{"id":"https://openalex.org/W4285325858","doi":"https://doi.org/10.2478/forma-2021-0016","title":"Duality Notions in Real Projective Plane","display_name":"Duality Notions in Real Projective Plane","publication_year":2021,"publication_date":"2021-12-01","ids":{"openalex":"https://openalex.org/W4285325858","doi":"https://doi.org/10.2478/forma-2021-0016"},"language":"en","primary_location":{"id":"doi:10.2478/forma-2021-0016","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2021-0016","pdf_url":"https://www.sciendo.com/pdf/10.2478/forma-2021-0016","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.sciendo.com/pdf/10.2478/forma-2021-0016","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5068514239","display_name":"Roland Coghetto","orcid":"https://orcid.org/0000-0002-4901-0766"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Roland Coghetto","raw_affiliation_strings":["cafr-MSA2P asbl , Rue de la Brasserie 5 , La Louvi\u00e8re , Belgium"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"cafr-MSA2P asbl , Rue de la Brasserie 5 , La Louvi\u00e8re , Belgium","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5068514239"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.29942328,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"29","issue":"4","first_page":"161","last_page":"173"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10304","display_name":"Geometric and Algebraic Topology","score":0.9776999950408936,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9628000259399414,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/duality","display_name":"Duality (order theory)","score":0.7741044759750366},{"id":"https://openalex.org/keywords/projective-plane","display_name":"Projective plane","score":0.6739773154258728},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6317172050476074},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.553785502910614},{"id":"https://openalex.org/keywords/projective-space","display_name":"Projective space","score":0.5365996360778809},{"id":"https://openalex.org/keywords/real-projective-plane","display_name":"Real projective plane","score":0.5259019732475281},{"id":"https://openalex.org/keywords/hol","display_name":"HOL","score":0.48275917768478394},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.44678062200546265},{"id":"https://openalex.org/keywords/pencil","display_name":"Pencil (optics)","score":0.41986584663391113},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.41795456409454346},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.40272951126098633},{"id":"https://openalex.org/keywords/projective-test","display_name":"Projective test","score":0.27665215730667114},{"id":"https://openalex.org/keywords/complex-projective-space","display_name":"Complex projective space","score":0.19215989112854004},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10537421703338623}],"concepts":[{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display_name":"Duality (order theory)","level":2,"score":0.7741044759750366},{"id":"https://openalex.org/C39290043","wikidata":"https://www.wikidata.org/wiki/Q1475760","display_name":"Projective plane","level":3,"score":0.6739773154258728},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6317172050476074},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.553785502910614},{"id":"https://openalex.org/C75280867","wikidata":"https://www.wikidata.org/wiki/Q877775","display_name":"Projective space","level":3,"score":0.5365996360778809},{"id":"https://openalex.org/C37470923","wikidata":"https://www.wikidata.org/wiki/Q633815","display_name":"Real projective plane","level":5,"score":0.5259019732475281},{"id":"https://openalex.org/C17435882","wikidata":"https://www.wikidata.org/wiki/Q17030435","display_name":"HOL","level":2,"score":0.48275917768478394},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.44678062200546265},{"id":"https://openalex.org/C134949993","wikidata":"https://www.wikidata.org/wiki/Q2068617","display_name":"Pencil (optics)","level":2,"score":0.41986584663391113},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.41795456409454346},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.40272951126098633},{"id":"https://openalex.org/C177846678","wikidata":"https://www.wikidata.org/wiki/Q1501864","display_name":"Projective test","level":2,"score":0.27665215730667114},{"id":"https://openalex.org/C132702353","wikidata":"https://www.wikidata.org/wiki/Q2538404","display_name":"Complex projective space","level":4,"score":0.19215989112854004},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10537421703338623},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C117220453","wikidata":"https://www.wikidata.org/wiki/Q5172842","display_name":"Correlation","level":2,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2478/forma-2021-0016","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2021-0016","pdf_url":"https://www.sciendo.com/pdf/10.2478/forma-2021-0016","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/13652","is_oa":true,"landing_page_url":"http://hdl.handle.net/11320/13652","pdf_url":null,"source":{"id":"https://openalex.org/S4306400933","display_name":"The Repository of University in Bialystok (University in Bialystok)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210094198","host_organization_name":"University Clinical Hospital In Bialystok","host_organization_lineage":["https://openalex.org/I4210094198"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"}],"best_oa_location":{"id":"doi:10.2478/forma-2021-0016","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2021-0016","pdf_url":"https://www.sciendo.com/pdf/10.2478/forma-2021-0016","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4285325858.pdf","grobid_xml":"https://content.openalex.org/works/W4285325858.grobid-xml"},"referenced_works_count":9,"referenced_works":["https://openalex.org/W191630440","https://openalex.org/W1572941424","https://openalex.org/W2054653818","https://openalex.org/W2295728669","https://openalex.org/W2527478776","https://openalex.org/W2608452728","https://openalex.org/W2770430340","https://openalex.org/W2887099240","https://openalex.org/W2899395665"],"related_works":["https://openalex.org/W1970976233","https://openalex.org/W4254177151","https://openalex.org/W4313240746","https://openalex.org/W589112568","https://openalex.org/W2012035037","https://openalex.org/W4285325858","https://openalex.org/W2015367641","https://openalex.org/W4300516845","https://openalex.org/W2592432152","https://openalex.org/W2953272308"],"abstract_inverted_index":{"Summary":[0],"In":[1,121,170,202],"this":[2,199,217,258],"article,":[3],"we":[4,106,108,124,176,206,227],"check":[5],"with":[6],"the":[7,12,18,24,34,37,98,134,141,171,184,187,190,195,203,222,237,246,267,273,280,283,287],"Mizar":[8],"system":[9],"[1],":[10],"[2],":[11],"converse":[13,19],"of":[14,20,23,39,54,78,136,140,161,168,189,211,224,242,282,289],"Desargues\u2019":[15],"theorem":[16,22,105],"and":[17,41,138,194,214,240],"Pappus\u2019":[21],"real":[25,191],"projective":[26,35,55,142,192,200],"plane.":[27,201],"It":[28],"is":[29,147,152,219,276],"well":[30],"known":[31],"that":[32,150,179,216,257,272],"in":[33,60,95,164,233,245,270],"plane,":[36],"notions":[38],"points":[40,188],"lines":[42,167,196],"are":[43,57],"dual":[44,79,113,156,162,220,238],"[11],":[45],"[9],":[46],"[15],":[47],"[8].":[48],"Some":[49],"results":[50,230],"(analytical,":[51],"synthetic,":[52],"combinatorial)":[53],"geometry":[56,291],"already":[58,85],"present":[59],"some":[61],"libraries":[62],"Lean/Hott":[63],"[5],":[64],"Isabelle/Hol":[65],"[3],":[66],"Coq":[67],"[13],":[68],"[14],":[69],"[4],":[70],"Agda":[71],"[6],":[72],".":[73,74,75,76],"Proofs":[77],"statements":[80],"by":[81,154],"proof":[82,268],"assistants":[83],"have":[84],"been":[86],"proposed,":[87],"using":[88,114,133],"an":[89,126],"axiomatic":[90],"method":[91],"(for":[92],"example":[93],"see":[94],"[13]":[96],"-":[97],"section":[99],"on":[100],"duality:":[101],"\u201c[...]":[102],"For":[103],"every":[104],"prove,":[107],"can":[109],"easily":[110],"derive":[111],"its":[112],"our":[115,122],"function":[116],"swap":[117],"[...]":[118],"2":[119],"\u201d).":[120],"formalisation,":[123],"use":[125],"analytical":[127],"rather":[128,174],"than":[129],"a":[130,165,208,234,277],"synthetic":[131],"approach":[132],"definitions":[135,157,178],"Leo\u0144czuk":[137],"Pra\u017cmowski":[139],"plane":[143,193,275,285],"[12].":[144],"Our":[145],"motivation":[146],"to":[148,158,182,198,221,231,263],"show":[149],"it":[151],"possible":[153],"developing":[155],"find":[159],"proofs":[160],"theorems":[163,241],"few":[166,235],"code.":[169],"first":[172],"part,":[173,205],"technical,":[175],"introduce":[177],"allow":[180,261],"us":[181,262],"construct":[183],"duality":[185],"between":[186],"associated":[197],"second":[204],"give":[207],"natural":[209],"definition":[210,218,223],"line":[212],"concurrency":[213],"prove":[215],"alignment.":[225],"Finally,":[226],"apply":[228],"these":[229],"find,":[232],"lines,":[236],"properties":[239],"those":[243],"defined":[244],"article":[247],"[12]":[248],"(transitive,":[249],"Vebleian,":[250],"at_least_3rank,":[251],"Fanoian,":[252],"Desarguesian,":[253],"2-dimensional).":[254],"We":[255],"hope":[256],"methodology":[259],"will":[260],"continued":[264],"more":[265],"quickly":[266],"started":[269],"[7]":[271],"Beltrami-Klein":[274],"model":[278],"satisfying":[279],"axioms":[281],"hyperbolic":[284],"(in":[286],"sense":[288],"Tarski":[290],"[10]).":[292]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
