{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,9]],"date-time":"2025-05-09T15:31:14Z","timestamp":1746804674061,"version":"3.40.5"},"reference-count":36,"publisher":"Wiley","license":[{"start":{"date-parts":[[2018,6,10]],"date-time":"2018-06-10T00:00:00Z","timestamp":1528588800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Advances in Fuzzy Systems"],"published-print":{"date-parts":[[2018,6,10]]},"abstract":"<jats:p>A new trend of using fuzzy algebraic structures in various applied sciences is becoming a central focus due to the accuracy and nondecoding nature. The aim of the present paper is to develop a new type of fuzzy subsystem of an ordered semigroup <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mi>S<\/mml:mi><mml:mo>.<\/mml:mo><\/mml:math> This new type of fuzzy subsystem will overcome the difficulties faced in fuzzy ideal theory of an ordered semigroup up to some extent. More precisely, we introduce <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mo>\u2208<\/mml:mo><mml:mo>,<\/mml:mo><mml:mo>\u2208<\/mml:mo><mml:mo>\u2228<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u03b4<\/mml:mi><\/mml:mrow><\/mml:msubsup><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:math>-fuzzy left (resp., right, quasi-) ideals of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mi>S<\/mml:mi><mml:mo>.<\/mml:mo><\/mml:math> These concepts are elaborated through appropriate examples. Further, we are bridging ordinary ideals and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mo>\u2208<\/mml:mo><mml:mo>,<\/mml:mo><mml:mo>\u2208<\/mml:mo><mml:mo>\u2228<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u03b4<\/mml:mi><\/mml:mrow><\/mml:msubsup><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:math>-fuzzy ideals of an ordered semigroup <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mi>S<\/mml:mi><\/mml:mrow><\/mml:math> through level subset and characteristic function<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mrow><mml:mo>.<\/mml:mo><\/mml:mrow><\/mml:math> Finally, we characterize regular ordered semigroups in terms of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mo>\u2208<\/mml:mo><mml:mo>,<\/mml:mo><mml:mo>\u2208<\/mml:mo><mml:mo>\u2228<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u03b4<\/mml:mi><\/mml:mrow><\/mml:msubsup><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:math>-fuzzy left (resp., right, quasi-) ideals.<\/jats:p>","DOI":"10.1155\/2018\/5304514","type":"journal-article","created":{"date-parts":[[2018,6,10]],"date-time":"2018-06-10T19:31:59Z","timestamp":1528659119000},"page":"1-10","source":"Crossref","is-referenced-by-count":2,"title":["Ordered Semigroups Based on <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mo>\u2208<\/mml:mo><mml:mo>,<\/mml:mo><mml:mo>\u2208<\/mml:mo><mml:mo>\u2228<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>k<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>\u03b4<\/mml:mi><\/mml:mrow><\/mml:msubsup><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:math>-Fuzzy Ideals"],"prefix":"10.1155","volume":"2018","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5555-6976","authenticated-orcid":true,"given":"Faiz Muhammad","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, School of Natural and Applied Sciences, Northwestern Polytechnical University, Xi\u2019an, Shaanxi, China"},{"name":"Department of Mathematics and Statistics, University of Swat, KP, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7881-5806","authenticated-orcid":true,"given":"Nie","family":"Yufeng","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, School of Natural and Applied Sciences, Northwestern Polytechnical University, Xi\u2019an, Shaanxi, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4218-8438","authenticated-orcid":true,"given":"Hidayat Ullah","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Malakand, Chakdara, KP, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bakht Muhammad","family":"Khan","sequence":"additional","affiliation":[{"name":"School of Electronics and Information, Northwestern Polytechnical University, Xi\u2019an, Shaanxi, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","first-page":"338","volume":"8","year":"1965","journal-title":"Information and Computation"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(71)90199-5"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-37125-0"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1007\/s002330010079"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2006.02.004"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1007\/BF02574349"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1142\/S1793005708001069"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1016\/0165-0114(95)00157-3"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1016\/0165-0114(95)00202-2"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1002\/malq.200810022"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-007-0165-z"},{"issue":"2","key":"12","first-page":"123","volume":"11","year":"2009","journal-title":"International Journal of Fuzzy Systems"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-007-0255-y"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-005-0472-1"},{"issue":"2","key":"15","first-page":"35","volume":"5","year":"2008","journal-title":"Iranian Journal of Fuzzy Systems"},{"issue":"3","key":"16","first-page":"391","volume":"32","year":"2009","journal-title":"Bulletin of the Malaysian Mathematical Sciences Society. 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