{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T03:24:21Z","timestamp":1648956261374},"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":5855,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1998,3]]},"abstract":"This paper explores algebraic aspects of two modifications of the usual account of first-order quantifiers.<\/jats:p>Standard first-order quantificational logic is modelled algebraically by cylindric algebras<\/jats:italic>. Prime examples of these are algebras whose members are sets of sequences: given a first-order model U<\/jats:italic> for a language that is based on the set {\u03c5\u03ba<\/jats:sub>: \u03ba < \u03b1} of variables, each formula \u03c6 is represented by the set<\/jats:p><\/jats:disp-formula><\/jats:p>of all those \u03b1-length sequences x<\/jats:italic> = \u3008x<\/jats:italic>\u03ba<\/jats:sub>: \u03ba < \u03b1\u3009 that satisfy \u03c6 in U<\/jats:italic>. Such a sequence provides a value-assignment to the variables (\u03c5\u03ba<\/jats:sub> is assigned value x<\/jats:italic>\u03ba<\/jats:sub>), but it may also be viewed geometrically as a point in the \u03b1-dimensional Cartesian space<\/jats:italic>\u03b1<\/jats:sup>U<\/jats:italic> of all \u03b1-length sequences whose terms come from the underlying set U<\/jats:italic> of U<\/jats:italic>. Then existential quantification is represented by the operation of cylindrification<\/jats:italic>. To explain this, define a binary relation T<\/jats:italic>\u03ba<\/jats:sub> on sequences by putting x<\/jats:italic>T<\/jats:italic>\u03ba<\/jats:sub>y<\/jats:italic> if and only if x<\/jats:italic> and y<\/jats:italic> differ at most at their \u03bath coordinate, i.e.,<\/jats:p><\/jats:disp-formula><\/jats:p>Then for any set X<\/jats:italic> \u2286 \u03b1<\/jats:sup>U<\/jats:italic>, the set<\/jats:p><\/jats:disp-formula><\/jats:p>is the \u201ccylinder\u201d generated by translation of X<\/jats:italic> parallel to the \u03bath coordinate axis in \u03b1<\/jats:sup>U<\/jats:italic>. Given the standard semantics for the existential quantifier \u2203\u03c5\u03ba<\/jats:sub> as<\/jats:p><\/jats:disp-formula><\/jats:p>it is evident that<\/jats:p><\/jats:disp-formula><\/jats:p>","DOI":"10.2307\/2586595","type":"journal-article","created":{"date-parts":[[2006,4,18]],"date-time":"2006-04-18T18:43:03Z","timestamp":1145385783000},"page":"163-184","source":"Crossref","is-referenced-by-count":4,"title":["Relativised quantification: Some canonical varieties of sequence-set algebras"],"prefix":"10.1017","volume":"63","author":[{"given":"Hajnal","family":"Andr\u00e9ka","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert","family":"Goldblatt","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Istv\u00e1n","family":"N\u00e9meti","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200015401_ref004","first-page":"671","volume":"309","year":"1988","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200015401_ref003","volume-title":"A finite equational axiomatization of Gn, manuscript","year":"1995"},{"key":"S0022481200015401_ref001","volume-title":"Correspondence and completeness for generalised quantifiers","year":"1994"},{"key":"S0022481200015401_ref018","first-page":"177","volume-title":"Logic colloquium '92","year":"1995"},{"key":"S0022481200015401_ref016","first-page":"561","volume-title":"Mathematical logic in computer science","volume":"26","year":"1981"},{"key":"S0022481200015401_ref015","first-page":"253","volume-title":"Algebraic methods in logic and in computer science","year":"1993"},{"key":"S0022481200015401_ref005","year":"1996","journal-title":"Journal of Philosophical Logic"},{"key":"S0022481200015401_ref011","volume-title":"Cylindric set algebras I","year":"1981"},{"key":"S0022481200015401_ref009","volume-title":"Cylindric algebras I","year":"1971"},{"key":"S0022481200015401_ref010","volume-title":"Cylindric algebras II","year":"1985"},{"key":"S0022481200015401_ref008","volume-title":"Handbook of Algebraic Logic","year":"1996"},{"key":"S0022481200015401_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BF01181878"},{"key":"S0022481200015401_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(89)90032-8"},{"key":"S0022481200015401_ref012","doi-asserted-by":"publisher","DOI":"10.2307\/2372123"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200015401","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,11]],"date-time":"2019-05-11T19:03:53Z","timestamp":1557601433000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200015401\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,3]]},"references-count":14,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1998,3]]}},"alternative-id":["S0022481200015401"],"URL":"https:\/\/doi.org\/10.2307\/2586595","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1998,3]]}}}