{"id":"https://openalex.org/W4390622025","doi":"https://doi.org/10.2478/forma-2023-0027","title":"Extensions of Orderings","display_name":"Extensions of Orderings","publication_year":2023,"publication_date":"2023-09-01","ids":{"openalex":"https://openalex.org/W4390622025","doi":"https://doi.org/10.2478/forma-2023-0027"},"language":"en","primary_location":{"id":"doi:10.2478/forma-2023-0027","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2023-0027","pdf_url":"https://sciendo.com/pdf/10.2478/forma-2023-0027","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://sciendo.com/pdf/10.2478/forma-2023-0027","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5048465549","display_name":"Christoph Schwarzweller","orcid":"https://orcid.org/0000-0001-9587-8737"},"institutions":[{"id":"https://openalex.org/I40413290","display_name":"University of Gda\u0144sk","ror":"https://ror.org/011dv8m48","country_code":"PL","type":"education","lineage":["https://openalex.org/I40413290"]}],"countries":["PL"],"is_corresponding":true,"raw_author_name":"Christoph Schwarzweller","raw_affiliation_strings":["Institute of Informatics , University of Gda\u0144sk , Poland"],"raw_orcid":"https://orcid.org/0000-0001-9587-8737","affiliations":[{"raw_affiliation_string":"Institute of Informatics , University of Gda\u0144sk , Poland","institution_ids":["https://openalex.org/I40413290"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5048465549"],"corresponding_institution_ids":["https://openalex.org/I40413290"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.34807653,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"31","issue":"1","first_page":"341","last_page":"352"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11703","display_name":"Rings, Modules, and Algebras","score":0.9987000226974487,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T11703","display_name":"Rings, Modules, and Algebras","score":0.9987000226974487,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T11151","display_name":"Advanced Topology and Set Theory","score":0.9941999912261963,"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/T11727","display_name":"Advanced Algebra and Logic","score":0.9908999800682068,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/combinatorics","display_name":"Combinatorics","score":0.5495100617408752},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5475120544433594},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.42417111992836},{"id":"https://openalex.org/keywords/crystallography","display_name":"Crystallography","score":0.3254983425140381},{"id":"https://openalex.org/keywords/chemistry","display_name":"Chemistry","score":0.1040097177028656}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5495100617408752},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5475120544433594},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.42417111992836},{"id":"https://openalex.org/C8010536","wikidata":"https://www.wikidata.org/wiki/Q160398","display_name":"Crystallography","level":1,"score":0.3254983425140381},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.1040097177028656}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2478/forma-2023-0027","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2023-0027","pdf_url":"https://sciendo.com/pdf/10.2478/forma-2023-0027","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:11320/15867","is_oa":true,"landing_page_url":"http://hdl.handle.net/11320/15867","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-2023-0027","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2023-0027","pdf_url":"https://sciendo.com/pdf/10.2478/forma-2023-0027","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":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4390622025.pdf"},"referenced_works_count":10,"referenced_works":["https://openalex.org/W1553126454","https://openalex.org/W1625130818","https://openalex.org/W1924950049","https://openalex.org/W1968315351","https://openalex.org/W2184838578","https://openalex.org/W2526342011","https://openalex.org/W2614270863","https://openalex.org/W3008758757","https://openalex.org/W4285326349","https://openalex.org/W4388142980"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W2935759653","https://openalex.org/W3105167352","https://openalex.org/W54078636","https://openalex.org/W2954470139","https://openalex.org/W1501425562","https://openalex.org/W2902782467","https://openalex.org/W3084825885","https://openalex.org/W2298861036","https://openalex.org/W2271181815"],"abstract_inverted_index":{"Summary":[0],"In":[1],"this":[2],"article":[3],"we":[4,153],"extend":[5],"the":[6,78,201],"algebraic":[7],"theory":[8],"of":[9,19,27,34,46,137,195,203],"ordered":[10],"fields":[11],"[6],":[12],"[8]":[13],"in":[14,67],"Mizar.":[15],"We":[16,51],"introduce":[17],"extensions":[18],"orderings:":[20],"if":[21,40,74,77,142,145,184,187,200],"E":[22,38,47,65,73,138,150,199,204],"is":[23,134,207],"a":[24,135,157,181],"field":[25],"extension":[26],"F":[28,35,159,196,206],",":[29,39,66],"then":[30],"an":[31,43],"ordering":[32,44,193],"P":[33,49,61,70,161,188,194],"extends":[36,71,162,197],"to":[37,64,72,163,198],"there":[41],"exists":[42],"O":[45],"containing":[48],".":[50,151],"first":[52],"prove":[53],"some":[54],"necessary":[55],"and":[56,75,143,185,189],"su":[57],"cient":[58],"conditions":[59],"for":[60,155],"being":[62],"extendable":[63],"particular":[68],"that":[69,160,191],"only":[76,144,186],"set":[79],"<m:math":[80,164],"xmlns:m=\"http://www.w3.org/1998/Math/MathML\">":[81,165],"<m:mrow>":[82,92,94,96,100,166,168,170],"<m:mi>Q</m:mi>":[83],"<m:mi>S</m:mi>":[84],"<m:mi>":[85,87,110,112],"</m:mi>":[86,88,111,113],"<m:mi>E</m:mi>":[89,116],"<m:mo>:</m:mo>":[90],"<m:mo>=</m:mo>":[91],"<m:mo>{</m:mo>":[93],"<m:mo>\u2211</m:mo>":[95],"<m:mi>a</m:mi>":[97,106,172],"<m:mo>*</m:mo>":[98],"<m:msup>":[99],"<m:mi>b</m:mi>":[101,114],"</m:mrow>":[102,117,118,120,121,174,176,177],"<m:mn>2</m:mn>":[103],"</m:msup>":[104],"<m:mo>|</m:mo>":[105],"<m:mo>\u2208</m:mo>":[107,115],"<m:mi>P</m:mi>":[108],"<m:mo>,</m:mo>":[109],"<m:mo>}</m:mo>":[119],"</m:math>":[122,178],"QS\\,\\,E:":[123],"=":[124],"\\left\\{":[125],"{\\sum":[126],"{a*{b^2}|a":[127],"\\in":[128,130],"P,\\,\\,b":[129],"E}":[131],"}":[132,182],"\\right\\}":[133],"preordering":[136],"\u2013":[139],"or":[140],"equivalently":[141],"\u22121":[146],"/":[147],"\u2209":[148],"QS":[149],"Then":[152],"show":[154],"non-square":[156],"\u2208":[158],"<m:mi>F</m:mi>":[167],"<m:mo>(</m:mo>":[169],"<m:msqrt>":[171],"</m:msqrt>":[173],"<m:mo>)</m:mo>":[175],"F\\left(":[179],"{\\sqrt":[180],"\\right)":[183],"finally":[190],"every":[192],"degree":[202],"over":[205],"odd.":[208]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}