{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T14:40:31Z","timestamp":1762353631302,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2019,10,15]],"date-time":"2019-10-15T00:00:00Z","timestamp":1571097600000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,10,15]],"date-time":"2019-10-15T00:00:00Z","timestamp":1571097600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,10,15]],"date-time":"2019-10-15T00:00:00Z","timestamp":1571097600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,3,31]]},"abstract":"<jats:p>For a regular cardinal $\\kappa$, a formula of the modal $\\mu$-calculus is $\\kappa$-continuous in a variable x if, on every model, its interpretation as a unary function of x is monotone and preserves unions of $\\kappa$-directed sets. We define the fragment $C_{\\aleph_1}(x)$ of the modal $\\mu$-calculus and prove that all the formulas in this fragment are $\\aleph_1$-continuous. For each formula $\\phi(x)$ of the modal $\\mu$-calculus, we construct a formula $\\psi(x) \\in C_{\\aleph_1 }(x)$ such that $\\phi(x)$ is $\\kappa$-continuous, for some $\\kappa$, if and only if $\\phi(x)$ is equivalent to $\\psi(x)$. Consequently, we prove that (i) the problem whether a formula is $\\kappa$-continuous for some $\\kappa$ is decidable, (ii) up to equivalence, there are only two fragments determined by continuity at some regular cardinal: the fragment $C_{\\aleph_0}(x)$ studied by Fontaine and the fragment $C_{\\aleph_1}(x)$. We apply our considerations to the problem of characterizing closure ordinals of formulas of the modal $\\mu$-calculus. An ordinal $\\alpha$ is the closure ordinal of a formula $\\phi(x)$ if its interpretation on every model converges to its least fixed-point in at most $\\alpha$ steps and if there is a model where the convergence occurs exactly in $\\alpha$ steps. We prove that $\\omega_1$, the least uncountable ordinal, is such a closure ordinal. Moreover we prove that closure ordinals are closed under ordinal sum. Thus, any formal expression built from 0, 1, $\\omega$, $\\omega_1$ by using the binary operator symbol + gives rise to a closure ordinal.<\/jats:p>","DOI":"10.23638\/lmcs-15(4:1)2019","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T17:36:58Z","timestamp":1743701818000},"source":"Crossref","is-referenced-by-count":1,"title":["$\\aleph_1$ and the modal $\\mu$-calculus"],"prefix":"10.23638","volume":"Volume 15, Issue 4","author":[{"given":"Maria Jo\u00e3o","family":"Gouveia","sequence":"first","affiliation":[]},{"given":"Luigi","family":"Santocanale","sequence":"additional","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2019,10,15]]},"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1704.03772v4","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1704.03772v4","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T17:36:58Z","timestamp":1743701818000},"score":1,"resource":{"primary":{"URL":"http:\/\/lmcs.episciences.org\/4356"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,15]]},"references-count":0,"URL":"https:\/\/doi.org\/10.23638\/lmcs-15(4:1)2019","relation":{"has-preprint":[{"id-type":"arxiv","id":"1704.03772v3","asserted-by":"subject"},{"id-type":"arxiv","id":"1704.03772v2","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"1704.03772","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1704.03772","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"type":"electronic","value":"1860-5974"}],"subject":[],"published":{"date-parts":[[2019,10,15]]},"article-number":"4356"}}