{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T15:18:05Z","timestamp":1648739885712},"reference-count":6,"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":13160,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,3]]},"abstract":"<jats:p>In [1] there were given postulates for an abstract \u201cprojective algebra\u201d which, in the words of the authors, represented a \u201cmodest beginning for a study of logic with quantifiers from a boolean point of view\u201d. In [5], D. Monk observed that the study initiated in [1] was an initial step in the development of <jats:italic>algebraic versions of logic<\/jats:italic> from which have evolved the cylindric and polyadic algebras.<\/jats:p><jats:p>Several years prior to the publication of [1], J. C. C. McKinsey [3] presented a set of postulates for the calculus of relations. Following the publication of [1], McKinsey [4] showed that every projective algebra is isomorphic to a subalgebra of a complete atomic projective algebra and thus, in view of the representation given in [1], every projective algebra is isomorphic to a projective algebra of subsets of a direct product, that is, to an <jats:italic>algebra of relations<\/jats:italic>.<\/jats:p><jats:p>Of course there has since followed an extensive development of projective algebra resulting in the multidimensional cylindric algebras [2]. However, what appears to have been overlooked is the correspondence between the Everett\u2013Ulam axiomatization and that of McKinsey.<\/jats:p><jats:p>It is the purpose of this paper to demonstrate the above, that is, we show that given a calculus of relations as defined by McKinsey it is possible to introduce projections and a partial product so that this algebra is a projective algebra and conversely, for a certain class of projective algebras it is possible to define a multiplication so that the resulting algebra is McKinsey's calculus of relations.<\/jats:p>","DOI":"10.2307\/2271948","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:46:34Z","timestamp":1146937594000},"page":"56-64","source":"Crossref","is-referenced-by-count":6,"title":["Projective algebra and the calculus of relations"],"prefix":"10.1017","volume":"43","author":[{"given":"A. R.","family":"Bednarek","sequence":"first","affiliation":[]},{"given":"S. M.","family":"Ulam","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049884_ref003","first-page":"85","volume":"5","author":"McKinsey","year":"1940","journal-title":"Postulates for the calculus of binary relations"},{"key":"S0022481200049884_ref006","first-page":"73","volume":"6","author":"Tarski","year":"1941","journal-title":"On the calculus of relations"},{"key":"S0022481200049884_ref001","doi-asserted-by":"publisher","DOI":"10.2307\/2371742"},{"key":"S0022481200049884_ref002","volume-title":"Cylindric algebras, Part I","author":"Henkin","year":"1971"},{"key":"S0022481200049884_ref004","doi-asserted-by":"publisher","DOI":"10.2307\/2372335"},{"key":"S0022481200049884_ref005","volume-title":"Stanislaw Ulam: Sets, Numbers, and Universes","author":"Beyer","year":"1974"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049884","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T16:19:50Z","timestamp":1558973990000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049884\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,3]]},"references-count":6,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1978,3]]}},"alternative-id":["S0022481200049884"],"URL":"https:\/\/doi.org\/10.2307\/2271948","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,3]]}}}