{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:53:11Z","timestamp":1753894391228,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2022,7,28]],"date-time":"2022-07-28T00:00:00Z","timestamp":1658966400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:p>We develop a uniform coalgebraic approach to J\\'onsson-Tarski and Thomason\ntype dualities for various classes of neighborhood frames and neighborhood\nalgebras. In the first part of the paper we construct an endofunctor on the\ncategory of complete and atomic Boolean algebras that is dual to the double\npowerset functor on $\\mathsf{Set}$. This allows us to show that Thomason\nduality for neighborhood frames can be viewed as an algebra-coalgebra duality.\nWe generalize this approach to any class of algebras for an endofunctor\npresented by one-step axioms in the language of infinitary modal logic. As a\nconsequence, we obtain a uniform approach to dualities for various classes of\nneighborhood frames, including monotone neighborhood frames, pretopological\nspaces, and topological spaces.\n  In the second part of the paper we develop a coalgebraic approach to\nJ\\'{o}nsson-Tarski duality for neighborhood algebras and descriptive\nneighborhood frames. We introduce an analogue of the Vietoris endofunctor on\nthe category of Stone spaces and show that descriptive neighborhood frames are\nisomorphic to coalgebras for this endofunctor. This allows us to obtain a\ncoalgebraic proof of the duality between descriptive neighborhood frames and\nneighborhood algebras. Using one-step axioms in the language of finitary modal\nlogic, we restrict this duality to other classes of neighborhood algebras\nstudied in the literature, including monotone modal algebras and contingency\nalgebras.\n  We conclude the paper by connecting the two types of dualities via canonical\nextensions, and discuss when these extensions are functorial.<\/jats:p>","DOI":"10.46298\/lmcs-18(3:4)2022","type":"journal-article","created":{"date-parts":[[2022,8,2]],"date-time":"2022-08-02T21:10:09Z","timestamp":1659474609000},"source":"Crossref","is-referenced-by-count":0,"title":["A Coalgebraic Approach to Dualities for Neighborhood Frames"],"prefix":"10.46298","volume":"Volume 18, Issue 3","author":[{"given":"Guram","family":"Bezhanishvili","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-6692-5051","authenticated-orcid":false,"given":"Nick","family":"Bezhanishvili","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jim","family":"de Groot","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"25203","published-online":{"date-parts":[[2022,7,28]]},"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/lmcs.episciences.org\/9870\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/lmcs.episciences.org\/9870\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,20]],"date-time":"2023-06-20T20:20:04Z","timestamp":1687292404000},"score":1,"resource":{"primary":{"URL":"https:\/\/lmcs.episciences.org\/7547"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,28]]},"references-count":0,"URL":"https:\/\/doi.org\/10.46298\/lmcs-18(3:4)2022","relation":{"has-preprint":[{"id-type":"arxiv","id":"2106.01628v2","asserted-by":"subject"},{"id-type":"arxiv","id":"2106.01628v1","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"2106.01628","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.2106.01628","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"type":"electronic","value":"1860-5974"}],"subject":[],"published":{"date-parts":[[2022,7,28]]},"article-number":"7547"}}