{"id":"https://openalex.org/W2999992290","doi":"https://doi.org/10.3390/sym12010168","title":"On Hypergroups with a \u03b2-Class of Finite Height","display_name":"On Hypergroups with a \u03b2-Class of Finite Height","publication_year":2020,"publication_date":"2020-01-15","ids":{"openalex":"https://openalex.org/W2999992290","doi":"https://doi.org/10.3390/sym12010168","mag":"2999992290"},"language":"en","primary_location":{"id":"doi:10.3390/sym12010168","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12010168","pdf_url":"https://www.mdpi.com/2073-8994/12/1/168/pdf?version=1579083904","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/12/1/168/pdf?version=1579083904","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5008815032","display_name":"Mario De Salvo","orcid":"https://orcid.org/0000-0001-5271-1742"},"institutions":[{"id":"https://openalex.org/I112862951","display_name":"University of Messina","ror":"https://ror.org/05ctdxz19","country_code":"IT","type":"education","lineage":["https://openalex.org/I112862951"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Mario De Salvo","raw_affiliation_strings":["Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Universit\u00e0 di Messina, 98166 Messina, Italy"],"raw_orcid":"https://orcid.org/0000-0001-5271-1742","affiliations":[{"raw_affiliation_string":"Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Universit\u00e0 di Messina, 98166 Messina, Italy","institution_ids":["https://openalex.org/I112862951"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5049942591","display_name":"Dario Fasino","orcid":"https://orcid.org/0000-0001-7682-0660"},"institutions":[{"id":"https://openalex.org/I129043915","display_name":"University of Udine","ror":"https://ror.org/05ht0mh31","country_code":"IT","type":"education","lineage":["https://openalex.org/I129043915"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Dario Fasino","raw_affiliation_strings":["Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Universit\u00e0 di Udine, 33100 Udine, Italy"],"raw_orcid":"https://orcid.org/0000-0001-7682-0660","affiliations":[{"raw_affiliation_string":"Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Universit\u00e0 di Udine, 33100 Udine, Italy","institution_ids":["https://openalex.org/I129043915"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5010201135","display_name":"Domenico Freni","orcid":"https://orcid.org/0000-0003-2895-7632"},"institutions":[{"id":"https://openalex.org/I129043915","display_name":"University of Udine","ror":"https://ror.org/05ht0mh31","country_code":"IT","type":"education","lineage":["https://openalex.org/I129043915"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Domenico Freni","raw_affiliation_strings":["Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Universit\u00e0 di Udine, 33100 Udine, Italy"],"raw_orcid":"https://orcid.org/0000-0003-2895-7632","affiliations":[{"raw_affiliation_string":"Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Universit\u00e0 di Udine, 33100 Udine, Italy","institution_ids":["https://openalex.org/I129043915"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5024988145","display_name":"Giovanni Lo Faro","orcid":"https://orcid.org/0000-0001-6174-8627"},"institutions":[{"id":"https://openalex.org/I112862951","display_name":"University of Messina","ror":"https://ror.org/05ctdxz19","country_code":"IT","type":"education","lineage":["https://openalex.org/I112862951"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Giovanni Lo Faro","raw_affiliation_strings":["Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Universit\u00e0 di Messina, 98166 Messina, Italy"],"raw_orcid":"https://orcid.org/0000-0001-6174-8627","affiliations":[{"raw_affiliation_string":"Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Universit\u00e0 di Messina, 98166 Messina, Italy","institution_ids":["https://openalex.org/I112862951"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5049942591"],"corresponding_institution_ids":["https://openalex.org/I129043915"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.3012,"has_fulltext":false,"cited_by_count":8,"citation_normalized_percentile":{"value":0.81677427,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"12","issue":"1","first_page":"168","last_page":"168"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11671","display_name":"Fuzzy and Soft Set Theory","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T11671","display_name":"Fuzzy and Soft Set Theory","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/1803","display_name":"Management Science and Operations Research"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11673","display_name":"Advanced Topics in Algebra","score":0.9900000095367432,"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/T11703","display_name":"Rings, Modules, and Algebras","score":0.9853000044822693,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.922860860824585},{"id":"https://openalex.org/keywords/modulo","display_name":"Modulo","score":0.7619756460189819},{"id":"https://openalex.org/keywords/cardinality","display_name":"Cardinality (data modeling)","score":0.757705569267273},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.7122116684913635},{"id":"https://openalex.org/keywords/characterization","display_name":"Characterization (materials science)","score":0.5695461630821228},{"id":"https://openalex.org/keywords/equivalence-relation","display_name":"Equivalence relation","score":0.5646499395370483},{"id":"https://openalex.org/keywords/equivalence-class","display_name":"Equivalence class (music)","score":0.5546424388885498},{"id":"https://openalex.org/keywords/equivalence","display_name":"Equivalence (formal languages)","score":0.5287415981292725},{"id":"https://openalex.org/keywords/property","display_name":"Property (philosophy)","score":0.502926766872406},{"id":"https://openalex.org/keywords/hierarchy","display_name":"Hierarchy","score":0.49280551075935364},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4208085536956787},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4004509747028351},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3686569333076477}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.922860860824585},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.7619756460189819},{"id":"https://openalex.org/C87117476","wikidata":"https://www.wikidata.org/wiki/Q362383","display_name":"Cardinality (data modeling)","level":2,"score":0.757705569267273},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.7122116684913635},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.5695461630821228},{"id":"https://openalex.org/C156103551","wikidata":"https://www.wikidata.org/wiki/Q130998","display_name":"Equivalence relation","level":2,"score":0.5646499395370483},{"id":"https://openalex.org/C2777044963","wikidata":"https://www.wikidata.org/wiki/Q5384707","display_name":"Equivalence class (music)","level":2,"score":0.5546424388885498},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.5287415981292725},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.502926766872406},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.49280551075935364},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4208085536956787},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4004509747028351},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3686569333076477},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C171250308","wikidata":"https://www.wikidata.org/wiki/Q11468","display_name":"Nanotechnology","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C34447519","wikidata":"https://www.wikidata.org/wiki/Q179522","display_name":"Market economy","level":1,"score":0.0},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym12010168","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12010168","pdf_url":"https://www.mdpi.com/2073-8994/12/1/168/pdf?version=1579083904","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:air.uniud.it:11390/1181908","is_oa":true,"landing_page_url":"http://hdl.handle.net/11390/1181908","pdf_url":null,"source":{"id":"https://openalex.org/S4306401163","display_name":"Institutional Research Information System (University of Udine)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I129043915","host_organization_name":"University of Udine","host_organization_lineage":["https://openalex.org/I129043915"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:doaj.org/article:c4e405e053db4f158ac2c7d33e2e6434","is_oa":true,"landing_page_url":"https://doaj.org/article/c4e405e053db4f158ac2c7d33e2e6434","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"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":"Symmetry, Vol 12, Iss 1, p 168 (2020)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym12010168","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym12010168","pdf_url":"https://www.mdpi.com/2073-8994/12/1/168/pdf?version=1579083904","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":23,"referenced_works":["https://openalex.org/W1709181003","https://openalex.org/W1996118974","https://openalex.org/W1996316893","https://openalex.org/W2003144175","https://openalex.org/W2016569951","https://openalex.org/W2033762726","https://openalex.org/W2037448807","https://openalex.org/W2040968849","https://openalex.org/W2047092778","https://openalex.org/W2052257421","https://openalex.org/W2080691036","https://openalex.org/W2087893536","https://openalex.org/W2144385241","https://openalex.org/W2394992476","https://openalex.org/W2573139712","https://openalex.org/W2773720373","https://openalex.org/W2919300592","https://openalex.org/W2941863369","https://openalex.org/W2947939890","https://openalex.org/W2965003316","https://openalex.org/W2974865961","https://openalex.org/W3023953025","https://openalex.org/W4214940931"],"related_works":["https://openalex.org/W2887267742","https://openalex.org/W4289681629","https://openalex.org/W4300027602","https://openalex.org/W2895616265","https://openalex.org/W1532854663","https://openalex.org/W2499765358","https://openalex.org/W2026098892","https://openalex.org/W2382232876","https://openalex.org/W2148887744","https://openalex.org/W2362545535"],"abstract_inverted_index":{"In":[0],"every":[1],"hypergroup,":[2],"the":[3,7,12,26,41,85,91,94],"equivalence":[4],"classes":[5,67],"modulo":[6],"fundamental":[8],"relation":[9],"\u03b2":[10,32,45,74],"are":[11],"union":[13],"of":[14,16,21,28,30,38,43,59,66,68,93],"hyperproducts":[15],"element":[17],"pairs.":[18],"Making":[19],"use":[20],"this":[22],"property,":[23],"we":[24,35,54,62,83],"introduce":[25],"notion":[27],"height":[29,42,77],"a":[31,44,52,56,64],"-class":[33,46,75],"and":[34,82],"analyze":[36],"properties":[37],"hypergroups":[39,69,95],"where":[40,70],"coincides":[47],"with":[48],"its":[49],"cardinality.":[50],"As":[51],"consequence,":[53],"obtain":[55],"new":[57],"characterization":[58],"1-hypergroups.":[60],"Moreover,":[61],"define":[63],"hierarchy":[65],"at":[71],"least":[72],"one":[73],"has":[76],"1":[78],"or":[79],"cardinality":[80],"1,":[81],"enumerate":[84],"elements":[86],"in":[87],"each":[88],"class":[89],"when":[90],"size":[92],"is":[96],"n":[97],"\u2264":[98],"4":[99],",":[100],"apart":[101],"from":[102],"isomorphisms.":[103]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2020-01-23T00:00:00"}