{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T08:28:38Z","timestamp":1777451318162,"version":"3.51.4"},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":13160,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,3]]},"abstract":"<jats:p>This paper continues the study of existentially complete nilpotent groups initiated in [6]. Following [6], we let <jats:italic>K<jats:sub>n<\/jats:sub><\/jats:italic> denote the theory of groups nilpotent of class \u2264 <jats:italic>n<\/jats:italic> and let <jats:italic>K<\/jats:italic><jats:sub arrange=\"stack\">n<\/jats:sub><jats:sup arrange=\"stack\">+<\/jats:sup> denote the theory of torsion-free groups nilpotent of class \u2264 <jats:italic>n<\/jats:italic>. The principal results of [6] were that for <jats:italic>n<\/jats:italic> \u2265 2, neither <jats:italic>K<jats:sub>n<\/jats:sub><\/jats:italic> nor <jats:italic>K<\/jats:italic><jats:sub arrange=\"stack\">n<\/jats:sub><jats:sup arrange=\"stack\">+<\/jats:sup> has a model companion, and the classes <jats:italic>E, F<\/jats:italic>, and <jats:italic>G<\/jats:italic> of existentially complete, finitely generic and infinitely generic models of <jats:italic>K<jats:sub>n<\/jats:sub><\/jats:italic> are all distinct. The question of the relationships between these classes in the context of <jats:italic>K<jats:sub>n<\/jats:sub><\/jats:italic> was left open, however, and the proof of their distinctness for <jats:italic>K<\/jats:italic><jats:sub arrange=\"stack\">n<\/jats:sub><jats:sup arrange=\"stack\">+<\/jats:sup> obviously did not carry over to <jats:italic>K<\/jats:italic><jats:sub arrange=\"stack\">n<\/jats:sub><jats:sup arrange=\"stack\">+<\/jats:sup>, because it made strong use of torsion elements.<\/jats:p><jats:p>In this paper we establish the relationships between <jats:italic>E, F<\/jats:italic>, and <jats:italic>G<\/jats:italic> for <jats:italic>K<\/jats:italic><jats:sub arrange=\"stack\">2<\/jats:sub><jats:sup arrange=\"stack\">+<\/jats:sup>. We show that all three classes are distinct. We also show that there is only one countable finitely generic model, and only one countable infinitely generic model, and that all the countable existentially complete models can be arranged in a sequence <jats:italic>N<\/jats:italic><jats:sub>1<\/jats:sub> \u2286 <jats:italic>N<\/jats:italic><jats:sub>2<\/jats:sub> \u2286 <jats:italic>N<\/jats:italic><jats:sub>3<\/jats:sub> \u2286 \u2026 \u2286 <jats:italic>N<\/jats:italic><jats:sub>\u03c9<\/jats:sub>, where <jats:italic>Z(N<jats:sub>n<\/jats:sub>)<\/jats:italic> is the direct sum of <jats:italic>n<\/jats:italic> copies of <jats:italic>Q<\/jats:italic>. Another result is that the finite and infinite forcing companions of <jats:italic>K<\/jats:italic><jats:sub arrange=\"stack\">2<\/jats:sub><jats:sup arrange=\"stack\">+<\/jats:sup> differ by an \u2200\u2203\u2200 sentence. Finally, we show that there exist finitely generic models of <jats:italic>K<\/jats:italic><jats:sub arrange=\"stack\">2<\/jats:sub><jats:sup arrange=\"stack\">+<\/jats:sup> in all infinite cardinalities.<\/jats:p>","DOI":"10.2307\/2271955","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:46:34Z","timestamp":1146937594000},"page":"126-134","source":"Crossref","is-referenced-by-count":7,"title":["Existentially complete torsion-free nilpotent groups"],"prefix":"10.1017","volume":"43","author":[{"given":"D.","family":"Saracino","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049951_ref004","volume-title":"Proceedings of the Second Scandinavian Logic Symposium","author":"Robinson","year":"1971"},{"key":"S0022481200049951_ref005","doi-asserted-by":"publisher","DOI":"10.4064\/cm-30-1-7-13"},{"key":"S0022481200049951_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF02757003"},{"key":"S0022481200049951_ref003","first-page":"69","volume-title":"Symposia Mathematica","volume":"5","author":"Robinson","year":"1969\/1970"},{"key":"S0022481200049951_ref007","first-page":"649","article-title":"Existentially complete torsion-free nilpotent groups","volume":"23","author":"Saracino","year":"1976","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200049951_ref001","doi-asserted-by":"publisher","DOI":"10.1080\/00927877608822120"},{"key":"S0022481200049951_ref002","volume-title":"Lecture notes on nilpotent groups","author":"Baumslag","year":"1971"},{"key":"S0022481200049951_ref008","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19720182502"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049951","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T16:19:29Z","timestamp":1558973969000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049951\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,3]]},"references-count":8,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1978,3]]}},"alternative-id":["S0022481200049951"],"URL":"https:\/\/doi.org\/10.2307\/2271955","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,3]]}}}