{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T14:32:28Z","timestamp":1649169148279},"reference-count":2,"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":7406,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1993,12]]},"abstract":"<jats:p>By a variety we mean a class of algebras in a language <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200020673_inline1\" \/>, containing only function symbols, which is closed under homomorphisms, submodels, and products. A variety <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200020673_inline2\" \/> is said to be strongly abelian if for any term <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200020673_inline3\" \/> in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200020673_inline1\" \/>, the quasi-identity<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200020673_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>holds in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200020673_inline2\" \/>.<\/jats:p><jats:p>In [1] it was proved that if a strongly abelian variety has less than the maximal possible uncountable spectrum, then it is equivalent to a multisorted unary variety. Using Shelah's Main Gap theorem one can conclude that if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200020673_inline2\" \/> is a classifiable (superstable without DOP or OTOP and shallow) strongly abelian variety then <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200020673_inline2\" \/> is a multisorted unary variety. In fact, it was known that this conclusion followed from the assumption of superstable without DOP alone.<\/jats:p><jats:p>This paper is devoted to the proof that the superstability assumption is enough to obtain the same structure result. This fulfills a promise made in [2]. Namely, we will prove the following<\/jats:p><jats:p>Theorem 0.1. <jats:italic>If <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200020673_inline2\" \/> is a superstable strongly abelian variety, then it is multisorted unary<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2275151","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:50:13Z","timestamp":1146955813000},"page":"1419-1425","source":"Crossref","is-referenced-by-count":1,"title":["Addendum to \u201cA structure theorem for strongly abelian varieties\u201d"],"prefix":"10.1017","volume":"58","author":[{"given":"Bradd","family":"Hart","sequence":"first","affiliation":[]},{"given":"Sergei","family":"Starchenko","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200020673_ref001","first-page":"832","volume":"56","author":"Hart","year":"1991","journal-title":"A structure theorem for strongly abelian varieties with few models"},{"key":"S0022481200020673_ref002","volume-title":"Transactions of the American Mathematical Society","author":"Hart"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200020673","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,15]],"date-time":"2019-05-15T21:03:26Z","timestamp":1557954206000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200020673\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1993,12]]},"references-count":2,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1993,12]]}},"alternative-id":["S0022481200020673"],"URL":"https:\/\/doi.org\/10.2307\/2275151","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1993,12]]}}}