154,"providerId":155,"preprint_url":53,"pdf_pmc":53,"pdf_unpaywall":53,"pdf_core":53,"pdf_publisher":53,"has_relations":6,"has_relation_labels":156,"is_relation_labels":157,"language":47,"hcp_subjects":158},"fd9980781abe440fa00a88d07ddee085","35424366","019e6262381470479d7d235dcba9bf26","019f25dea1f070b4989360b91849e16c","Categoricity results for L∞κ",1988,22,"Annals of Pure and Applied Logic",12292,{"identifier":204,"id":205,"publisher_group_identifier":146,"container_group_identifier":53,"type":53,"title":206,"abstract":53,"abstract_source":53,"abstract_display_right":53,"keywords":53,"pubdate":47,"pubyear":207,"pubdate_print":47,"volume":47,"issue":47,"fpage":47,"lpage":47,"is_oa":53,"doi":53,"publisher":53,"publisherId":150,"workContainer":53,"workContainerId":53,"conference":53,"conferenceId":53,"authors":153,"citations":129,"references":154,"providerId":155,"preprint_url":53,"pdf_pmc":53,"pdf_unpaywall":53,"pdf_core":53,"pdf_publisher":53,"has_relations":6,"has_relation_labels":156,"is_relation_labels":157,"language":47,"hcp_subjects":158},"649497921255a914c700f4d5430035a5","14151142","Some varieties of semidistributive lattices",1985,{"identifier":209,"id":210,"publisher_group_identifier":146,"container_group_identifier":175,"type":53,"title":211,"abstract":53,"abstract_source":53,"abstract_display_right":53,"keywords":53,"pubdate":47,"pubyear":212,"pubdate_print":47,"volume":47,"issue":47,"fpage":47,"lpage":47,"is_oa":53,"doi":53,"publisher":53,"publisherId":150,"workContainer":178,"workContainerId":179,"conference":53,"conferenceId":53,"authors":153,"citations":129,"references":154,"providerId":155,"preprint_url":53,"pdf_pmc":53,"pdf_unpaywall":53,"pdf_core":53,"pdf_publisher":53,"has_relations":6,"has_relation_labels":156,"is_relation_labels":157,"language":47,"hcp_subjects":158},"fc5edebb8960d6c46a6b30ff70dee2ab","43343524","Splitting lattices generate all lattices",1977,{"identifier":214,"id":215,"publisher_group_identifier":146,"container_group_identifier":175,"type":53,"title":216,"abstract":53,"abstract_source":53,"abstract_display_right":53,"keywords":53,"pubdate":47,"pubyear":217,"pubdate_print":47,"volume":47,"issue":47,"fpage":47,"lpage":47,"is_oa":53,"doi":53,"publisher":53,"publisherId":150,"workContainer":178,"workContainerId":179,"conference":53,"conferenceId":53,"authors":153,"citations":129,"references":154,"providerId":155,"preprint_url":53,"pdf_pmc":53,"pdf_unpaywall":53,"pdf_core":53,"pdf_publisher":53,"has_relations":6,"has_relation_labels":156,"is_relation_labels":157,"language":47,"hcp_subjects":158},"adfc05e22e49e4c9329e47eb71cd0e04","43360654","From a lattice to its ideal lattice",1974,["Reactive",219],{"$snuxt-seo-utils:routeRules":220,"$stoast-ui":221,"$sauth-advanced-feature":6,"$sadd-article-drawer-ui":223,"$ssite-config":224},{"head":-1,"seoMeta":-1},{"showToast":6,"toastMessage":53,"toastVariant":222,"nonDismissive":6},"neutral",{"isVisible":6,"publicationId":47},{"_priority":225,"defaultLocale":228,"env":229,"name":230,"url":231},{"name":226,"env":227,"url":226,"defaultLocale":226},-3,-15,"en","production","Scilit","https:\u002F\u002Fwww.scilit.com\u002F",["Set"],["ShallowReactive",234],{"fetch-publication-6ce36eb08296bb8bbc318c8e6709880a":-1,"publication-editors-229524481":-1,"fetch-articles-analytics-publications-229524481":-1,"fetch-citations-analytics-publications-229524481":-1,"shared-reference-count-229524481":-1,"preprints-229524481-JOURNAL-ARTICLE":-1,"funders-229524481":-1,"references-10-90078324-4563728-43617515-43343524-43360654-14151142-45722729-45323288-120850442-35424366-55884942-35319196-35191675-93640273-106156264-3231073-13983626-14053476-17809749-99200338-23083865":-1,"shared-reference-teaser-229524481":-1},"\u002Fpublications\u002F6ce36eb08296bb8bbc318c8e6709880a",["Reactive",237],{"global-settings":238,"maintenance":254,"auth":255,"navigation":258,"solr-single":367,"solr-api":368,"userRole":369,"search-cache":372,"user-library":373,"solr-analytics":381,"global":426,"scifeed":427,"scitation":443,"solr-references":454},{"data":239,"pending":6,"error":47,"client_ip":253,"mdpiAuthorization":6,"isMdpiRefreshed":40,"isMdpiPromise":47},{"totals":240,"date_rankings":250,"date_full_index":251,"date_delta_index":252},["Reactive",241],{"articles":242,"oa_articles":243,"containers":244,"publishers":245,"authors":246,"institutes":247,"preprint":248,"citations":249},193821916,50787247,137242,31033,25135466,146716,7978620,2370975473,1794745378,1784301319,1794759542,"216.73.217.21",{"maintenanceText":53,"onlyShowMaintenanceAnnouncement":6},{"loginData":256,"error":47,"profileStatus":257,"profilePromise":47},{},"idle",{"navigation":259},[260,264,275,279,283,293,297],{"to":261,"title":263},{"name":262},"rankings","Rankings",{"title":265,"subItems":266},"Publications",[267,271],{"to":268,"title":270},{"name":269},"publications","Search Publications",{"to":272,"title":274},{"name":273},"publications-cited-by-search","Cited-by Search",{"to":276,"title":278},{"name":277},"sources","Sources",{"to":280,"title":282},{"name":281},"publishers","Publishers",{"title":284,"subItems":285},"Scholars",[286,289],{"to":287,"title":284},{"name":288},"scholars",{"to":290,"title":292},{"name":291},"scholars-top-cited-scholars","Top Cited Scholars",{"to":294,"title":296},{"name":295},"organizations","Organizations",{"title":298,"subItems":299},"About",[300,322,341,356],{"title":230,"subItems":301},[302,306,310,314,318],{"to":303,"title":305},{"name":304},"about-about-us","About Us",{"to":307,"title":309},{"name":308},"about-advisory-board","Advisory Board",{"to":311,"title":313},{"name":312},"about-data-sources","Data Sources",{"to":315,"title":317},{"name":316},"about-career","Career",{"to":319,"title":321},{"name":320},"about-contact","Support \u002F Contact",{"title":323,"subItems":324},"User Guide",[325,329,333,337],{"to":326,"title":328},{"name":327},"about-user-guide-publishers","Publisher User",{"to":330,"title":332},{"name":331},"about-user-guide-scholars","Search Function",{"to":334,"title":336},{"name":335},"about-user-guide-rankings","Rankings Function",{"to":338,"title":340},{"name":339},"about-user-guide-api","API Resource",{"title":342,"subItems":343},"Scilit Knowledge",[344,348,352],{"to":345,"title":347},{"name":346},"about-knowledge-metrics","Metrics",{"to":349,"title":351},{"name":350},"about-knowledge-recognitions","Scilit Recognitions",{"to":353,"title":355},{"name":354},"about-knowledge-publications-classification","Publications Classification",{"title":357,"subItems":358},"Media",[359,363],{"to":360,"title":362},{"name":361},"about-media-news","News",{"to":364,"title":366},{"name":365},"about-media-blog","Blog",{},{},{"userRoles":370,"currentRole":371},[],"unknown",{"lastSearchKey":53,"staleWhileRevalidate":6,"configModule":269},{"publications":374,"totalPublications":129,"pending":6,"sort":375,"page":7,"limit":378,"savedInlibraryList":380},[],{"id":376,"name":377,"default":40},"newest_first","Newest first",{"limit":379,"default":40},20,[],{"buckets":382,"configModule":385,"analyticsConfig":386,"loaded":415,"dateRange":416,"dateChangeState":424},["Ref",383],["Reactive",384],{},["Ref",269],["Ref",387],["Reactive",388],[389],{"index":390,"idField":391,"facet":392},"citations","cited_article_id",{"year":393,"sources":399,"publishers":409},{"type":394,"field":395,"start":396,"end":397,"gap":398},"range","citing_article_pub_date","2016-01-01T00:00:00Z","2026-12-31T00:00:00Z","+1YEAR",{"type":400,"query":401,"facet":402},"query","*:*",{"total_number":403,"graph":406},{"type":404,"func":405},"func","unique(citing_work_container_group)",{"type":407,"refine":40,"missing":6,"field":408,"limit":55},"terms","citing_work_container_group",{"type":400,"query":401,"facet":410},{"total_number":411,"graph":413},{"type":404,"func":412},"unique(citing_publisher_group)",{"type":407,"refine":40,"missing":6,"field":414,"limit":55},"citing_publisher_group",["Ref",40],["Ref",417],["Reactive",418],{"startYear":419,"endYear":421},{"value":163,"label":420},"2016",{"value":422,"label":423},2026,"2026",["EmptyRef",425],"false",{},{"isDrawerVisible":428,"createSciFeedForm":429,"loading":434,"sciFeedList":435,"total":438,"currentPage":440,"limit":441},["EmptyRef",425],["Ref",430],["Reactive",431],{"title":53,"frequency":432,"day_of_week":7,"day_of_month":7,"validated":40,"fieldSearch":433},"daily",[],["EmptyRef",425],["Ref",436],["Reactive",437],[],["EmptyRef",439],"0",["Ref",7],["Ref",442],10,{"loading":444,"scitationList":445,"total":448,"currentPage":449,"limit":450,"unsubscribeSelfData":451},["EmptyRef",425],["Ref",446],["Reactive",447],[],["EmptyRef",439],["Ref",7],["Ref",442],["Ref",452],["Reactive",453],{},{"results":455},["Ref",141]]