{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:17:05Z","timestamp":1740107825603,"version":"3.37.3"},"reference-count":11,"publisher":"Springer Science and Business Media LLC","issue":"15","license":[{"start":{"date-parts":[[2021,6,15]],"date-time":"2021-06-15T00:00:00Z","timestamp":1623715200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,6,15]],"date-time":"2021-06-15T00:00:00Z","timestamp":1623715200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"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"}]},{"DOI":"10.13039\/501100001824","name":"Grantov\u00e1 Agentura Cesk\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\/501100005203","name":"OeAD-GmbH","doi-asserted-by":"publisher","award":["CZ 02\/2019","CZ 02\/2019"],"award-info":[{"award-number":["CZ 02\/2019","CZ 02\/2019"]}],"id":[{"id":"10.13039\/501100005203","id-type":"DOI","asserted-by":"publisher"}]},{"name":"IGA","award":["PrF 2021 030"],"award-info":[{"award-number":["PrF 2021 030"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Soft Comput"],"published-print":{"date-parts":[[2021,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We introduce so-called consistent posets which are bounded posets with an antitone involution<jats:inline-formula><jats:alternatives><jats:tex-math>$$'$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mrow\/><mml:mo>\u2032<\/mml:mo><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>where the lower cones of<jats:inline-formula><jats:alternatives><jats:tex-math>$$x,x'$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>,<\/mml:mo><mml:msup><mml:mi>x<\/mml:mi><mml:mo>\u2032<\/mml:mo><\/mml:msup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and of<jats:inline-formula><jats:alternatives><jats:tex-math>$$y,y'$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>y<\/mml:mi><mml:mo>,<\/mml:mo><mml:msup><mml:mi>y<\/mml:mi><mml:mo>\u2032<\/mml:mo><\/mml:msup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>coincide provided that<jats:italic>x<\/jats:italic>,\u00a0<jats:italic>y<\/jats:italic>are different from 0,\u00a01 and, moreover, if<jats:italic>x<\/jats:italic>,\u00a0<jats:italic>y<\/jats:italic>are different from 0, then their lower cone is different from 0, too. We show that these posets can be represented by means of commutative meet-directoids with an antitone involution satisfying certain identities and implications. In the case of a finite distributive or strongly modular consistent poset, this poset can be converted into a residuated structure and hence it can serve as an algebraic semantics of a certain non-classical logic with unsharp conjunction and implication. Finally we show that the Dedekind\u2013MacNeille completion of a consistent poset is a consistent lattice, i.e., a bounded lattice with an antitone involution satisfying the above-mentioned properties.<\/jats:p>","DOI":"10.1007\/s00500-021-05906-y","type":"journal-article","created":{"date-parts":[[2021,6,15]],"date-time":"2021-06-15T15:04:11Z","timestamp":1623769451000},"page":"9765-9772","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Consistent posets"],"prefix":"10.1007","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3840-3879","authenticated-orcid":false,"given":"Ivan","family":"Chajda","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7030-4080","authenticated-orcid":false,"given":"Helmut","family":"L\u00e4nger","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,6,15]]},"reference":[{"key":"5906_CR1","volume-title":"Directoids. 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Asian-Eur J Math 11:1850097","journal-title":"Asian-Eur J Math"},{"key":"5906_CR5","doi-asserted-by":"crossref","unstructured":"Chajda I, L\u00e4nger H (2019) Residuation in modular lattices and posets. Asian-Eur J Math 12:1950092","DOI":"10.1142\/S179355711950092X"},{"key":"5906_CR6","doi-asserted-by":"crossref","unstructured":"Chajda I, L\u00e4nger H (2021a) Residuation in finite posets. Math Slovaca (to appear). arXiv:1910.09009","DOI":"10.1515\/ms-2021-0021"},{"key":"5906_CR7","doi-asserted-by":"crossref","unstructured":"Chajda I, L\u00e4nger H (2021b) Kleene posets and pseudo-Kleene posets. Miskolc Math Notes (to appear). arXiv:2006.04417","DOI":"10.18514\/MMN.2022.3475"},{"key":"5906_CR8","doi-asserted-by":"publisher","unstructured":"Chajda I, Kola\u0159\u00edk M, L\u00e4nger H (2021a) Extensions of posets with an antitone involution to residuated structures. 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