{"id":"https://openalex.org/W3000475255","doi":"https://doi.org/10.1109/tfuzz.2020.2966154","title":"Two General Construction Ways Toward Unified Framework of Ordinal Sums of Fuzzy Implications","display_name":"Two General Construction Ways Toward Unified Framework of Ordinal Sums of Fuzzy Implications","publication_year":2020,"publication_date":"2020-01-13","ids":{"openalex":"https://openalex.org/W3000475255","doi":"https://doi.org/10.1109/tfuzz.2020.2966154","mag":"3000475255"},"language":"en","primary_location":{"id":"doi:10.1109/tfuzz.2020.2966154","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tfuzz.2020.2966154","pdf_url":null,"source":{"id":"https://openalex.org/S134177497","display_name":"IEEE Transactions on Fuzzy Systems","issn_l":"1063-6706","issn":["1063-6706","1941-0034"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Fuzzy Systems","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5034528379","display_name":"Hongjun Zhou","orcid":"https://orcid.org/0000-0001-8364-7739"},"institutions":[{"id":"https://openalex.org/I88830068","display_name":"Shaanxi Normal University","ror":"https://ror.org/0170z8493","country_code":"CN","type":"education","lineage":["https://openalex.org/I88830068"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Hongjun Zhou","raw_affiliation_strings":["College of Mathematics and Information Science, Shaanxi Normal University, Xi\u2019an, China","College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, China"],"raw_orcid":"https://orcid.org/0000-0001-8364-7739","affiliations":[{"raw_affiliation_string":"College of Mathematics and Information Science, Shaanxi Normal University, Xi\u2019an, China","institution_ids":["https://openalex.org/I88830068"]},{"raw_affiliation_string":"College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, China","institution_ids":["https://openalex.org/I88830068"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5034528379"],"corresponding_institution_ids":["https://openalex.org/I88830068"],"apc_list":null,"apc_paid":null,"fwci":4.1446,"has_fulltext":false,"cited_by_count":39,"citation_normalized_percentile":{"value":0.94195188,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"29","issue":"4","first_page":"846","last_page":"860"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10050","display_name":"Multi-Criteria Decision Making","score":0.9993000030517578,"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/T10050","display_name":"Multi-Criteria Decision Making","score":0.9993000030517578,"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/T11063","display_name":"Rough Sets and Fuzzy Logic","score":0.9941999912261963,"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"}},{"id":"https://openalex.org/T10820","display_name":"Fuzzy Logic and Control Systems","score":0.9894000291824341,"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/mathematics","display_name":"Mathematics","score":0.7453352212905884},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.66140216588974},{"id":"https://openalex.org/keywords/fuzzy-logic","display_name":"Fuzzy logic","score":0.6384298801422119},{"id":"https://openalex.org/keywords/diagonal","display_name":"Diagonal","score":0.6383662223815918},{"id":"https://openalex.org/keywords/ordinal-regression","display_name":"Ordinal regression","score":0.5456181764602661},{"id":"https://openalex.org/keywords/fuzzy-number","display_name":"Fuzzy number","score":0.5269854068756104},{"id":"https://openalex.org/keywords/fuzzy-set","display_name":"Fuzzy set","score":0.5126739740371704},{"id":"https://openalex.org/keywords/unit-square","display_name":"Unit square","score":0.49315279722213745},{"id":"https://openalex.org/keywords/square","display_name":"Square (algebra)","score":0.49199435114860535},{"id":"https://openalex.org/keywords/complement","display_name":"Complement (music)","score":0.4584396183490753},{"id":"https://openalex.org/keywords/defuzzification","display_name":"Defuzzification","score":0.4363619089126587},{"id":"https://openalex.org/keywords/fuzzy-set-operations","display_name":"Fuzzy set operations","score":0.41777920722961426},{"id":"https://openalex.org/keywords/type-2-fuzzy-sets-and-systems","display_name":"Type-2 fuzzy sets and systems","score":0.4117889404296875},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.39000338315963745},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.21401458978652954},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.2134096920490265},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.1774628758430481},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.12459394335746765},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.05307254195213318}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7453352212905884},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.66140216588974},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.6384298801422119},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.6383662223815918},{"id":"https://openalex.org/C110313322","wikidata":"https://www.wikidata.org/wiki/Q7100793","display_name":"Ordinal regression","level":2,"score":0.5456181764602661},{"id":"https://openalex.org/C1883856","wikidata":"https://www.wikidata.org/wiki/Q3407463","display_name":"Fuzzy number","level":4,"score":0.5269854068756104},{"id":"https://openalex.org/C42011625","wikidata":"https://www.wikidata.org/wiki/Q1055058","display_name":"Fuzzy set","level":3,"score":0.5126739740371704},{"id":"https://openalex.org/C7356347","wikidata":"https://www.wikidata.org/wiki/Q1307838","display_name":"Unit square","level":2,"score":0.49315279722213745},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.49199435114860535},{"id":"https://openalex.org/C112313634","wikidata":"https://www.wikidata.org/wiki/Q7886648","display_name":"Complement (music)","level":5,"score":0.4584396183490753},{"id":"https://openalex.org/C170260401","wikidata":"https://www.wikidata.org/wiki/Q5251735","display_name":"Defuzzification","level":5,"score":0.4363619089126587},{"id":"https://openalex.org/C148671577","wikidata":"https://www.wikidata.org/wiki/Q5511133","display_name":"Fuzzy set operations","level":4,"score":0.41777920722961426},{"id":"https://openalex.org/C17350324","wikidata":"https://www.wikidata.org/wiki/Q7860605","display_name":"Type-2 fuzzy sets and systems","level":5,"score":0.4117889404296875},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.39000338315963745},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.21401458978652954},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.2134096920490265},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.1774628758430481},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.12459394335746765},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.05307254195213318},{"id":"https://openalex.org/C188082640","wikidata":"https://www.wikidata.org/wiki/Q1780899","display_name":"Complementation","level":4,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C127716648","wikidata":"https://www.wikidata.org/wiki/Q104053","display_name":"Phenotype","level":3,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tfuzz.2020.2966154","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tfuzz.2020.2966154","pdf_url":null,"source":{"id":"https://openalex.org/S134177497","display_name":"IEEE Transactions on Fuzzy Systems","issn_l":"1063-6706","issn":["1063-6706","1941-0034"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Fuzzy Systems","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.5699999928474426,"id":"https://metadata.un.org/sdg/11","display_name":"Sustainable cities and communities"}],"awards":[{"id":"https://openalex.org/G3122535420","display_name":null,"funder_award_id":"61473336","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":68,"referenced_works":["https://openalex.org/W189069394","https://openalex.org/W590777244","https://openalex.org/W608460879","https://openalex.org/W656973277","https://openalex.org/W657218221","https://openalex.org/W1503075583","https://openalex.org/W1535673504","https://openalex.org/W1543820402","https://openalex.org/W1553483187","https://openalex.org/W1788095458","https://openalex.org/W1987898498","https://openalex.org/W1987918021","https://openalex.org/W1997358452","https://openalex.org/W2011601021","https://openalex.org/W2025021415","https://openalex.org/W2037443063","https://openalex.org/W2038675095","https://openalex.org/W2050233652","https://openalex.org/W2052215587","https://openalex.org/W2053519120","https://openalex.org/W2066218705","https://openalex.org/W2073212647","https://openalex.org/W2084440709","https://openalex.org/W2099647936","https://openalex.org/W2102476898","https://openalex.org/W2120459411","https://openalex.org/W2136946629","https://openalex.org/W2154274300","https://openalex.org/W2163737009","https://openalex.org/W2221355491","https://openalex.org/W2221981724","https://openalex.org/W2227574681","https://openalex.org/W2288230525","https://openalex.org/W2294278623","https://openalex.org/W2312494561","https://openalex.org/W2324744822","https://openalex.org/W2343324289","https://openalex.org/W2344918565","https://openalex.org/W2385088083","https://openalex.org/W2476082904","https://openalex.org/W2481995907","https://openalex.org/W2491118651","https://openalex.org/W2506954980","https://openalex.org/W2566789387","https://openalex.org/W2580548604","https://openalex.org/W2615540658","https://openalex.org/W2736479549","https://openalex.org/W2749788382","https://openalex.org/W2750831055","https://openalex.org/W2753199002","https://openalex.org/W2758502165","https://openalex.org/W2760513823","https://openalex.org/W2763522707","https://openalex.org/W2770421795","https://openalex.org/W2794986269","https://openalex.org/W2798648560","https://openalex.org/W2885239430","https://openalex.org/W2899122236","https://openalex.org/W2905043480","https://openalex.org/W2910958968","https://openalex.org/W2942177650","https://openalex.org/W3148090786","https://openalex.org/W3213572188","https://openalex.org/W4234672050","https://openalex.org/W4235727405","https://openalex.org/W4249816584","https://openalex.org/W4250965767","https://openalex.org/W6689644826"],"related_works":["https://openalex.org/W4361984403","https://openalex.org/W2184441312","https://openalex.org/W2960454922","https://openalex.org/W1975005191","https://openalex.org/W1485268022","https://openalex.org/W2674573885","https://openalex.org/W1487720066","https://openalex.org/W2078650680","https://openalex.org/W2331106272","https://openalex.org/W1542808721"],"abstract_inverted_index":{"The":[0,30,92],"present":[1],"article":[2],"proposes":[3],"two":[4],"construction":[5,34],"ways":[6],"to":[7,42,45,51,101,104,154],"study":[8,43,102],"the":[9,19,23,28,52,70,74,80,85,120,124,134,150,175,186,189,192,203,206,222,228],"general":[10],"forms":[11],"of":[12,15,21,55,69,73,119,127,164,167,196,209,227],"ordinal":[13,24,32,82,165,177,194],"sums":[14,25,166,178,195],"fuzzy":[16,49,57,90,108,130,140,156,168,197,211],"implications":[17,58,131,157,169,198,212],"with":[18,123],"intent":[20],"unifying":[22],"existing":[26,176],"in":[27,77,201],"literature.":[29],"first":[31],"sum":[33,83],"way,":[35,94,191],"which":[36,95,172,202],"we":[37,96],"call":[38,97],"\u201cImplication":[39,98],"Complementing,\u201d":[40],"is":[41,88,100,137],"how":[44,103],"complement":[46],"a":[47,89,138],"specific":[48],"implication":[50,109],"linear":[53,125,207],"transformations":[54,126,208],"given":[56,116,129,159,210],"defined":[59,214],"on":[60,84,115],"respective":[61,128],"disjoint":[62],"subsquares":[63,218],"whose":[64],"principal":[65,71,223],"diagonals":[66],"are":[67,158,170,199,213,215],"segments":[68],"diagonal":[72,226],"unit":[75,86,121,229],"square,":[76],"order":[78],"that":[79,133],"resulting":[81],"square":[87,122],"implication.":[91,141],"second":[93,190],"Reconstructing,\u201d":[99],"reconstruct":[105],"an":[106],"initial":[107],"through":[110],"replacing":[111],"its":[112],"some":[113],"values":[114],"rectangular":[117],"regions":[118,204],"such":[132],"redefined":[135],"function":[136],"new":[139,162],"In":[142,182],"both":[143],"ways,":[144],"necessary":[145],"and":[146,160],"sufficient":[147],"conditions":[148],"for":[149],"final":[151],"reconstructed":[152],"functions":[153],"be":[155],"several":[161,180],"constructions":[163],"obtained,":[171],"would":[173],"generalize":[174],"from":[179],"aspects.":[181],"particular,":[183],"by":[184],"adopting":[185],"idea":[187],"behind":[188],"generalized":[193],"proposed,":[200],"where":[205],"neither":[216],"necessarily":[217,220],"nor":[219],"along":[221],"or":[224],"minor":[225],"square.":[230]},"counts_by_year":[{"year":2026,"cited_by_count":5},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":7},{"year":2023,"cited_by_count":7},{"year":2022,"cited_by_count":8},{"year":2021,"cited_by_count":6},{"year":2020,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
