{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,18]],"date-time":"2026-07-18T05:10:09Z","timestamp":1784351409005,"version":"3.55.0"},"reference-count":16,"publisher":"Springer Science and Business Media LLC","issue":"19","license":[{"start":{"date-parts":[[2020,7,26]],"date-time":"2020-07-26T00:00:00Z","timestamp":1595721600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,7,26]],"date-time":"2020-07-26T00:00:00Z","timestamp":1595721600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100005203","name":"OeAD-GmbH","doi-asserted-by":"publisher","award":["CZ 02\/2019"],"award-info":[{"award-number":["CZ 02\/2019"]}],"id":[{"id":"10.13039\/501100005203","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005203","name":"OeAD-GmbH","doi-asserted-by":"publisher","award":["CZ 02\/2019"],"award-info":[{"award-number":["CZ 02\/2019"]}],"id":[{"id":"10.13039\/501100005203","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001824","name":"Grantov\u00e1 Agentura \u010cesk\u00e9 Republiky","doi-asserted-by":"publisher","award":["20-09869L"],"award-info":[{"award-number":["20-09869L"]}],"id":[{"id":"10.13039\/501100001824","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["I 4579-N"],"award-info":[{"award-number":["I 4579-N"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"name":"IGA","award":["PrF 2020 014"],"award-info":[{"award-number":["PrF 2020 014"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Soft Comput"],"published-print":{"date-parts":[[2020,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Effect algebras form an algebraic formalization of the logic of quantum mechanics. For lattice effect algebras <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbf {E}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n<mml:mi>E<\/mml:mi>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula>, we investigate a natural implication and prove that the implication reduct of <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbf {E}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n<mml:mi>E<\/mml:mi>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> is term equivalent to <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbf {E}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n<mml:mi>E<\/mml:mi>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Then, we present a simple axiom system in Gentzen style in order to axiomatize the logic induced by lattice effect algebras. For effect algebras which need not be lattice-ordered, we introduce a certain kind of implication which is everywhere defined but whose result need not be a single element. Then, we study effect implication algebras and prove the correspondence between these algebras and effect algebras satisfying the ascending chain condition. We present an axiom system in Gentzen style also for not necessarily lattice-ordered effect algebras and prove that it is an algebraic semantics for the logic induced by finite effect algebras.<\/jats:p>","DOI":"10.1007\/s00500-020-05188-w","type":"journal-article","created":{"date-parts":[[2020,7,26]],"date-time":"2020-07-26T10:02:37Z","timestamp":1595757757000},"page":"14275-14286","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["The logic induced by effect algebras"],"prefix":"10.1007","volume":"24","author":[{"given":"Ivan","family":"Chajda","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Radom\u00edr","family":"Hala\u0161","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7030-4080","authenticated-orcid":false,"given":"Helmut","family":"L\u00e4nger","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2020,7,26]]},"reference":[{"key":"5188_CR1","doi-asserted-by":"publisher","first-page":"314","DOI":"10.1016\/j.jmp.2009.04.005","volume":"53","author":"D Aerts","year":"2009","unstructured":"Aerts D (2009) Quantum structure in cognition. J Math Psychol 53:314\u2013348","journal-title":"J Math Psychol"},{"key":"5188_CR2","doi-asserted-by":"publisher","first-page":"1699","DOI":"10.1007\/BF02057883","volume":"25","author":"MK Bennett","year":"1995","unstructured":"Bennett MK, Foulis DJ (1995) Phi-symmetric effect algebras. Found Phys 25:1699\u20131722","journal-title":"Found Phys"},{"key":"5188_CR3","volume-title":"Algebraizable logics","author":"WJ Blok","year":"1989","unstructured":"Blok WJ, Pigozzi D (1989) Algebraizable logics, vol 77. Memoirs of the American Mathematical Society, Providence, RI"},{"issue":"434","key":"5188_CR4","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1016\/j.ins.2017.12.049","volume":"433","author":"RA Borzooei","year":"2018","unstructured":"Borzooei RA, Dvure\u010denskij A, Shafari AH (2018) Material implications in lattice effect algebras. Inform Sci 433(434):233\u2013240","journal-title":"Inform Sci"},{"key":"5188_CR5","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1007\/s00153-009-0125-7","volume":"48","author":"M Botur","year":"2009","unstructured":"Botur M, Hala\u0161 R (2009) Commutative basic algebras and non-associative fuzzy logics. Arch Math Logic 48:243\u2013255","journal-title":"Arch Math Logic"},{"key":"5188_CR6","doi-asserted-by":"publisher","first-page":"4306","DOI":"10.1007\/s10773-014-2500-3","volume":"54","author":"I Chajda","year":"2015","unstructured":"Chajda I (2015) The propositional logic induced by means of basic algebras. Int J Theor Phys 54:4306\u20134312","journal-title":"Int J Theor Phys"},{"key":"5188_CR7","doi-asserted-by":"crossref","unstructured":"Chajda I, L\u00e4nger H (2019) Residuation in lattice effect algebras. Fuzzy Sets Syst","DOI":"10.1016\/j.fss.2019.11.008"},{"key":"5188_CR8","unstructured":"Chajda I, L\u00e4nger H (submitted) Unsharp residuation in effect algebras. J Multiple-Valued Logic Soft Comput. arxiv:1907.02738"},{"key":"5188_CR9","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1007\/s00012-009-0010-6","volume":"61","author":"I Chajda","year":"2009","unstructured":"Chajda I, Hala\u0161 R, K\u00fchr J (2009) Every effect algebra can be made into a total algebra. Algebra Univ 61:139\u2013150","journal-title":"Algebra Univ"},{"key":"5188_CR10","doi-asserted-by":"crossref","unstructured":"Chajda I, L\u00e4nger H, Paseka J (2017) The groupoid-based logic for lattice effect algebras. In: Proceedings 2017 IEEE 47th international symposium on multiple-valued logic (ISMVL). Springer, pp 230\u2013235. ISSN 2378-2226","DOI":"10.1109\/ISMVL.2017.15"},{"key":"5188_CR11","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-017-2422-7","volume-title":"New trends in quantum structures","author":"A Dvure\u010denskij","year":"2000","unstructured":"Dvure\u010denskij A, Pulmannov\u00e1 S (2000) New trends in quantum structures. Kluwer, Dordrecht ISBN 0-7923-6471-6"},{"key":"5188_CR12","doi-asserted-by":"publisher","first-page":"1331","DOI":"10.1007\/BF02283036","volume":"24","author":"DJ Foulis","year":"1994","unstructured":"Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24:1331\u20131352","journal-title":"Found Phys"},{"key":"5188_CR13","doi-asserted-by":"publisher","first-page":"1291","DOI":"10.1007\/s11225-012-9454-3","volume":"100","author":"DJ Foulis","year":"2012","unstructured":"Foulis DJ, Pulmannov\u00e1 S (2012) Logical connectives on lattice effect algebras. Stud Logica 100:1291\u20131315","journal-title":"Stud Logica"},{"key":"5188_CR14","unstructured":"Rad SR, Sharafi AH, Smets S (2019) A complete axiomatisation for the logic of lattice effect algebras. Int J Theor Phys"},{"key":"5188_CR15","volume-title":"An algebraic approach to non-classical logics","author":"H Rasiowa","year":"1974","unstructured":"Rasiowa H (1974) An algebraic approach to non-classical logics. North-Holland, Amsterdam"},{"key":"5188_CR16","volume-title":"The mathematics of metamathematics","author":"H Rasiowa","year":"1970","unstructured":"Rasiowa H, Sikorski R (1970) The mathematics of metamathematics. PWN, Warsaw"}],"container-title":["Soft Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-020-05188-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00500-020-05188-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-020-05188-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,25]],"date-time":"2021-07-25T23:17:48Z","timestamp":1627255068000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00500-020-05188-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,26]]},"references-count":16,"journal-issue":{"issue":"19","published-print":{"date-parts":[[2020,10]]}},"alternative-id":["5188"],"URL":"https:\/\/doi.org\/10.1007\/s00500-020-05188-w","relation":{},"ISSN":["1432-7643","1433-7479"],"issn-type":[{"value":"1432-7643","type":"print"},{"value":"1433-7479","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,26]]},"assertion":[{"value":"26 July 2020","order":1,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Compliance with ethical standards"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"This article does not contain any studies with human participants or animals performed by any of the authors.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Human participants or animals performed"}}]}}