{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T10:28:14Z","timestamp":1775557694522,"version":"3.50.1"},"reference-count":6,"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":8593,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,9]]},"abstract":"<jats:p>Let <jats:italic>L<\/jats:italic> be a countable language which contains only constant and relation symbols but no function symbols. All theories considered here will be <jats:italic>L<\/jats:italic>-theories. A theory is <jats:italic>coinductive<\/jats:italic> if it can be axiomatized by a set of \u2203\u2200 sentences, and a structure is coinductive if its theory is.<\/jats:p><jats:p>The object of this paper is to show that coinductive \u2135<jats:sub>0<\/jats:sub>-categorical structures are especially simple. First, they are <jats:italic>\u03c9<\/jats:italic>-stable with Morley rank \u2264 1; and second, they have <jats:italic>simple<\/jats:italic> algebraic closures, by which is meant that the algebraic closure of the union of two sets is the union of their algebraic closures. While these two properties do not characterize coinductive \u2135<jats:sub>0<\/jats:sub>-categorical structures, they do characterize those structures which are cellular.<\/jats:p><jats:p>We will say that a countable structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025603_inline1\"\/> is <jats:italic>cellular<\/jats:italic> if there is a finite subset <jats:italic>A<\/jats:italic><jats:sub>0<\/jats:sub> \u2286 <jats:italic>A<\/jats:italic> and there are equivalence relations <jats:italic>E<\/jats:italic> and <jats:italic>F<\/jats:italic> on <jats:italic>A\u2216A<\/jats:italic><jats:sub>0<\/jats:sub> such that the following hold:<\/jats:p><jats:p>(1) There are only finitely many <jats:italic>E<\/jats:italic>-classes.<\/jats:p><jats:p>(2) If <jats:italic>C<\/jats:italic> is an <jats:italic>E<\/jats:italic>-class and <jats:italic>D<\/jats:italic> an <jats:italic>F<\/jats:italic>-class, then \u2223<jats:italic>C<\/jats:italic> \u2229 <jats:italic>D<\/jats:italic> \u2223 = 1.<\/jats:p><jats:p>(3) If <jats:italic>a<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>a<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic> \u2212 1<\/jats:sub>, <jats:italic>b<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>b<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>b<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic> \u2212 1<\/jats:sub> \u0404 <jats:italic>A<\/jats:italic>, then \u3008<jats:italic>a<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>a<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026,<jats:italic>a<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic> \u2212 1<\/jats:sub>\u3009 and \u3008<jats:italic>b<\/jats:italic><jats:sub>0<\/jats:sub>,<jats:italic>b<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>b<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic> \u2212 1<\/jats:sub>\u3009, satisfy the same quantifier-free formulas provided that:<\/jats:p><jats:p>(a) if <jats:italic>i<\/jats:italic> &lt; <jats:italic>k<\/jats:italic> and either <jats:italic>a<jats:sub>i<\/jats:sub><\/jats:italic> \u0404 <jats:italic>A<\/jats:italic><jats:sub>0<\/jats:sub> or <jats:italic>b<jats:sub>i<\/jats:sub><\/jats:italic> \u0404 <jats:italic>A<\/jats:italic><jats:sub>0<\/jats:sub>, then <jats:italic>a<jats:sub>i<\/jats:sub><\/jats:italic> = <jats:italic>b<jats:sub>i<\/jats:sub><\/jats:italic>;<\/jats:p><jats:p>(b) if <jats:italic>i<\/jats:italic> &lt; <jats:italic>k<\/jats:italic>, then <jats:italic>a<jats:sub>i<\/jats:sub><\/jats:italic>; and <jats:italic>b<jats:sub>i<\/jats:sub><\/jats:italic> are <jats:italic>E<\/jats:italic>-equivalent; and<\/jats:p><jats:p>(c) if <jats:italic>i, j<\/jats:italic> &lt; <jats:italic>k<\/jats:italic>, then <jats:italic>a<jats:sub>i<\/jats:sub><\/jats:italic>; is <jats:italic>F<\/jats:italic>-equivalent to <jats:italic>a<jats:sub>j<\/jats:sub><\/jats:italic> if <jats:italic>b<jats:sub>j<\/jats:sub><\/jats:italic> is <jats:italic>F<\/jats:italic>-equivalent to <jats:italic>b<jats:sub>j<\/jats:sub><\/jats:italic>.<\/jats:p><jats:p>The main result of this paper is the following theorem.<\/jats:p><jats:p>Theorem 1. <jats:italic>If<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025603_inline1\"\/><jats:italic>is coinductive and<\/jats:italic> \u2135<jats:sub>0<\/jats:sub>-<jats:italic>categorical, then<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200025603_inline1\"\/><jats:italic>is cellular<\/jats:italic>.<\/jats:p><jats:p>The results of this paper were obtained independently of similar results obtained by Lachlan which can be found in [4] and [5]. The proofs here are quite different.<\/jats:p>","DOI":"10.2307\/2274478","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:36:50Z","timestamp":1146955010000},"page":"1130-1137","source":"Crossref","is-referenced-by-count":7,"title":["Coinductive \u2135<sub>0<\/sub>-categorical theories"],"prefix":"10.1017","volume":"55","author":[{"given":"James H.","family":"Schmerl","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200025603_ref006","doi-asserted-by":"publisher","DOI":"10.1111\/j.1755-2567.1964.tb01088.x"},{"key":"S0022481200025603_ref003","first-page":"222","volume":"53","author":"Hodkinson","year":"1988","journal-title":"Relational structures determined by their finite induced structures"},{"key":"S0022481200025603_ref001","first-page":"494","volume":"37","author":"Henson","year":"1972","journal-title":"Countable homogeneous relational structures and \u21350-categorical theories"},{"key":"S0022481200025603_ref002","volume-title":"Building models by games","author":"Hodges","year":"1985"},{"key":"S0022481200025603_ref004","first-page":"698","volume":"52","author":"Lachlan","year":"1987","journal-title":"Complete theories with only universal and existential axioms"},{"key":"S0022481200025603_ref005","volume-title":"Transactions of the American Mathematical Society","author":"Lachlan"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200025603","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T20:24:39Z","timestamp":1558211079000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200025603\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,9]]},"references-count":6,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1990,9]]}},"alternative-id":["S0022481200025603"],"URL":"https:\/\/doi.org\/10.2307\/2274478","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,9]]}}}