{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T08:09:03Z","timestamp":1770451743556,"version":"3.49.0"},"reference-count":24,"publisher":"SAGE Publications","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IFS"],"published-print":{"date-parts":[[2021,8,11]]},"abstract":"<jats:p>In this paper, we first point out some mistakes in [12]. Especially the Theorem 3.9 [12] showed that: Let A be residuated lattice and \u2205\u00a0\u2260\u00a0X\u00a0\u2286\u00a0A, then the least ideal containing X can be expressed as: \u2329X\u232a\u00a0=\u00a0{a\u00a0\u2208\u00a0A|a\u00a0\u2264\u00a0(\u00b7\u00b7\u00a0\u00b7\u00a0((x1\u00a0\u2295\u00a0x2)\u00a0\u2295\u00a0x3)\u00a0\u2295\u00a0\u00b7\u00b7\u00a0\u00b7)\u00a0\u2295\u00a0xn, xi\u00a0\u2208\u00a0X, i\u00a0=\u00a01, 2\u00a0\u00b7\u00b7\u00a0\u00b7\u00a0, n}. But we present an example to illustrate the ideal generation formula may not hold on residuated lattices. Further we give the correct ideal generation formula on residuated lattices. Moreover, we extend the concepts of annihilators and \u03b1-ideals to MTL-algebras and focus on studying the relations between them. Furthermore, we show that the set I\u03b1\u00a0(M) of all \u03b1-ideals on a linear MTL-algebra M only contains two trivial \u03b1-ideals {0} and M. However, the authors [24] studied the structure of I\u03b1\u00a0(M) in a linear BL-algebra M, which means some results with respect to I\u03b1\u00a0(M) given in [24] are trivial. Unlike that, we investigate the lattice structure of I\u03b1\u00a0(M) on general MTL-algebras.<\/jats:p>","DOI":"10.3233\/jifs-202417","type":"journal-article","created":{"date-parts":[[2021,6,30]],"date-time":"2021-06-30T05:09:09Z","timestamp":1625029749000},"page":"613-623","source":"Crossref","is-referenced-by-count":0,"title":["On ideals of residuated lattices"],"prefix":"10.1177","volume":"41","author":[{"given":"Yan Yan","family":"Dong","sequence":"first","affiliation":[{"name":"Beijing Key Laboratory on MCAACI, School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, P.R.China"}]},{"given":"Jun Tao","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Science, Xi\u2019an Shiyou University, Xi\u2019an, Shaanxi, China"}]}],"member":"179","reference":[{"key":"10.3233\/JIFS-202417_ref1","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1023\/A:1022935811257","article-title":"Some properties of residuated lattices","volume":"53","author":"Belohl\u00e1vek","year":"2003","journal-title":"Czechoslovak Math J"},{"key":"10.3233\/JIFS-202417_ref2","unstructured":"Bikhoff G. , Lattice theory, Amer Maths Soc, 1973."},{"key":"10.3233\/JIFS-202417_ref3","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1017\/S1446788700012775","article-title":"Annulets and \u03b1-ideals in a distributive lattice","volume":"15","author":"Cornish","year":"1973","journal-title":"J Austral Math Soc"},{"key":"10.3233\/JIFS-202417_ref4","doi-asserted-by":"crossref","first-page":"3207","DOI":"10.1007\/s00500-018-3195-9","article-title":"\u03b1-filters and prime \u03b1-filter spaces in residuated lattices","volume":"23","author":"Dong","year":"2019","journal-title":"Soft Computing"},{"key":"10.3233\/JIFS-202417_ref5","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":"10.3233\/JIFS-202417_ref6","doi-asserted-by":"crossref","unstructured":"Gr\u00e4tzer G. , Lattice theory, W. 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