{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T01:12:51Z","timestamp":1778202771064,"version":"3.51.4"},"reference-count":17,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,2,23]],"date-time":"2026-02-23T00:00:00Z","timestamp":1771804800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,2,23]],"date-time":"2026-02-23T00:00:00Z","timestamp":1771804800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100003086","name":"Eusko Jaurlaritza","doi-asserted-by":"publisher","award":["IT1483-22"],"award-info":[{"award-number":["IT1483-22"]}],"id":[{"id":"10.13039\/501100003086","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Order"],"published-print":{"date-parts":[[2026,4]]},"abstract":"Abstract<\/jats:title>\n \n Given a locale\n L<\/jats:italic>\n , the ordered collection\n \n \n $$\\textsf{S}_c(L)$$<\/jats:tex-math>\n \n \n \n S<\/mml:mi>\n c<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n L<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n of joins of closed sublocales forms a frame\u2014somewhat unexpectedly, as it is naturally embedded in the coframe of all sublocales of\n L<\/jats:italic>\n , where by coframe we mean the order-theoretic dual of a frame. This construction has attracted attention in point-free topology: as a maximal essential extension in the category of frames, for its (non-)functorial properties, its relation to canonical extensions and exact filters of frames, etc. A central open question of the theory, posed by Picado, Pultr, and Tozzi in 2019, asked whether\n \n \n $$\\textsf{S}_c(L)$$<\/jats:tex-math>\n \n \n \n S<\/mml:mi>\n c<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n L<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is always a coframe, or whether there exists a locale for which this fails. In this paper, we resolve this question in the negative by constructing a locale\n L<\/jats:italic>\n such that\n \n \n $$\\textsf{S}_c(L)$$<\/jats:tex-math>\n \n \n \n S<\/mml:mi>\n c<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n L<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is not a coframe. The main challenge in such question lies in the difficulty of understanding exact infima in\n \n \n $$\\textsf{S}_c(L)$$<\/jats:tex-math>\n \n \n \n S<\/mml:mi>\n c<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n L<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ; we circumvent this by analysing a certain separation property satisfied by\n \n \n $$\\textsf{S}_c(L)$$<\/jats:tex-math>\n \n \n \n S<\/mml:mi>\n c<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n L<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n .\n <\/jats:p>","DOI":"10.1007\/s11083-025-09725-w","type":"journal-article","created":{"date-parts":[[2026,2,23]],"date-time":"2026-02-23T08:03:19Z","timestamp":1771833799000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Joins of Closed Sublocales are not Always A Coframe"],"prefix":"10.1007","volume":"43","author":[{"given":"Igor","family":"Arrieta","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,2,23]]},"reference":[{"key":"9725_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00012-021-00757-y","volume":"83","author":"I Arrieta","year":"2022","unstructured":"Arrieta, I.: On joins of complemented sublocales. 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