{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,2,10]],"date-time":"2023-02-10T09:19:46Z","timestamp":1676020786871},"reference-count":6,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":5945,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1997,12]]},"abstract":"<jats:p>Let <jats:italic>L<jats:sub>0<\/jats:sub><\/jats:italic>, <jats:italic>L<\/jats:italic><jats:sub>1<\/jats:sub> and <jats:italic>L<\/jats:italic><jats:sub>2<\/jats:sub> be countable languages with <jats:italic>L<\/jats:italic> \u2229 <jats:italic>L<\/jats:italic><jats:sub>1<\/jats:sub> = <jats:italic>L<\/jats:italic><jats:sub>0<\/jats:sub>. Let <jats:italic>M<\/jats:italic><jats:sub>0<\/jats:sub> be an <jats:italic>L<\/jats:italic><jats:sub>0<\/jats:sub>-structure and <jats:italic>M<jats:sub>i<\/jats:sub><\/jats:italic>, an expansion of <jats:italic>M<\/jats:italic><jats:sub>0<\/jats:sub> to an <jats:italic>L<jats:sub>i<\/jats:sub><\/jats:italic>,-structure (<jats:italic>i<\/jats:italic> = 1,2). We will call an <jats:italic>L<\/jats:italic><jats:sub>1<\/jats:sub> \u222a <jats:italic>L<\/jats:italic><jats:sub>2<\/jats:sub>-structure <jats:italic>M<\/jats:italic> an amalgamation of <jats:italic>M<\/jats:italic><jats:sub>1<\/jats:sub> and <jats:italic>M<\/jats:italic><jats:sub>2<\/jats:sub> if <jats:italic>M<\/jats:italic>\u2223<jats:italic>L<jats:sub>i<\/jats:sub><\/jats:italic> \u2245 <jats:italic>M<jats:sub>i<\/jats:sub><\/jats:italic>, (<jats:italic>i<\/jats:italic> = 1,2). Let's consider the following problem.<\/jats:p><jats:p>(*) Suppose that both <jats:italic>M<\/jats:italic><jats:sub>1<\/jats:sub> and <jats:italic>M<\/jats:italic><jats:sub>2<\/jats:sub> belong to the class <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015590_inline1\" \/>. Can we always find an amalgamation <jats:italic>M<\/jats:italic> in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015590_inline1\" \/>?<\/jats:p><jats:p>Of course the existence of such an amalgamation depends on the class <jats:italic>L<\/jats:italic>. Some examples of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015590_inline1\" \/> and the answers are given below.<\/jats:p><jats:p>1. <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015590_inline1\" \/> = Countably saturated strongly minimal structures with the DMP In [3], Hrushovski showed that any two strongly minimal theories formulated in totally different languages have a common extension which is still strongly minimal and with the DMP (DMP is the property that states that if a point <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015590_inline2\" \/> is sufficiently close to <jats:italic>\u0101<\/jats:italic>, then <jats:italic>\u03c6<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015590_inline3\" \/>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015590_inline2\" \/>) has the same rank and the same degree as <jats:italic>\u03c6<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015590_inline3\" \/>, <jats:italic>\u0101<\/jats:italic>).) His proof essentially shows that if <jats:italic>L<\/jats:italic><jats:sub>0<\/jats:sub> = \u2205 then any two countably saturated strongly minimal structures with the DMP have a strongly minimal amalgamation. Also he gave an example that shows the condition <jats:italic>L<\/jats:italic><jats:sub>0<\/jats:sub> = \u2205 is necessary.<\/jats:p><jats:p>2. <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200015590_inline1\" \/>= \u2135<jats:sub>1<\/jats:sub>-categorical countable structures. Let <jats:italic>M<\/jats:italic><jats:sub>1<\/jats:sub> be the structure (\u211a, +) and let <jats:italic>M<\/jats:italic><jats:sub>2<\/jats:sub> be the {<jats:italic>E, F<\/jats:italic>}-structure defined by: (i) <jats:italic>E<\/jats:italic> is an equivalence relation which divides the universe into two infinite classes <jats:italic>A<\/jats:italic> and <jats:italic>B<\/jats:italic>, (ii) <jats:italic>F<\/jats:italic> is a bijection between <jats:italic>A<\/jats:italic> and <jats:italic>B<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2275627","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T23:03:06Z","timestamp":1146956586000},"page":"1070-1074","source":"Crossref","is-referenced-by-count":2,"title":["Amalgamations preserving \u2135<sub>1<\/sub>-categoricity"],"prefix":"10.1017","volume":"62","author":[{"given":"Anand","family":"Pillay","sequence":"first","affiliation":[]},{"given":"Akito","family":"Tsuboi","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200015590_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(93)90168-D"},{"key":"S0022481200015590_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF02808211"},{"key":"S0022481200015590_ref006","first-page":"585","volume":"45","author":"Schmerl","year":"1980","journal-title":"Decidability and \u21350-categoricity of theories of partially ordered sets"},{"key":"S0022481200015590_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1989-0943605-1"},{"key":"S0022481200015590_ref005","first-page":"1003","volume":"56","author":"Pillay","year":"1991","journal-title":"Some remarks on modular regular types"},{"key":"S0022481200015590_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BF02758643"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200015590","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,11]],"date-time":"2019-05-11T19:54:38Z","timestamp":1557604478000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200015590\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997,12]]},"references-count":6,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1997,12]]}},"alternative-id":["S0022481200015590"],"URL":"https:\/\/doi.org\/10.2307\/2275627","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1997,12]]}}}