{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,25]],"date-time":"2025-02-25T13:58:24Z","timestamp":1740491904569,"version":"3.38.0"},"reference-count":10,"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":16082,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1970,3]]},"abstract":"The algebras studied in this paper were suggested to the author by William Craig as a possible substitute for cylindric algebras. Both kinds of algebras may be considered as algebraic versions of first-order logic. Cylindric algebras can be introduced as follows. Let \u2112 be a first-order language, and let be an \u2112-structure. We assume that \u2112 has a simple infinite sequence \u03bd<\/jats:italic>0<\/jats:sub>, \u03bd<\/jats:italic>1<\/jats:sub>, \u2026 of individual variables, and we take as known what it means for a sequence \u03bd<\/jats:italic>0<\/jats:sub>, \u03bd<\/jats:italic>1<\/jats:sub>, \u2026 of individual variables, and we take as known what it means for a sequence x<\/jats:italic> = \u3008x<\/jats:italic>0<\/jats:sub>, x<\/jats:italic>1<\/jats:sub>, \u2026\u3009 of elements of to satisfy a formula \u03d5<\/jats:italic> of \u2112<\/jats:italic> in . Let \u03d5<\/jats:italic><\/jats:sup> be the collection of all sequences x<\/jats:italic> which satisfy \u03d5<\/jats:italic> in . We can perform certain natural operations on the sets \u03d5<\/jats:italic><\/jats:sup>, of basic model-theoretic significance: Boolean operations = cylindrifications diagonal elements (0-ary operations) . In this way we make the class of all sets \u03d5<\/jats:italic><\/jats:sup> into an algebra; a natural abstraction gives the class of all cylindric set algebras (of dimension \u03c9<\/jats:italic>). Thus this method of constructing an algebraic counterpart of first-order logic is based upon the notion of satisfaction of a formula by an infinite sequence of elements. Since, however, a formula has only finitely many variables occurring in it, it may seem more natural to consider satisfaction by a finite sequence of elements; then \u03d5<\/jats:italic><\/jats:sup> becomes a collection of finite sequences of varying ranks (cf. Tarski [10]). In forming an algebra of sets of finite sequences it turns out to be possible to get by with only finitely many operations instead of the infinitely many ci<\/jats:sub><\/jats:italic>'s and dij<\/jats:sub><\/jats:italic>'s of cylindric algebras. Let be the class of all algebras of sets of finite sequences (an exact definition is given in \u00a71).<\/jats:p>","DOI":"10.1017\/s0022481200092185","type":"journal-article","created":{"date-parts":[[2014,3,13]],"date-time":"2014-03-13T08:46:30Z","timestamp":1394700390000},"page":"19-28","source":"Crossref","is-referenced-by-count":1,"title":["On an algebra of sets of finite sequences"],"prefix":"10.1017","volume":"35","author":[{"given":"J.","family":"Donald Monk","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200092185_ref009","first-page":"331","article-title":"Nonfinitizability of classes of representable cylindric algebras","volume":"34","author":"Monk","year":"1969","journal-title":"this Journal"},{"key":"S0022481200092185_ref007","first-page":"344","article-title":"Nonfinitizability of classes of representable polyadic algebras","volume":"34","author":"Johnson","year":"1969","journal-title":"this Journal"},{"key":"S0022481200092185_ref001","first-page":"1","volume-title":"Constructivity in mathematics","author":"Bernays","year":"1957"},{"key":"S0022481200092185_ref004","doi-asserted-by":"publisher","DOI":"10.4064\/fm-51-3-195-228"},{"key":"S0022481200092185_ref002","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1307\/mmj\/1028990028","article-title":"A note on cylindric and polyadic algebras","volume":"3","author":"Copeland","year":"1955","journal-title":"The Michigan mathematical journal"},{"key":"S0022481200092185_ref003","first-page":"55","volume-title":"Theory of models","author":"Craig","year":"1965"},{"key":"S0022481200092185_ref006","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/002\/0124250"},{"key":"S0022481200092185_ref010","first-page":"261","article-title":"Der Wahrheitsbegriff in den formatierten Sprachen","volume":"1","author":"Tarski","year":"1935","journal-title":"Studia phil."},{"key":"S0022481200092185_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/002\/0124250"},{"key":"S0022481200092185_ref008","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1307\/mmj\/1028999131","article-title":"On representable relation algebras","volume":"11","author":"Monk","year":"1964","journal-title":"The Michigan mathematical journal"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200092185","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,1]],"date-time":"2019-06-01T15:24:27Z","timestamp":1559402667000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200092185\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1970,3]]},"references-count":10,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1970,3]]}},"alternative-id":["S0022481200092185"],"URL":"https:\/\/doi.org\/10.1017\/s0022481200092185","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"type":"print","value":"0022-4812"},{"type":"electronic","value":"1943-5886"}],"subject":[],"published":{"date-parts":[[1970,3]]}}}