{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T10:27:29Z","timestamp":1775644049465,"version":"3.50.1"},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":3663,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2004,3]]},"abstract":"<jats:p>The purpose of this paper is to establish some basic points in the model theory of comodules over a coalgebra. It is not even immediately apparent that there is a model theory of comodules since these are not structures in the usual sense of model theory. Let us give the definitions right away so that the reader can see what we mean.<\/jats:p><jats:p>Fix a field <jats:italic>k<\/jats:italic>. A <jats:italic>k-coalgebra C<\/jats:italic> is a <jats:italic>k<\/jats:italic>-vector space equipped with a <jats:italic>k<\/jats:italic>-linear map \u0394: <jats:italic>C<\/jats:italic> \u2192 <jats:italic>C<\/jats:italic> \u2297 <jats:italic>C<\/jats:italic>, called the <jats:italic>comultiplication<\/jats:italic> (by \u2297 we always mean tensor product over <jats:italic>k<\/jats:italic>), and a <jats:italic>k<\/jats:italic>-linear map \u03b5: <jats:italic>C<\/jats:italic> \u2192 <jats:italic>k<\/jats:italic>, called the <jats:italic>counit<\/jats:italic>, such that \u0394\u22971<jats:sub><jats:italic>C<\/jats:italic><\/jats:sub> = 1<jats:sub><jats:italic>C<\/jats:italic><\/jats:sub> \u2297 \u0394 (coassociativity) and (1<jats:sub><jats:italic>C<\/jats:italic><\/jats:sub> \u2297 <jats:italic>\u03b5<\/jats:italic>)\u0394 = 1<jats:sub><jats:italic>C<\/jats:italic><\/jats:sub> = (<jats:italic>\u03b5<\/jats:italic> \u2297 1<jats:sub><jats:italic>C<\/jats:italic><\/jats:sub>)\u0394, where we identify <jats:italic>C<\/jats:italic> with both <jats:italic>k<\/jats:italic> \u2297 <jats:italic>C<\/jats:italic> and <jats:italic>C<\/jats:italic> \u2297 <jats:italic>k<\/jats:italic>. These definitions are literally the duals of those for a <jats:italic>k<\/jats:italic>-algebra: express the axioms for <jats:italic>C\u2032<\/jats:italic> to be a <jats:italic>k<\/jats:italic>-algebra in terms of the multiplication map <jats:italic>\u03bc<\/jats:italic>: <jats:italic>C<\/jats:italic>\u2032 \u2297 <jats:italic>C<\/jats:italic>\u2032 \u2192 <jats:italic>C<\/jats:italic>\u2032 and the \u201cunit\u201d (embedding of <jats:italic>k<\/jats:italic> into <jats:italic>C<\/jats:italic>\u2032), <jats:italic>\u03b4<\/jats:italic>: <jats:italic>k<\/jats:italic> \u2192 <jats:italic>C<\/jats:italic>\u2032 in the form that certain diagrams commute and then just turn round all the arrows. See [5] or more recent references such as [7] for more.<\/jats:p>","DOI":"10.2178\/jsl\/1080938832","type":"journal-article","created":{"date-parts":[[2005,3,2]],"date-time":"2005-03-02T21:27:39Z","timestamp":1109798859000},"page":"137-142","source":"Crossref","is-referenced-by-count":2,"title":["Model theory of comodules"],"prefix":"10.1017","volume":"69","author":[{"given":"Septimiu","family":"Crivei","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mike","family":"Prest","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Geert","family":"Reynders","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200008070_ref003","unstructured":"Prest M. and Wisbauer R. , Finite presentation and purity in categories \u03c3[M], Colloquium Mathematicum , to appear."},{"key":"S0022481200008070_ref004","unstructured":"Reynders G. . Ziegler spectra over serial rings and coalgebras, Doctoral thesis . University of Manchester, 1998."},{"key":"S0022481200008070_ref001","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511600579"},{"key":"S0022481200008070_ref002","doi-asserted-by":"publisher","DOI":"10.1080\/00927879408824927"},{"key":"S0022481200008070_ref006","volume-title":"Foundations of module and ring theory","author":"Wisbauer","year":"1991"},{"key":"S0022481200008070_ref007","first-page":"277","volume-title":"Proceedings of the mathematics conference (Birzeit University 1998)","author":"Wisbauer","year":"2000"},{"key":"S0022481200008070_ref005","volume-title":"Hopf algebras","author":"Sweedler","year":"1969"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200008070","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,6]],"date-time":"2019-05-06T21:28:17Z","timestamp":1557178097000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200008070\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,3]]},"references-count":7,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2004,3]]}},"alternative-id":["S0022481200008070"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1080938832","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,3]]}}}