{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T10:30:30Z","timestamp":1785407430948,"version":"3.56.0"},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2025,11,20]],"date-time":"2025-11-20T00:00:00Z","timestamp":1763596800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2026,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    The Lindenbaum lemma saying that\n                    <jats:italic>completely meet-irreducible closed sets form a basis of any finitary closure system<\/jats:italic>\n                    is an easy-to-prove yet crucial result transcending algebraic logic. While the finitarity restriction is crucial for its usual proof, it is not necessary: there are indeed works proving it (or its variant for a larger class of\n                    <jats:italic>finitely<\/jats:italic>\n                    meet-irreducible closed sets) for non-finitary closure systems arising from particular\n                    <jats:italic>infinitary<\/jats:italic>\n                    logics (i.e., substitution-invariant consequence relations). There is also a general result proving it for a wide class of logics with\n                    <jats:italic>strong p-disjunction<\/jats:italic>\n                    and a\n                    <jats:italic>countable<\/jats:italic>\n                    Hilbert-style axiomatization. Identifying the essential properties of strong p-disjunctions we prove a variant of the Lindenbaum lemma for closure systems which are 1) defined over countable sets, 2) countably axiomatized, and 3) frames (in the order-theoretic sense) but not necessarily substitution-invariant.\n                  <\/jats:p>","DOI":"10.1017\/s1755020325101019","type":"journal-article","created":{"date-parts":[[2025,11,20]],"date-time":"2025-11-20T05:32:22Z","timestamp":1763616742000},"page":"41-45","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["A VARIANT OF THE LINDENBAUM LEMMA FOR INFINITARY CLOSURE SYSTEMS"],"prefix":"10.1017","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3617-1392","authenticated-orcid":false,"given":"PETR","family":"CINTULA","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/0496n6574","id-type":"ROR","asserted-by":"publisher"}],"name":"INSTITUTE OF COMPUTER SCIENCE OF THE CZECH ACADEMY OF SCIENCES"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2025,11,20]]},"reference":[{"key":"S1755020325101019_r1","first-page":"134","volume-title":"WOLLIC 2018","volume":"10944","author":"B\u00edlkov\u00e1","year":"2018"},{"key":"S1755020325101019_r8","doi-asserted-by":"publisher","DOI":"10.1007\/s00500-016-2338-0"},{"key":"S1755020325101019_r3","volume-title":"Mathematics of Modality","volume":"43","author":"Goldblatt","year":"1993"},{"key":"S1755020325101019_r7","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1977.tb00778.x"},{"key":"S1755020325101019_r6","doi-asserted-by":"publisher","DOI":"10.1007\/BF01048686"},{"key":"S1755020325101019_r2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-85675-5"},{"key":"S1755020325101019_r4","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-75939-3_15"},{"key":"S1755020325101019_r5","first-page":"161","article-title":"Hybrid logics with infinitary proof systems","volume":"16","author":"Renardel de Lavalette","year":"2006","journal-title":"JLC"}],"container-title":["The Review of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1755020325101019","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,18]],"date-time":"2026-02-18T08:57:57Z","timestamp":1771405077000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1755020325101019\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,20]]},"references-count":8,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["S1755020325101019"],"URL":"https:\/\/doi.org\/10.1017\/s1755020325101019","relation":{},"ISSN":["1755-0203","1755-0211"],"issn-type":[{"value":"1755-0203","type":"print"},{"value":"1755-0211","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,20]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}