{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T09:06:43Z","timestamp":1777367203042,"version":"3.51.4"},"reference-count":15,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2012,1,11]],"date-time":"2012-01-11T00:00:00Z","timestamp":1326240000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2012,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as a simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets.<\/jats:p>","DOI":"10.1002\/malq.201020098","type":"journal-article","created":{"date-parts":[[2012,1,11]],"date-time":"2012-01-11T09:12:47Z","timestamp":1326273167000},"page":"188-207","source":"Crossref","is-referenced-by-count":12,"title":["On the free implicative semilattice extension of a Hilbert algebra"],"prefix":"10.1002","volume":"58","author":[{"given":"Sergio A.","family":"Celani","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ramon","family":"Jansana","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2012,1,11]]},"reference":[{"key":"e_1_2_9_2_1","unstructured":"G.BezhanishviliandR.Jansana Duality for distributive and implicative semi\u2010lattices preprint available on the second author's webpage."},{"key":"e_1_2_9_3_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11225-011-9323-5"},{"key":"e_1_2_9_4_1","article-title":"Esakia style duality for implicative semilattices","author":"Bezhanishvili G.","journal-title":"Appl. Categ. Struct."},{"key":"e_1_2_9_5_1","doi-asserted-by":"publisher","DOI":"10.2478\/s11533-009-0070-z"},{"key":"e_1_2_9_6_1","doi-asserted-by":"publisher","DOI":"10.1155\/S0161171202011134"},{"key":"e_1_2_9_7_1","doi-asserted-by":"publisher","DOI":"10.2478\/BF02475182"},{"key":"e_1_2_9_8_1","volume-title":"Sur les alg\u00e9bres de Hilbert, in: Collection de Logique Math\u00e9matique, S\u00e9rie A Vol. 21","author":"Diego A.","year":"1966"},{"key":"e_1_2_9_9_1","first-page":"1","article-title":"Prime and maximal ideals of partially ordered sets","volume":"56","author":"Ern\u00e9 M.","year":"2006","journal-title":"Math. Slovaca"},{"key":"e_1_2_9_10_1","doi-asserted-by":"publisher","DOI":"10.2307\/2306387"},{"key":"e_1_2_9_11_1","first-page":"104","article-title":"Priestley duality for distributive semilattices","volume":"77","author":"Hansoul G.","year":"2008","journal-title":"Bull. Soc. Roy. Sci. Li\u00e8ge"},{"key":"e_1_2_9_12_1","doi-asserted-by":"publisher","DOI":"10.2307\/1998339"},{"key":"e_1_2_9_13_1","first-page":"1","article-title":"Sur les alg\u00e9bres de Heyting sym\u00e9triques","volume":"39","author":"Monteiro A.","year":"1980","journal-title":"Port. Math."},{"key":"e_1_2_9_14_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1965-0176944-9"},{"key":"e_1_2_9_15_1","first-page":"41","article-title":"Sur quelques alg\u00e8bres de la logique","volume":"40","author":"Porta H.","year":"1981","journal-title":"Port. Math."},{"key":"e_1_2_9_16_1","volume-title":"An algebraic approach to non\u2010classical logics, Studies in Logic and the Foundations of Mathematics Vol. 78","author":"Rasiowa H.","year":"1974"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.201020098","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.201020098","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,30]],"date-time":"2023-10-30T08:15:48Z","timestamp":1698653748000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.201020098"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,1,11]]},"references-count":15,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2012,5]]}},"alternative-id":["10.1002\/malq.201020098"],"URL":"https:\/\/doi.org\/10.1002\/malq.201020098","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,1,11]]}}}