{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T15:29:16Z","timestamp":1773242956360,"version":"3.50.1"},"reference-count":8,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":8685,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,6]]},"abstract":"<jats:p>In this paper we discuss cylindric algebras with terms. The setting is two\u2014sorted algebras\u2014one sort for terms and one for Boolean elements. As with cylindric algebras, a cylindric algebra with terms has its roots in first order predicate logic [HMT1].<\/jats:p><jats:p>Let <jats:italic>\u03a3<\/jats:italic> be a set of sentences in a first order language with terms, equality and variables <jats:italic>u<\/jats:italic><jats:sub>0<\/jats:sub>,<jats:italic>u<\/jats:italic><jats:sub>1<\/jats:sub>,<jats:italic>u<\/jats:italic><jats:sub>2<\/jats:sub>, \u2026, Define a relation \u2261<jats:sub><jats:italic>\u03a3<\/jats:italic><\/jats:sub> on Fm, the set of formulas, by <jats:italic>\u03c6<\/jats:italic> \u2261<jats:sub><jats:italic>\u03a3<\/jats:italic><\/jats:sub><jats:italic>\u03b8<\/jats:italic> if and only if <jats:italic>\u03a3<\/jats:italic> \u22a2 <jats:italic>\u03c6<\/jats:italic> \u2194 <jats:italic>\u03b8<\/jats:italic>, and on Tm, the set of terms, by <jats:italic>\u03c4<\/jats:italic> \u2261<jats:sub><jats:italic>\u03a3<\/jats:italic><\/jats:sub><jats:italic>\u03c3<\/jats:italic> if and only if <jats:italic>\u03a3<\/jats:italic> \u22a2 <jats:italic>\u03c4 \u2248 \u03c3<\/jats:italic>. The operations +, <jats:italic>\u00b7<\/jats:italic>, <jats:italic>c<\/jats:italic><jats:sub><jats:italic>\u03ba<\/jats:italic><\/jats:sub>, 0, 1 are defined as usual on equivalence classes. Define <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026190_inline1\"\/>, where <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026190_inline2\"\/> is <jats:italic>\u03c3<\/jats:italic> with \u03c4 substituted for all occurrences of <jats:italic>u<\/jats:italic><jats:sub>\u03ba<\/jats:sub>. That the operation *<jats:sub><jats:italic>\u03ba<\/jats:italic><\/jats:sub>, for <jats:italic>\u03ba &lt; \u03b1<\/jats:italic>, is well defined follows from the first order axioms of equality. Let <jats:italic>v<\/jats:italic><jats:sub><jats:italic>\u03ba<\/jats:italic><\/jats:sub> = <jats:italic>[<\/jats:italic><jats:italic>u<\/jats:italic><jats:sub><jats:italic>\u03ba<\/jats:italic><\/jats:sub><jats:italic>]<\/jats:italic>. To establish the link between terms and Booleans, define operations <jats:italic>o<\/jats:italic><jats:sub><jats:italic>\u03ba<\/jats:italic><\/jats:sub> as follows: <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026190_inline3\"\/>, where <jats:italic>\u03c6'<\/jats:italic> is a variant of <jats:italic>\u03c6<\/jats:italic> such that <jats:italic>u<\/jats:italic><jats:sub><jats:italic>\u03ba<\/jats:italic><\/jats:sub> is free for <jats:italic>\u03c4<\/jats:italic> in <jats:italic>\u03c6\u2032<\/jats:italic> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026190_inline4\"\/> is <jats:italic>\u03c6\u2032<\/jats:italic> with <jats:italic>\u03c4<\/jats:italic> substituted for all free occurrences of <jats:italic>u<\/jats:italic><jats:sub><jats:italic>\u03ba<\/jats:italic><\/jats:sub> in <jats:italic>\u03c6\u2032<\/jats:italic>. From the first order axioms it follows that <jats:italic>o<\/jats:italic><jats:sub><jats:italic>\u03ba<\/jats:italic><\/jats:sub>, for <jats:italic>\u03ba &lt; \u03b1<\/jats:italic>, is well defined. Finally, instead of diagonal elements, we define a Boolean-valued operation on terms as follows: [<jats:italic>\u03c4<\/jats:italic>] <jats:italic>e<\/jats:italic> [<jats:italic>\u03c3<\/jats:italic>] = [<jats:italic>\u03c4 \u2248 \u03c3<\/jats:italic>].<\/jats:p>","DOI":"10.2307\/2274669","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:35:29Z","timestamp":1146954929000},"page":"854-866","source":"Crossref","is-referenced-by-count":6,"title":["Cylindric algebras with terms"],"prefix":"10.1017","volume":"55","author":[{"given":"Norman","family":"Feldman","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200026190_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0095612"},{"key":"S0022481200026190_ref005","first-page":"677","volume":"49","author":"Feldman","year":"1984","journal-title":"Representation of substitution algebras"},{"key":"S0022481200026190_ref003","first-page":"39","volume-title":"Izvestiya Vysshikh Vchebnykh Zavedeni\u012d Matematika","author":"Cirulis","year":"1988"},{"key":"S0022481200026190_ref002","volume-title":"Algebraic logic (Proceedings, Budapest, 1988)","author":"Cirulis"},{"key":"S0022481200026190_ref001","first-page":"25","volume-title":"Mathematical logic in computer science","volume":"26","author":"Andr\u00e9ka","year":"1981"},{"key":"S0022481200026190_ref007","volume-title":"Cylindric algebras","author":"Henkin","year":"1985"},{"key":"S0022481200026190_ref004","first-page":"481","volume":"47","author":"Feldman","year":"1982","journal-title":"Axiomatization of polynomial substitution algebras"},{"key":"S0022481200026190_ref006","volume-title":"Cylindric algebras","author":"Henkin","year":"1971"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200026190","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T21:13:54Z","timestamp":1558214034000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200026190\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,6]]},"references-count":8,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1990,6]]}},"alternative-id":["S0022481200026190"],"URL":"https:\/\/doi.org\/10.2307\/2274669","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,6]]}}}