{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,19]],"date-time":"2024-09-19T16:20:18Z","timestamp":1726762818172},"reference-count":30,"publisher":"Oxford University Press (OUP)","issue":"7","license":[{"start":{"date-parts":[[2022,6,8]],"date-time":"2022-06-08T00:00:00Z","timestamp":1654646400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,10,20]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The main aim of this paper is to investigate the topologies that constructed by some ideals on residuated lattices and some topologies which induced by lattice ideals and distance functions on involutive residuated lattices. To begin with, we present that prime $\\oplus $-ideals and prime $\\boxplus $-ideals are coincident on $MTL$-algebras and give some new results about ideals on residuated lattices. In the following, we study an $i$-topology which is induced by an $i$-system on a residuated lattice $A$ and get that $A$ equipped with such an $i$-topology is a topological residuated lattice and give a characterization for such topological involutive residuated lattices. Meanwhile, we give a notion of $\\mathcal {I}$-completion of a residuated lattice $A$ with respect to the $i$-topology induced by an $i$-system $\\mathcal {I}$ and characterize the $\\mathcal {I}$-completion of $A$ by means of the inverse limit of an inverse system. Finally, we show that the topology that induced by a lattice ideal and a distance function on an involutive residuated lattice is a semitopological residuated lattice and which coincides with some $i$-topological residuated lattice when the lattice ideal is an ideal of $A$.<\/jats:p>","DOI":"10.1093\/logcom\/exac037","type":"journal-article","created":{"date-parts":[[2022,4,29]],"date-time":"2022-04-29T11:23:11Z","timestamp":1651231391000},"page":"1375-1400","source":"Crossref","is-referenced-by-count":2,"title":["Topologies on residuated lattices"],"prefix":"10.1093","volume":"32","author":[{"given":"Wei","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics , Shaanxi Normal University, Xi\u2019an 710119, China"}]},{"given":"Bin","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics , Shaanxi Normal University, Xi\u2019an 710119, China"}]}],"member":"286","published-online":{"date-parts":[[2022,6,8]]},"reference":[{"key":"2022102505283677300_ref1","doi-asserted-by":"crossref","DOI":"10.2991\/978-94-91216-35-0","volume-title":"Topological Groups and Related Structures","author":"Arhangelskii","year":"2008"},{"key":"2022102505283677300_ref2","volume-title":"Introduction to the Theory of Topological Rings and Modules","author":"Arnautov","year":"1996"},{"key":"2022102505283677300_ref3","first-page":"59","article-title":"On (semi)topological BL-algebras","volume":"6","author":"Borzooei","year":"2011","journal-title":"Iranian Journal of Mathematical Sciences and Informatics"},{"key":"2022102505283677300_ref4","doi-asserted-by":"crossref","first-page":"53","DOI":"10.37193\/CJM.2021.01.06","article-title":"Ideals in residuated lattices","volume":"37","author":"Busneag","year":"2021","journal-title":"Carpathian Journal of Mathematics"},{"key":"2022102505283677300_ref5","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/S0165-0114(01)00098-7","article-title":"Monoidal t-norm based logictowards a logic for left-continuous t-norms","volume":"124","author":"Esteva","year":"2001","journal-title":"Fuzzy Sets and Systems"},{"key":"2022102505283677300_ref6","volume-title":"Residuated Lattices: An Algebraic Glimpse at Structural Logic","author":"Galatos","year":"2007"},{"key":"2022102505283677300_ref7","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1016\/j.fss.2014.11.007","article-title":"Algebraic and topological results on lifting properties in residuated lattices","volume":"271","author":"Georgescu","year":"2015","journal-title":"Fuzzy Sets and Systems"},{"key":"2022102505283677300_ref8","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-77487-9","volume-title":"Universal Algebra","author":"Gratzer","year":"2008"},{"key":"2022102505283677300_ref9","first-page":"443","article-title":"On ideals in De Morgen residuated lattices","volume":"54","author":"Holdon","year":"2018","journal-title":"Kybernetika"},{"key":"2022102505283677300_ref10","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/0165-0114(94)00351-7","article-title":"The socle and fuzzy socle of a BCI-algebra","volume":"72","author":"Hoo","year":"1995","journal-title":"Fuzzy Sets and Systems"},{"key":"2022102505283677300_ref11","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/S0166-8641(97)00027-8","article-title":"Topological MV-algebras","volume":"81","author":"Hoo","year":"1997","journal-title":"Topology and its Applications"},{"key":"2022102505283677300_ref12","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.fss.2020.03.011","article-title":"Solutions to two open problems in topological residuated lattices","volume":"405","author":"He","year":"2021","journal-title":"Fuzzy Sets and Systems"},{"key":"2022102505283677300_ref13","volume-title":"Stone Spaces","author":"Johnstone","year":"1982"},{"key":"2022102505283677300_ref14","first-page":"15","article-title":"On (semi)topological residuated lattices","volume":"41","author":"Kouhestani","year":"2014","journal-title":"Annals of the University of Craiova, Mathematics and Computer Science Series"},{"key":"2022102505283677300_ref15","first-page":"183","article-title":"Topological residuated lattices","volume":"9","author":"Kouhestani","year":"2017","journal-title":"International Journal of Industrial Mathematics"},{"key":"2022102505283677300_ref16","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1007\/s13042-014-0317-2","article-title":"Ideals and fuzzy ideals on residuated lattices","volume":"8","author":"Liu","year":"2017","journal-title":"International Journal of Machine Learning and Cybernetics"},{"key":"2022102505283677300_ref17","doi-asserted-by":"crossref","first-page":"2531","DOI":"10.1007\/s00500-017-2574-y","article-title":"Filter topologies on MV-algebras","volume":"21","author":"Luan","year":"2017","journal-title":"Soft Computing"},{"key":"2022102505283677300_ref18","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.fss.2016.04.012","article-title":"On (para, quasi) topological MV-algebras","volume":"313","author":"Najafi","year":"2017","journal-title":"Fuzzy Sets and Systems"},{"key":"2022102505283677300_ref19","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/978-94-017-3598-8_8","article-title":"Substructural logics and residuated lattices\u2014an introuduction","author":"Ono","year":"2003","journal-title":"50 Years of Studia Logica"},{"key":"2022102505283677300_ref20","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/j.fss.2020.04.009","article-title":"Prime, minimal prime and maximal ideals spaces in residuated lattices","volume":"405","author":"Piciu","year":"2021","journal-title":"Fuzzy Sets and Systems"},{"key":"2022102505283677300_ref21","doi-asserted-by":"crossref","first-page":"3179","DOI":"10.1007\/s00500-020-04709-x","article-title":"Topological residuated lattices","volume":"24","author":"Rasouli","year":"2020","journal-title":"Soft Computing"},{"key":"2022102505283677300_ref22","doi-asserted-by":"crossref","first-page":"10519","DOI":"10.1007\/s00500-021-05985-x","article-title":"The hull-kernel topology on prime filters in residuated lattices","volume":"25","author":"Rasouli","year":"2021","journal-title":"Soft Computing"},{"key":"2022102505283677300_ref23","first-page":"1","article-title":"A topology on BCK-algebras via left and right stabilizers","volume":"4","author":"Roudabri","year":"2009","journal-title":"Iranian Journal of Mathematical Sciences and Informatics"},{"key":"2022102505283677300_ref24","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1007\/s00500-007-0182-y","article-title":"States on semi-divisible residuated lattices","volume":"12","author":"Turunen","year":"2008","journal-title":"Soft Computing"},{"key":"2022102505283677300_ref25","volume-title":"Non-Classical Mathematical Logic and Approximate Reasoning","author":"Wang","year":"2000"},{"key":"2022102505283677300_ref26","doi-asserted-by":"crossref","first-page":"812","DOI":"10.1093\/jigpal\/jzz005","article-title":"Monadic NM-algebras","volume":"27","author":"Wang","year":"2019","journal-title":"Logic Journal of the IGPL"},{"key":"2022102505283677300_ref27","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1090\/S0002-9947-1939-1501995-3","article-title":"Residuated lattices","volume":"45","author":"Ward","year":"1939","journal-title":"Transactions of the American Mathematical Society"},{"key":"2022102505283677300_ref28","first-page":"145","article-title":"On topological EQ-algebras","volume":"15","author":"Yang","year":"2018","journal-title":"Iranian Journal of Fuzzy Systems"},{"key":"2022102505283677300_ref29","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1016\/j.fss.2019.03.014","article-title":"Some weaker versions of topological residuated lattices","volume":"373","author":"Yang","year":"2019","journal-title":"Fuzzy Sets and Systems"},{"key":"2022102505283677300_ref30","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.fss.2014.11.014","article-title":"Topology on BL-algebras","volume":"289","author":"Zahiri","year":"2016","journal-title":"Fuzzy Sets and Systems"}],"container-title":["Journal of Logic and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/logcom\/article-pdf\/32\/7\/1375\/46616052\/exac037.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/logcom\/article-pdf\/32\/7\/1375\/46616052\/exac037.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,25]],"date-time":"2022-10-25T05:55:00Z","timestamp":1666677300000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/logcom\/article\/32\/7\/1375\/6603982"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,8]]},"references-count":30,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,6,8]]},"published-print":{"date-parts":[[2022,10,20]]}},"URL":"https:\/\/doi.org\/10.1093\/logcom\/exac037","relation":{},"ISSN":["0955-792X","1465-363X"],"issn-type":[{"value":"0955-792X","type":"print"},{"value":"1465-363X","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2022,10]]},"published":{"date-parts":[[2022,6,8]]}}}