{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:21:57Z","timestamp":1760145717440,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,21]],"date-time":"2024-08-21T00:00:00Z","timestamp":1724198400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In the study of algebraic structures related to logical systems, ideals and filters have different meanings and are algebraic notions related to logical provable formulas. Unlike the classical Boolean lattice theory, ideals and filters are not dual notions in residuated lattices. An interesting subclass of residuated lattices is the class of triangle algebras, which is an equational representation of interval-valued residuated lattices that provides an algebraic framework for using closed intervals as truth values in fuzzy logic. The main aim of this article is to introduce and study the concept of ideals in triangle algebras and investigate the connection between ideals and filters. We first point out that the construction procedure for the filter generated by a subset of a triangle algebra established by another study is incorrect, and we proceed to give an alternative characterization.<\/jats:p>","DOI":"10.3390\/axioms13080566","type":"journal-article","created":{"date-parts":[[2024,8,23]],"date-time":"2024-08-23T12:58:07Z","timestamp":1724417887000},"page":"566","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An Exploration of Ideals and Filters in Triangle Algebras"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-6756-2522","authenticated-orcid":false,"given":"Euclide","family":"Noumen","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science, University of Dschang, Dschang P.O.Box 67, Cameroon"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2105-0563","authenticated-orcid":false,"given":"Fabrice","family":"Tchoua Yinga","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science, University of Dschang, Dschang P.O.Box 67, Cameroon"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9035-6753","authenticated-orcid":false,"given":"Blaise Bl\u00e9riot","family":"Koguep Njionou","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science, University of Dschang, Dschang P.O.Box 67, Cameroon"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6852-4041","authenticated-orcid":false,"given":"Chris","family":"Cornelis","sequence":"additional","affiliation":[{"name":"Computational Web Intelligence, Department of Applied Mathematics, Computer Science and Statistics, Ghent University, 9000 Ghent, Belgium"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,21]]},"reference":[{"key":"ref_1","unstructured":"Gratzer, G. (2009). Lattice Theory: First Concepts and Distributive Lattices, Dover Publications."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"468","DOI":"10.1002\/malq.200810012","article-title":"Commutative rings whose ideals form an MV-algebra","volume":"55","author":"Belluce","year":"2009","journal-title":"Math. Log. Q."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1093\/jigpal\/jzy001","article-title":"BL-rings","volume":"26","author":"Lele","year":"2018","journal-title":"Log. J. IGPL"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.fss.2012.09.014","article-title":"MV-algebras derived from ideals in BL-algebras","volume":"218","author":"Lele","year":"2013","journal-title":"Fuzzy Sets Syst."},{"key":"ref_5","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":"Int. J. Mach. Learn. Cybern."},{"key":"ref_6","unstructured":"Piciu, D. (2007). Algebras of Fuzzy Logic, University of Craiova. Editura Universitaria."},{"key":"ref_7","first-page":"493","article-title":"On characterizations of BL-algebras via implicative ideals","volume":"37","author":"Yongwei","year":"2017","journal-title":"Ital. J. Pure Appl. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1142\/S1793005719500248","article-title":"N-fold boolean, implicative and integral ideals on bounded commutative residuated lattices","volume":"15","author":"Tchoua","year":"2019","journal-title":"New Math. Nat. Comput."},{"key":"ref_9","unstructured":"Tchoua, Y.F., Koguep, N.B.B., Lele, C., and Temgoua, A.E.R. (Kybernetika, 2024). Ideals and N-Involutive filters on residuated lattices, Kybernetika, submitted."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Pal, A., and Pal, M. (2010). Some Results on Interval-Valued Fuzzy Matrices. 1st International Conference on E-Business Intelligence (ICEBI 2010), Atlantis Press. Advances in Intelligent Systems Research.","DOI":"10.2991\/icebi.2010.39"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Van Gasse, B., Cornelis, C., Deschrijver, G., and Kerre, E.E. (2006). Triangle Algebras: Towards an Axiomatization of Interval-Valued Residuated Lattices. International Conference on Rough Sets and Current Trends in Computing (RSCTC 2006), Springer. Lecture Notes in Computer Science.","DOI":"10.1007\/11908029_14"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1042","DOI":"10.1016\/j.fss.2007.09.003","article-title":"Triangle algebras: A formal logic approach to interval-valued residuated lattices","volume":"159","author":"Deschrijver","year":"2008","journal-title":"Fuzzy Sets Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1016\/j.ins.2008.11.005","article-title":"The pseudo-linear semantics of interval-valued fuzzy logics","volume":"179","author":"Deschrijver","year":"2009","journal-title":"Inf. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3006","DOI":"10.1016\/j.ins.2010.04.010","article-title":"Filters in residuated lattice and triangle algebras","volume":"180","author":"Deschrijver","year":"2010","journal-title":"Inf. Sci."},{"key":"ref_15","first-page":"1181","article-title":"On maximal filters in triangle algebras","volume":"30","author":"Zahiri","year":"2016","journal-title":"J. Intell. Fuzzy Syst. Appl. Eng. Technol."},{"key":"ref_16","first-page":"267","article-title":"A new approach to filters in triangle algebras","volume":"101","author":"Zahiri","year":"2017","journal-title":"Mathematics"},{"key":"ref_17","first-page":"91","article-title":"An Investigation on the Co-annihilators in Triangle Algebras","volume":"15","author":"Zahiri","year":"2018","journal-title":"Iran. J. Fuzzy Syst."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Zhang, X., and Liang, R. (2022). Interval-Valued General Residuated Lattice-Ordered Groupoids and Expanded Triangle Algebras. Axioms, 12.","DOI":"10.3390\/axioms12010042"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"53","DOI":"10.37193\/CJM.2021.01.06","article-title":"Ideals in residuated lattices","volume":"37","author":"Piciu","year":"2021","journal-title":"Carpanthian J. Math."},{"key":"ref_20","first-page":"443","article-title":"On ideals in De Morgan residuated lattices","volume":"54","author":"Holdon","year":"2018","journal-title":"Kybernetika"},{"key":"ref_21","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":"Trans. Am. Math. Soc."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1007\/s13226-022-00258-1","article-title":"Relative annihilator in bounded commutative residuated lattices","volume":"54","author":"Tallee","year":"2023","journal-title":"Ind. J. Pure Appl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"75","DOI":"10.21136\/MB.2019.0104-17","article-title":"Kerman An Investigation On THE n-Fold IVRL-Filters In Triangle Algebras","volume":"1","author":"Zahiri","year":"2020","journal-title":"Math. Bohem."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.fss.2020.07.001","article-title":"On local triangle algebras","volume":"418","author":"Zahiri","year":"2021","journal-title":"Fuzzy Sets Syst."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Oner, T., Senturk, I., and Oner, G. (2017). An Independent Set of Axioms of MV-Algebras and Solutions to the Set-Theoretical Yang-Baxter Equation. Axioms, 6.","DOI":"10.3390\/axioms6030017"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"15","DOI":"10.7151\/dmgaa.1431","article-title":"Theoretical Solutions for the Yang-Baxter Equation in Triangle Algebras","volume":"44","author":"Senturk","year":"2024","journal-title":"Discuss. Math. Gen. Algebra Appl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/566\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:40:07Z","timestamp":1760110807000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/566"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,21]]},"references-count":26,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2024,8]]}},"alternative-id":["axioms13080566"],"URL":"https:\/\/doi.org\/10.3390\/axioms13080566","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,8,21]]}}}