{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T03:59:39Z","timestamp":1781236779545,"version":"3.54.1"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"10","license":[{"start":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T00:00:00Z","timestamp":1781222400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T00:00:00Z","timestamp":1781222400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100011033","name":"Agencia Estatal de Investigaci\u00f3n","doi-asserted-by":"publisher","award":["PID2021-127870OB-I00"],"award-info":[{"award-number":["PID2021-127870OB-I00"]}],"id":[{"id":"10.13039\/501100011033","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100011033","name":"Agencia Estatal de Investigaci\u00f3n","doi-asserted-by":"publisher","award":["PID2022-140630NB-I00"],"award-info":[{"award-number":["PID2022-140630NB-I00"]}],"id":[{"id":"10.13039\/501100011033","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100009473","name":"Universidad de M\u00e1laga","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100009473","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2026,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Efficient closure computation is a key challenge in fuzzy Formal Concept Analysis. Direct implicational systems, which allow for one-step closure calculation, offer a powerful solution, but their formalization and practical computation in the fuzzy setting have remained largely open problems. This paper provides a comprehensive contribution to fill this gap. We first extend the concept of a direct system to fuzzy attribute implications and establish a complete theoretical characterization via a novel Fuzzy Exchange Condition. Building on this, we define the notion of a direct-optimal system and develop the  algorithm, a provably correct, interleaved strategy for its computation. A rigorous experimental evaluation demonstrates that our proposed algorithm is orders of magnitude more efficient than naive sequential approaches, and we diagnose the distinct computational bottlenecks that cause these simpler methods to fail. This work thus delivers a complete and empirically validated pipeline, from theory to practice, for the efficient computation of direct-optimal bases.<\/jats:p>","DOI":"10.1007\/s40314-026-03823-0","type":"journal-article","created":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T03:47:59Z","timestamp":1781236079000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Direct-optimal systems of attribute implications in fuzzy formal concept analysis"],"prefix":"10.1007","volume":"45","author":[{"given":"Manuel","family":"Ojeda-Hern\u00e1ndez","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0172-1585","authenticated-orcid":false,"given":"Domingo","family":"L\u00f3pez-Rodr\u00edguez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,6,12]]},"reference":[{"key":"3823_CR1","doi-asserted-by":"publisher","unstructured":"Arag\u00f3n RG, Medina J, Ram\u00edrez-Poussa E (2024) Factorizing formal contexts from closures of necessity operators. Comput Appl Math 43:1\u201324. https:\/\/doi.org\/10.1007\/s40314-024-02590-0","DOI":"10.1007\/s40314-024-02590-0"},{"key":"3823_CR2","doi-asserted-by":"publisher","unstructured":"Arag\u00f3n RG, Medina J, Ram\u00edrez-Poussa E (2025) Decomposition of contexts into independent subcontexts based on thresholds. Comput Appl Math 44:1\u201322. https:\/\/doi.org\/10.1007\/s40314-025-03302-y","DOI":"10.1007\/s40314-025-03302-y"},{"key":"3823_CR3","unstructured":"B\u011blohl\u00e1vek R (1998) Fuzzy concepts and conceptual structures: induced similarities. Joint conference on information sciences proceedings, vol 1. Assoc. Intel. Machinery, pp 179\u2013182"},{"key":"3823_CR4","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-0633-1","author":"R B\u011blohl\u00e1vek","year":"2002","unstructured":"B\u011blohl\u00e1vek R (2002) Fuzzy relational systems. Springer. https:\/\/doi.org\/10.1007\/978-1-4615-0633-1","journal-title":"Springer"},{"key":"3823_CR5","doi-asserted-by":"publisher","unstructured":"B\u011blohl\u00e1vek R, Vychodil V (2005) Fuzzy attribute implications: computing non-redundant bases using maximal independent sets. Australasian joint conference on artificial intelligence. LNAI 3809, Springer-Verlag Berlin Heidelberg, pp 1126\u20131129. https:\/\/doi.org\/10.1007\/11589990153","DOI":"10.1007\/11589990153"},{"key":"3823_CR6","doi-asserted-by":"publisher","unstructured":"B\u011blohl\u00e1vek R, Vychodil V (2006) Attribute implications in a fuzzy setting. In: Formal concept analysis: 4th international conference, ICFCA 2006, Dresden, Germany, February 13\u201317. Proceedings, pp 45\u201360 (2006). Springer. https:\/\/doi.org\/10.1007\/116714043","DOI":"10.1007\/116714043"},{"issue":"1","key":"3823_CR7","doi-asserted-by":"publisher","first-page":"66","DOI":"10.1080\/03081079.2016.1205712","volume":"46","author":"R B\u011blohl\u00e1vek","year":"2017","unstructured":"B\u011blohl\u00e1vek R, Vychodil V (2017) Attribute dependencies for data with grades II. Int J Gen Syst 46(1):66\u201392. https:\/\/doi.org\/10.1080\/03081079.2016.1205712","journal-title":"Int J Gen Syst"},{"key":"3823_CR8","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1016\/j.ijar.2015.12.007","volume":"70","author":"R B\u011blohl\u00e1vek","year":"2016","unstructured":"B\u011blohl\u00e1vek R, Cordero P, Enciso M, Mora A, Vychodil V (2016) Automated prover for attribute dependencies in data with grades. Int J Approx Reason 70:51\u201367. https:\/\/doi.org\/10.1016\/j.ijar.2015.12.007","journal-title":"Int J Approx Reason"},{"issue":"22\u201324","key":"3823_CR9","doi-asserted-by":"publisher","first-page":"2155","DOI":"10.1016\/J.TCS.2009.12.021","volume":"411","author":"K Bertet","year":"2010","unstructured":"Bertet K, Monjardet B (2010) The multiple facets of the canonical direct unit implicational basis. Theor Comput Sci 411(22\u201324):2155\u20132166. https:\/\/doi.org\/10.1016\/J.TCS.2009.12.021","journal-title":"Theor Comput Sci"},{"key":"3823_CR10","doi-asserted-by":"publisher","unstructured":"Bertet K, Nebut M (2004) Efficient algorithms on the family associated to an implicational system. Discret Math Theor Comput Sci 6:315\u2013338. https:\/\/doi.org\/10.46298\/dmtcs.330","DOI":"10.46298\/dmtcs.330"},{"key":"3823_CR11","first-page":"209","volume":"3","author":"A Burusco","year":"1994","unstructured":"Burusco A, Fuentes-Gonz\u00e1lez R (1994) The study of the $$L$$-fuzzy concept lattice. Mathw Soft Comput 3:209\u2013218","journal-title":"Mathw Soft Comput"},{"key":"3823_CR12","unstructured":"\u010cech E (1966) Topological spaces. Interscience Publishers,(Revised edition prepared by Z. Frol\u00edk and M. Kat\u011btov, based on the original manuscripts)"},{"key":"3823_CR13","doi-asserted-by":"publisher","unstructured":"Cordero P, Enciso M, Mora A (2018) Directness in fuzzy formal concept analysis. International conference on information processing and management of uncertainty in knowledge-based systems. Springer, pp 585\u2013595. https:\/\/doi.org\/10.1007\/978-3-319-91473-250","DOI":"10.1007\/978-3-319-91473-250"},{"issue":"7","key":"3823_CR14","doi-asserted-by":"publisher","first-page":"724","DOI":"10.1080\/03081079.2020.1831484","volume":"49","author":"P Cordero","year":"2020","unstructured":"Cordero P, Enciso M, Mora A, Vychodil V (2020) Parameterized simplification logic I: reasoning with implications and classes of closure operators. Int J Gen Syst 49(7):724\u2013746. https:\/\/doi.org\/10.1080\/03081079.2020.1831484","journal-title":"Int J Gen Syst"},{"key":"3823_CR15","doi-asserted-by":"publisher","unstructured":"Cornejo ME, Medina J, Oca\u00f1a FJ (2026) Theories, models and bases of attribute implications in multi-adjoint concept lattices with hedges. Comput Appl Math 45(30):1\u201333. https:\/\/doi.org\/10.1007\/s40314-025-03391-9","DOI":"10.1007\/s40314-025-03391-9"},{"key":"3823_CR16","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-49291-8","author":"B Ganter","year":"2016","unstructured":"Ganter B, Obiedkov S (2016) Conceptual exploration. Springer Berlin. https:\/\/doi.org\/10.1007\/978-3-662-49291-8","journal-title":"Springer Berlin"},{"key":"3823_CR17","first-page":"5","volume":"95","author":"JL Guigues","year":"1986","unstructured":"Guigues JL, Duquenne V (1986) Familles minimales d\u2019implications informatives r\u00e9sultant d\u2019une tables de donn\u00e9es binaires. Math\u00e9matiques et Sciences Humaines 95:5\u201318","journal-title":"Math\u00e9matiques et Sciences Humaines"},{"key":"3823_CR18","unstructured":"H\u00e1jek P (2013) Metamathematics of fuzzy logic. Trends in logic. Springer"},{"issue":"2","key":"3823_CR19","doi-asserted-by":"publisher","first-page":"130","DOI":"10.1016\/j.fss.2008.05.004","volume":"160","author":"J Medina","year":"2009","unstructured":"Medina J, Ojeda-Aciego M, Ruiz-Calvi\u00f1o J (2009) Formal concept analysis via multi-adjoint concept lattices. Fuzzy Sets Syst 160(2):130\u2013144. https:\/\/doi.org\/10.1016\/j.fss.2008.05.004","journal-title":"Fuzzy Sets Syst"},{"key":"3823_CR20","doi-asserted-by":"publisher","first-page":"510","DOI":"10.1080\/00207160.2011.644275","volume":"89","author":"A Mora","year":"2012","unstructured":"Mora A, Enciso M, Cordero P, Fortes I (2012) Closure via functional dependence simplification. Int J Comput Math 89:510\u2013526. https:\/\/doi.org\/10.1080\/00207160.2011.644275","journal-title":"Int J Comput Math"},{"key":"3823_CR21","doi-asserted-by":"crossref","unstructured":"Ojeda-Hern\u00e1ndez M, L\u00f3pez-Rodr\u00edguez D (2025) On direct systems of implications with graded attributes. The 14th conference of the European Society for Fuzzy Logic and Technology (EUSFLAT). https:\/\/doi.org\/10.1007\/978-3-031-97228-713","DOI":"10.1007\/978-3-031-97228-7_13"},{"key":"3823_CR22","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2021.113390","volume":"404","author":"M Ojeda-Hern\u00e1ndez","year":"2022","unstructured":"Ojeda-Hern\u00e1ndez M, Cabrera IP, Cordero P (2022) Quasi-closed elements in fuzzy posets. J Comput Appl Math 404:113390. https:\/\/doi.org\/10.1016\/j.cam.2021.113390","journal-title":"J Comput Appl Math"},{"key":"3823_CR23","doi-asserted-by":"publisher","unstructured":"Ojeda-Hern\u00e1ndez M, Cabrera IP, Cordero P, Mu\u00f1oz-Velasco E (2023) On pseudointents in fuzzy formal concept analysis. International conference on conceptual structures. Springer, pp 36\u201340. https:\/\/doi.org\/10.1007\/978-3-031-40960-84","DOI":"10.1007\/978-3-031-40960-84"},{"key":"3823_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-60460-7","volume-title":"Fuzzy Begriffe: Formale Begriffsanalyse Von Unscharfen Daten","author":"S Pollandt","year":"1997","unstructured":"Pollandt S (1997) Fuzzy Begriffe: Formale Begriffsanalyse Von Unscharfen Daten. Springer, Berlin"},{"key":"3823_CR25","doi-asserted-by":"publisher","first-page":"106","DOI":"10.1016\/j.dam.2017.12.031","volume":"249","author":"E Rodr\u00edguez-Lorenzo","year":"2018","unstructured":"Rodr\u00edguez-Lorenzo E, Bertet K, Cordero P, Enciso M, Mora A (2018) Direct-optimal basis computation by means of the fusion of simplification rules. Discret Appl Math 249:106\u2013119. https:\/\/doi.org\/10.1016\/j.dam.2017.12.031","journal-title":"Discret Appl Math"},{"key":"3823_CR26","unstructured":"Vychodil V, B\u011blohl\u00e1vek R (2005) Fuzzy attribute logic: attribute implications, their validity, entailment, and non-redundant basis. In: Proceedings of the eleventh international fuzzy systems association world congress, vol I"},{"key":"3823_CR27","doi-asserted-by":"publisher","unstructured":"Wille R (1982) Restructuring lattice theory: an approach based on hierarchies of concepts. Springer, pp 445\u2013470. https:\/\/doi.org\/10.1007\/978-94-009-7798-315","DOI":"10.1007\/978-94-009-7798-315"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-026-03823-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-026-03823-0","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-026-03823-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T03:48:07Z","timestamp":1781236087000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-026-03823-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,12]]},"references-count":27,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2026,12]]}},"alternative-id":["3823"],"URL":"https:\/\/doi.org\/10.1007\/s40314-026-03823-0","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,12]]},"assertion":[{"value":"14 October 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 February 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 May 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 June 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Authors declare to have no competing interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"All authors consent to the submission of this work to the journal.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval and consent to participate"}},{"value":"All authors consent to the submission of this work to the journal.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}}],"article-number":"433"}}