{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T10:28:34Z","timestamp":1772447314744,"version":"3.50.1"},"reference-count":12,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":2384,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2007,9]]},"abstract":"This paper contains a result on the reconstruction of certain homogeneous transitive \u03c9-categorical structures from their automorphism group. The structures treated are relational. In the proof it is shown that their automorphism group contains a generic pair<\/jats:italic> (in a slightly non-standard sense, coming from Baire category).<\/jats:p>Reconstruction results give conditions under which the abstract group structure of the automorphism group Aut() of an \u03c9-categorical structure determines the topology on Aut(), and hence determines up to bi-interpretability, by [1]; they can also give conditions under which the abstract group Aut() determines the permutation group \u27e8Aut (), \u27e9. so determines up to bi-definability. One such condition has been identified by M. Rubin in [12], and it is related to the definability, in Aut(), of point stabilisers. If the condition holds, the structure is said to have a weak<\/jats:italic> \u2200\u2203 interpretation, and Aut() determines up to bi-interpretability or, in some cases, up to bi-definability.<\/jats:p>A better-known approach to reconstruction is via the \u2018small index property\u2019: an \u03c9-categorical stucture has the small index property<\/jats:italic> if any subgroup of Aut() of index less than is open. This guarantees that the abstract group structure of Aut() determines the topology, so if is \u03c9-categorical with Aut() \u2245 Aut() then and are bi-interpretable.<\/jats:p>","DOI":"10.2178\/jsl\/1191333842","type":"journal-article","created":{"date-parts":[[2007,12,19]],"date-time":"2007-12-19T16:25:04Z","timestamp":1198081504000},"page":"792-802","source":"Crossref","is-referenced-by-count":8,"title":["Reconstruction of homogeneous relational structures"],"prefix":"10.1017","volume":"72","author":[{"given":"Silvia","family":"Barbina","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dugald","family":"Macpherson","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200005089_ref012","first-page":"225\u2013249","article-title":"On the reconstruction of \u21350-categorical structures from their autmorphism groups","volume":"69","author":"Rubin","year":"1992","journal-title":"Proceedings of the London Mathematical Society"},{"key":"S0022481200005089_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BF01305233"},{"key":"S0022481200005089_ref005","first-page":"494\u2013500","volume":"37","author":"Henson","year":"1972","journal-title":"Countable homogeneous relational structures and \u21350-categorical theories"},{"key":"S0022481200005089_ref004","first-page":"33\u201372","volume-title":"Automorphisms of first order structures","author":"Evans","year":"1994"},{"key":"S0022481200005089_ref003","article-title":"Reconstruction of classical geometries from their automorphism groups","author":"Barbina","journal-title":"Journal of the London Mathematical Society"},{"key":"S0022481200005089_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(86)90037-0"},{"key":"S0022481200005089_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BF02764005"},{"key":"S0022481200005089_ref008","first-page":"204\u2013218","article-title":"The small index property for \u03c9-stable \u03c9-categorical structures and for the random graph","volume":"48","author":"Hodges","year":"1993","journal-title":"Journal of the London Mathematical Society"},{"key":"S0022481200005089_ref002","unstructured":"[2] Barbina S. , Automorphism groups of omega-categorical structures, Ph.D. thesis, University of Leeds, 2004."},{"key":"S0022481200005089_ref009","doi-asserted-by":"publisher","DOI":"10.2178\/bsl\/1058448678"},{"key":"S0022481200005089_ref011","volume-title":"Classical descriptive set theory","author":"Kechris","year":"1994"},{"key":"S0022481200005089_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF01299742"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200005089","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,1]],"date-time":"2019-05-01T21:20:40Z","timestamp":1556745640000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200005089\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,9]]},"references-count":12,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2007,9]]}},"alternative-id":["S0022481200005089"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1191333842","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,9]]}}}