{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T03:35:58Z","timestamp":1777692958022,"version":"3.51.4"},"reference-count":3,"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":8228,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1991,9]]},"abstract":"<jats:p>For a structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S002248120002404X_inline4.png\"\/> let \u03c6(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S002248120002404X_inline4.png\"\/>) be the number of nonisomorphic, countably infinite substructures of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S002248120002404X_inline4.png\"\/>. The problem considered here, suggested by M. Pouzet, is that of characterizing those countable <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S002248120002404X_inline4.png\"\/> for which \u03c6(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S002248120002404X_inline4.png\"\/>) \u2264 \u2135<jats:sub>0<\/jats:sub>. In this paper we will deal exclusively with structures in a finite, binary relational language <jats:italic>L<\/jats:italic>. The characterization of those <jats:italic>L<\/jats:italic>-structures for which \u03c6(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S002248120002404X_inline4.png\"\/>) \u2264 \u2135<jats:sub>0<\/jats:sub> (which turns out to be equivalent to <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S002248120002404X_inline1.png\"\/>) is given in Theorem 3. It is the culmination of a three-step process. The first step, resulting in Theorem 1, shows that for a countable stable <jats:italic>L<\/jats:italic>-structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S002248120002404X_inline4.png\"\/>, \u03c6(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S002248120002404X_inline4.png\"\/>) \u2264 \u2135<jats:sub>0<\/jats:sub> iff <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S002248120002404X_inline4.png\"\/> is cellular. (See Definition 0.1.) In the second step we consider linearly ordered sets <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S002248120002404X_inline4.png\"\/> = (<jats:italic>A<\/jats:italic>, \u2264 \u2135<jats:sub>0<\/jats:sub>), and characterize in Theorem 2 the order types of those <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S002248120002404X_inline4.png\"\/> for which \u03c6(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S002248120002404X_inline4.png\"\/>) \u2264 \u2135<jats:sub>0<\/jats:sub>. Finally, in Theorem 3, we amalgamate Theorems 1 and 2 to get the classification of all countable <jats:italic>L<\/jats:italic>-structures for which \u03c6(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S002248120002404X_inline4.png\"\/>) \u2264 \u2135<jats:sub>0<\/jats:sub>.<\/jats:p>","DOI":"10.2307\/2275056","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:41:03Z","timestamp":1146955263000},"page":"876-884","source":"Crossref","is-referenced-by-count":2,"title":["Binary relational structures having only countably many nonisomorphic substructures"],"prefix":"10.1017","volume":"56","author":[{"given":"Dugald","family":"Macpherson","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"James H.","family":"Schmerl","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120002404X_ref002","volume-title":"Linear orderings","author":"Rosenstein","year":"1982"},{"key":"S002248120002404X_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760639"},{"key":"S002248120002404X_ref003","first-page":"1130","volume":"55","author":"Schmerl","year":"1990","journal-title":"Coinductive \u21350-categorical theories"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120002404X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T10:34:36Z","timestamp":1679481276000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120002404X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1991,9]]},"references-count":3,"aliases":["10.2178\/jsl\/1183743735"],"journal-issue":{"issue":"3","published-print":{"date-parts":[[1991,9]]}},"alternative-id":["S002248120002404X"],"URL":"https:\/\/doi.org\/10.2307\/2275056","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1991,9]]}}}