{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T16:40:49Z","timestamp":1648917649661},"reference-count":9,"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":12611,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1979,9]]},"abstract":"<jats:p>The relation <jats:underline>\u226a<\/jats:underline> (is a homomorphic image of) between (linear) order types has properties similar to those of the better known relation \u2264 (is embeddable in). For example, the order type \u03b7 of the rationals not only embeds every countable order type <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline1\" \/> but also maps homomorphically onto <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline1\" \/>. If <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline1\" \/> is scattered, then <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline1\" \/> can be embedded in (\u03c9* + \u03c9)<jats:sup>\u03b1<\/jats:sup> for some \u03b1 &lt; \u03c9<jats:sub>1<\/jats:sub>. In that case, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline1\" \/> is also a homomorphic image of (\u03c9* + \u03c9)<jats:sup>\u03b1<\/jats:sup> [Lan 2]. If <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline1\" \/> is uncountable, then for some uncountable ordinal \u03b1, \u03b1 <jats:underline>\u226a<\/jats:underline><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline1\" \/>, \u03b1* <jats:underline>\u226a<\/jats:underline><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline1\" \/>, or \u03b7 <jats:underline>\u226a<\/jats:underline><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline1\" \/>. Proofs of these facts are much the same for \u2264 and <jats:underline>\u226a<\/jats:underline>.<\/jats:p><jats:p>The main theorem of [Lav 1] implies that the embedding relation better-quasiorders the set of countable order types. Our main theorem (\u00a73) states the analogous result for the homomorphism relation. As a consequence, if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline1\" \/><jats:sub>0<\/jats:sub>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline1\" \/><jats:sub>1<\/jats:sub>, \u2026 is an infinite sequence of countable order types, then there are <jats:italic>i<\/jats:italic>, <jats:italic>j<\/jats:italic>, <jats:italic>i<\/jats:italic> &lt; <jats:italic>j<\/jats:italic>, such that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline1\" \/><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>, is a homomorphic image of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline1\" \/><jats:sub><jats:italic>j<\/jats:italic><\/jats:sub>. We observed in [Lan 1] that if this is true, then for each countable order type <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline1\" \/> there is a sentence <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline3\" \/> of <jats:italic>L<\/jats:italic><jats:sub>\u03c91\u03c9<\/jats:sub> such that if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline2\" \/> is a countable order type, then <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline2\" \/> satisfies <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline3\" \/> if and only if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline2\" \/> is a homomorphic image of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048313_inline1\" \/>. In fact, the motivation for the work leading to this paper came from this observation.<\/jats:p><jats:p>On the negative side, it is pointed out (\u00a73) that our theorem cannot be extended as far as that of [Lav 1].<\/jats:p>","DOI":"10.2307\/2273132","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:50:56Z","timestamp":1146952256000},"page":"403-411","source":"Crossref","is-referenced-by-count":6,"title":["A combinatorial property of the homomorphism relation between countable order types"],"prefix":"10.1017","volume":"44","author":[{"given":"Charles","family":"Landraitis","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200048313_ref007","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1017\/S0305004100039062","article-title":"On well quasi-ordering infinite trees","volume":"61","author":"Laver","year":"1965","journal-title":"Proceedings of the Cambridge Philosophical Society"},{"key":"S0022481200048313_ref006","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100038603"},{"key":"S0022481200048313_ref005","volume-title":"Studies in foundations and combinatorics","author":"Laver","year":"1978"},{"key":"S0022481200048313_ref003","unstructured":"Landraitis C. , Ph. D. dissertation, Dartmouth College, 1975."},{"key":"S0022481200048313_ref002","first-page":"289","volume":"42","author":"Landraitis","year":"1977","journal-title":"Definability in well quasi-ordered sets of structures"},{"key":"S0022481200048313_ref004","doi-asserted-by":"publisher","DOI":"10.2307\/1970754"},{"key":"S0022481200048313_ref009","volume-title":"Cardinal and ordinal numbers","author":"Sierpinski","year":"1965"},{"key":"S0022481200048313_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF01451165"},{"key":"S0022481200048313_ref008","unstructured":"Rosenstein J. G. , Linear orderings (to appear)."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200048313","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T20:52:58Z","timestamp":1558903978000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200048313\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,9]]},"references-count":9,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1979,9]]}},"alternative-id":["S0022481200048313"],"URL":"https:\/\/doi.org\/10.2307\/2273132","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,9]]}}}