{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T22:02:49Z","timestamp":1649196169100},"reference-count":14,"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":10603,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1985,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>S. Ulam asked about the number of nonisomorphic projective algebras with <jats:italic>\u03ba<\/jats:italic> generators. This paper answers his question for projective algebras of finite dimension at least three and shows that there are the maximum possible number, continuum many, of nonisomorphic one-generated structures of finite dimension <jats:italic>n<\/jats:italic>, where <jats:italic>n<\/jats:italic> is at least three, of the following kinds: projective set algebras, projective algebras, diagonal-free cylindric set algebras, diagonal-free cylindric algebras, cylindric set algebras, and cylindric algebras. The results of this paper extend earlier results to the collection of cylindric set algebras and provide a uniform proof for all the results. Extensions of these results for dimension two are discussed where some modifications on the hypotheses are needed. Furthermore for <jats:italic>\u03b1<\/jats:italic> \u2265 2, the number of isomorphism classes of regular locally finite cylindric set algebras of dimension <jats:italic>\u03b1<\/jats:italic> of the following two kinds are computed: ones of power <jats:italic>\u03ba<\/jats:italic> for infinite <jats:italic>\u03ba<\/jats:italic> \u2265 \u2223<jats:italic>\u03b1<\/jats:italic>\u2223, and ones with a single generator.<\/jats:p>","DOI":"10.2307\/2273789","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:12:14Z","timestamp":1146953534000},"page":"59-71","source":"Crossref","is-referenced-by-count":3,"title":["The number of one-generated cylindric set algebras of dimension greater than two"],"prefix":"10.1017","volume":"50","author":[{"given":"Jean A.","family":"Larson","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200033077_ref008","first-page":"A","article-title":"The number of one-generated finite dimensional cylindric set algebras (abstract 78T-A51)","volume":"25","author":"Larson","year":"1978","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200033077_ref006","volume-title":"Cylindric algebras","author":"Henkin","year":"1971"},{"key":"S0022481200033077_ref011","volume-title":"Classification theory and the number of nonisomorphic models","author":"Shelah","year":"1978"},{"key":"S0022481200033077_ref002","first-page":"95","volume-title":"Universal algebra (Esztergom, 1977)","volume":"29","author":"Bergman","year":"1982"},{"key":"S0022481200033077_ref012","volume-title":"A collection of mathematical problems","author":"Ulam","year":"1960"},{"key":"S0022481200033077_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0095613"},{"key":"S0022481200033077_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/BF01191750"},{"key":"S0022481200033077_ref013","volume-title":"Transactions of the American Mathematical Society","author":"Comer"},{"key":"S0022481200033077_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BF02483866"},{"key":"S0022481200033077_ref010","article-title":"The number of finitely generated infinite cylindric algebras of dimension two","author":"Larson","journal-title":"Algebra Universalis"},{"key":"S0022481200033077_ref001","unstructured":"Andr\u00e9ka H. and N\u00e9meti I. , On the number of generators of cylindric algebras, preprint, Institute of Mathematics, Hungarian Academy of Sciences, Budapest, 1979."},{"key":"S0022481200033077_ref014","first-page":"311","volume":"45","author":"Maddux","year":"1980","journal-title":"The equational theory of CA3 is undecidable"},{"key":"S0022481200033077_ref005","doi-asserted-by":"publisher","DOI":"10.2307\/2371742"},{"key":"S0022481200033077_ref003","first-page":"80","article-title":"Remarks on projective algebras (abstract)","volume":"54","author":"Chin","year":"1948","journal-title":"Bulletin of the American Mathematical Society"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200033077","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T23:22:45Z","timestamp":1558567365000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200033077\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1985,3]]},"references-count":14,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1985,3]]}},"alternative-id":["S0022481200033077"],"URL":"https:\/\/doi.org\/10.2307\/2273789","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1985,3]]}}}