{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,6,24]],"date-time":"2023-06-24T12:10:18Z","timestamp":1687608618810},"reference-count":36,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2020,4,1]],"date-time":"2020-04-01T00:00:00Z","timestamp":1585699200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,4,20]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We define an order polarity to be a polarity $(X,Y,{\\operatorname{R}})$ where $X$ and $Y$ are partially ordered, and we define an extension polarity to be a triple $(e_X,e_Y,{\\operatorname{R}})$ such that $e_X:P\\to X$ and $e_Y:P\\to Y$ are poset extensions and $(X,Y,{\\operatorname{R}})$ is an order polarity. We define a hierarchy of increasingly strong coherence conditions for extension polarities, each equivalent to the existence of a preorder structure on $X\\cup Y$ such that the natural embeddings, $\\iota _X$ and $\\iota _Y$, of $X$ and $Y$, respectively, into $X\\cup Y$ preserve the order structures of $X$ and $Y$ in increasingly strict ways. We define a Galois polarity to be an extension polarity satisfying the strongest of these coherence conditions and where $e_X$ and $e_Y$ are meet- and join-extensions, respectively. We show that for such polarities the corresponding preorder on $X\\cup Y$ is unique. We define morphisms for polarities, providing the class of Galois polarities with the structure of a category, and we define an adjunction between this category and the category of $\\varDelta _1$-completions and appropriate homomorphisms.<\/jats:p>","DOI":"10.1093\/logcom\/exaa024","type":"journal-article","created":{"date-parts":[[2020,2,27]],"date-time":"2020-02-27T12:29:18Z","timestamp":1582806558000},"page":"785-833","source":"Crossref","is-referenced-by-count":1,"title":["Order polarities"],"prefix":"10.1093","volume":"30","author":[{"given":"Rob","family":"Egrot","sequence":"first","affiliation":[{"name":"Faculty of ICT, Mahidol University, 999 Phutthamonthon Sai 4 Rd. Salaya, Nakhon Pathom 73170, Thailand"}]}],"member":"286","published-online":{"date-parts":[[2020,4,15]]},"reference":[{"key":"2020050200393497500_ref1","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1007\/BF01898828","article-title":"Categorical characterization of the MacNeille completion","volume":"18","author":"Banaschewski","year":"1967","journal-title":"Archiv der Mathematik"},{"key":"2020050200393497500_ref2","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511895968","volume-title":"Information Flow: The Logic of Distributed Systems","author":"Barwise","year":"1997"},{"key":"2020050200393497500_ref3","volume-title":"Lattice Theory","author":"Birkhoff","year":"1940"},{"key":"2020050200393497500_ref4","volume-title":"Lattice Theory","author":"Birkhoff","year":"1995"},{"key":"2020050200393497500_ref5","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781107050884","volume-title":"Modal 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