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Logic"],"published-print":{"date-parts":[[2026,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We prove that for every antichain\n                    <jats:italic>A<\/jats:italic>\n                    in the poset\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\langle [\\omega ]^{&lt;\\omega },\\subseteq \\rangle $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mo>\u27e8<\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow>\n                                <mml:mo>[<\/mml:mo>\n                                <mml:mi>\u03c9<\/mml:mi>\n                                <mml:mo>]<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:mo>&lt;<\/mml:mo>\n                                <mml:mi>\u03c9<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\u2286<\/mml:mo>\n                            <mml:mo>\u27e9<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    the set of maximal antichains which extend\n                    <jats:italic>A<\/jats:italic>\n                    is either finite or has the size of the continuum. As a consequence we prove a conjecture of de Jongh and Vargas-Sandoval about nepfi families of finite languages [2, 10].\n                  <\/jats:p>","DOI":"10.1007\/s00153-025-00976-3","type":"journal-article","created":{"date-parts":[[2025,5,21]],"date-time":"2025-05-21T15:07:40Z","timestamp":1747840060000},"page":"1-12","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Extending antichains in the poset $$\\langle [\\omega ]^{&lt;\\omega },\\subseteq \\rangle $$"],"prefix":"10.1007","volume":"65","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-4720-574X","authenticated-orcid":false,"given":"Francisco Santiago","family":"Nieto-de la Rosa","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2019-6077","authenticated-orcid":false,"given":"Ulises Ariet","family":"Ramos-Garc\u00eda","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4643-2023","authenticated-orcid":false,"given":"Ana Luc\u00eda","family":"Vargas-Sandoval","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dick","family":"de Jongh","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,5,21]]},"reference":[{"key":"976_CR1","unstructured":"Bringsjord, Selmer, Govindarajulu, Naveen\u00a0Sundar: Artificial intelligence, The Stanford Encyclopedia of Philosophy (Edward\u00a0N. 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