{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T11:56:05Z","timestamp":1648986965102},"reference-count":12,"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":10328,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1985,12]]},"abstract":"<jats:p>The theory of cylindric algebras (CA's) is the algebraic theory of first order logics. Several ideas about logic are easier to formulate in the frame of CA-theory. Such are e.g. some concepts of abstract model theory (cf. [1] and [10]\u2013[12]) as well as ideas about relationships between several axiomatic theories of different similarity types (cf. [4] and [10]). In contrast with the relationship between Boolean algebras and classical propositional logic, CA's correspond not only to classical first order logic but also to several other ones. Hence CA-theoretic results contain more information than their counterparts in first order logic. For more about this see [1], [3], [5], [9], [10] and [12].<\/jats:p><jats:p>Here we shall use the notation and concepts of the monographs Henkin-Monk-Tarski [7] and [8]. <jats:italic>\u03c9<\/jats:italic> denotes the set of natural numbers. CA<jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub> denotes the class of all cylindric algebras of dimension <jats:italic>\u03b1<\/jats:italic>; by \u201ca CA<jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub>\u201d we shall understand an element of the class CA<jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub>. The class Dc<jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub> \u2286 CA<jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub> was defined in [7]. Note that Dc<jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub> = 0 for <jats:italic>\u03b1<\/jats:italic> \u2208 <jats:italic>\u03c9<\/jats:italic>. The classes Ws<jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120003187X_inline1\" \/> and Cs<jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub> were defined in 1.1.1 of [8], p. 4. They are called the classes of all weak cylindric set algebras, regular cylindric set algebras and cylindric set algebras respectively. It is proved in [8] (I.7.13, I.1.9) that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120003187X_inline2\" \/> \u2286 CA<jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub>. (These inclusions are proper by 7.3.7, 1.4.3 and 1.5.3 of [8].)<\/jats:p><jats:p>It was proved in 2.3.22 and 2.3.23 of [7] that every simple, finitely generated Dc<jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub> is generated by a single element. This is the algebraic counterpart of a property of first order logics (cf. 2.3.23 of [7]). The question arose: <jats:italic>for which simple CA<jats:sub><jats:italic>\u03b1<\/jats:italic><\/jats:sub>'s does \u201cfinitely generated\u201d imply \u201cgenerated by a single element\u201d<\/jats:italic> (see p. 291 and Problem 2.3 in [7]). In terms of abstract model theory this amounts to asking the question: For which logics does the property described in 2.3.23 of [7] hold? This property is roughly the following. In any maximal theory any finite set of concepts is definable in terms of a single concept. The connection with CA-theory is that maximal theories correspond to simple CA's (the elements of which are the concepts of the original logic) and definability corresponds to generation.<\/jats:p>","DOI":"10.2307\/2273976","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:15:32Z","timestamp":1146939332000},"page":"865-873","source":"Crossref","is-referenced-by-count":0,"title":["On the number of generators of cylindric algebras"],"prefix":"10.1017","volume":"50","author":[{"given":"H.","family":"Andr\u00e9ka","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"I.","family":"N\u00e9meti","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120003187X_ref012","article-title":"Improving a theorem of Andr\u00e9ka, Monk and N\u00e9meti in cylindric algebric model theory","author":"Sain","journal-title":"Notre Dame Journal of Formal Logic"},{"key":"S002248120003187X_ref011","first-page":"195","article-title":"There are general rules for specifying semantics: Observations on abstract model theory","volume":"13","author":"Sain","year":"1979","journal-title":"CLandCL\u2014Computational Linguistics and Computer Languages"},{"key":"S002248120003187X_ref005","first-page":"#21599","article-title":"Review of [1]","volume":"58","author":"Comer","year":"1979","journal-title":"Mathematical Reviews"},{"key":"S002248120003187X_ref004","first-page":"1045","volume-title":"Proceedings of the Fifth International Joint Conference on Artificial Intelligence (Cambridge, Massachusetts, 1977)","volume":"2","author":"Burstall","year":"1977"},{"key":"S002248120003187X_ref006","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/025\/0376346"},{"key":"S002248120003187X_ref002","first-page":"23","volume-title":"Finite algebra and multiple-valued logic (Proceedings of the Colloquium, Szeged, 1979)","volume":"28","author":"Andr\u00e9ka","year":"1981"},{"key":"S002248120003187X_ref009","first-page":"43","article-title":"Some constructions of cylindric algebra theory applied to dynamic algebras of programs","volume":"14","author":"N\u00e9meti","year":"1980","journal-title":"CLandCL\u2014Computational Linguistics and Computer Languages"},{"key":"S002248120003187X_ref010","first-page":"561","volume-title":"Mathematical logic in computer science (Papers from the Colloquium, Salg\u00f3tarj\u00e1n, 1978)","volume":"26","author":"N\u00e9meti","year":"1981"},{"key":"S002248120003187X_ref007","volume-title":"Cylindric algebras","author":"Henkin","year":"1971"},{"key":"S002248120003187X_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF02121113"},{"key":"S002248120003187X_ref003","first-page":"25","volume-title":"Mathematical logic in computer science (Papers from the Colloquium, Salg\u00f3tarjan, 1978)","volume":"26","author":"Andr\u00e9ka","year":"1981"},{"key":"S002248120003187X_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0095613"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120003187X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T16:56:05Z","timestamp":1558544165000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120003187X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1985,12]]},"references-count":12,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1985,12]]}},"alternative-id":["S002248120003187X"],"URL":"https:\/\/doi.org\/10.2307\/2273976","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1985,12]]}}}