{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T08:13:58Z","timestamp":1777536838699,"version":"3.51.4"},"reference-count":18,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2017,10,13]],"date-time":"2017-10-13T00:00:00Z","timestamp":1507852800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We introduce a new class of operations called asymmetric equivalences. Several properties of asymmetric equivalence operations have been investigated. Based on the asymmetric equivalence, quasi-metric spaces are constructed on [0, 1]. Finally, we discuss symmetrization of asymmetric equivalences.<\/jats:p>","DOI":"10.3390\/sym9100224","type":"journal-article","created":{"date-parts":[[2017,10,13]],"date-time":"2017-10-13T11:34:09Z","timestamp":1507894449000},"page":"224","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Asymmetric Equivalences in Fuzzy Logic"],"prefix":"10.3390","volume":"9","author":[{"given":"Bo","family":"Hu","sequence":"first","affiliation":[{"name":"School of Mechanical and Electrical Engineering, Guizhou Normal University, Guiyang 550025, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lvqing","family":"Bi","sequence":"additional","affiliation":[{"name":"School of Electronics and Communication Engineering, Yulin Normal University, Yulin 537000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sizhao","family":"Li","sequence":"additional","affiliation":[{"name":"School of Information Science and Engineering, Xiamen University, Xiamen 361005, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2283-9459","authenticated-orcid":false,"given":"Songsong","family":"Dai","sequence":"additional","affiliation":[{"name":"School of Information Science and Engineering, Xiamen University, Xiamen 361005, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,10,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/S0165-0114(01)00098-7","article-title":"Monoidal t-norm based logic: Towards a logic for left-continuous t-norms","volume":"124","author":"Esteva","year":"2001","journal-title":"Fuzzy Sets Syst."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"H\u00e1jek, P. (1998). Metamathematics of Fuzzy Logic, Kluwer.","DOI":"10.1007\/978-94-011-5300-3"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1314","DOI":"10.1016\/j.fss.2007.01.009","article-title":"Similarity of fuzzy choice functions","volume":"158","author":"Georgescu","year":"2007","journal-title":"Fuzzy Sets Syst."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"5103","DOI":"10.1016\/j.ins.2007.05.014","article-title":"Robustness of fuzzy reasoning via logically equivalence measure","volume":"177","author":"Jin","year":"2007","journal-title":"Inf. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"653","DOI":"10.1016\/j.ijar.2012.11.007","article-title":"Robustness analysis of full implication inference method","volume":"54","author":"Dai","year":"2013","journal-title":"Int. J. Approx. Reason."},{"key":"ref_6","first-page":"623","article-title":"Two types of fuzzy metric spaces suitable for fuzzy reasoning","volume":"44","author":"Wang","year":"2014","journal-title":"Sci. China Inf. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.ijar.2015.04.002","article-title":"Robustness analysis of logic metrics on F(X)","volume":"61","author":"Duan","year":"2015","journal-title":"Int. J. Approx. Reason."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/j.fss.2010.11.011","article-title":"EQ-logic: Non-commutative fuzzy logic based on fuzzy equality","volume":"172","author":"Dyba","year":"2011","journal-title":"Fuzzy Sets Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/S0165-0114(01)00099-9","article-title":"The L\u03a0 and L\u03a0\n            \n              \n                \n                    1\n                     2\n                  \n              \n            \n           propositional and predicate logic","volume":"124","author":"Cintula","year":"2001","journal-title":"Fuzzy Sets Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/s001530050173","article-title":"The L\u03a0 and L\u03a0\n            \n              \n                \n                    1\n                     2\n                  \n              \n            \n           logic: Two complete fuzzy systems joining \u0141ukasiewicz and product logic","volume":"40","author":"Esteva","year":"2001","journal-title":"Arch. Math. Logic"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2337","DOI":"10.1023\/A:1026461417704","article-title":"Equations, state, and lattices of infinite-dimensional hilbert spaces","volume":"39","author":"Megill","year":"2000","journal-title":"Int. J. Theor. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2797","DOI":"10.1023\/B:IJTP.0000006006.18494.1c","article-title":"Equivalences, identities, symmetric differences, and congruences in orthomodular lattices","volume":"42","author":"Megill","year":"2003","journal-title":"Int. J. Theor. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1023\/A:1014857228326","article-title":"Deduction, ordering, and operations in quantum logic","volume":"32","author":"Megill","year":"2002","journal-title":"Found. Phys."},{"key":"ref_14","unstructured":"Baczy\u0144ski, M., and Jayaram, B. (2008). Fuzzy Implications, Springer."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Klement, E.P., Mesiar, R., and Pap, E. (2000). Triangular Norms, Kluwer.","DOI":"10.1007\/978-94-015-9540-7"},{"key":"ref_16","unstructured":"Fletcher, P., and Lindgren, W.F. (1982). Quasi-Uniform Spaces, Dekker."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1016\/j.fss.2004.12.002","article-title":"Simplification and independence of axioms of fuzzy logic systems IMTL and NM","volume":"152","author":"Pei","year":"2005","journal-title":"Fuzzy Sets Syst."},{"key":"ref_18","first-page":"18","article-title":"Simplification of axiom systems of fuzzy logics NM and L\u2217","volume":"2","author":"Wang","year":"2006","journal-title":"Fuzzy Syst. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/10\/224\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:47:12Z","timestamp":1760208432000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/10\/224"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,10,13]]},"references-count":18,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2017,10]]}},"alternative-id":["sym9100224"],"URL":"https:\/\/doi.org\/10.3390\/sym9100224","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,10,13]]}}}