{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T08:28:00Z","timestamp":1648628880652},"reference-count":2,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":23203,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1950,9]]},"abstract":"<jats:p>The object of this note is to classify the isomorphism types of all complete atomic proper relation algebras. In particular we find that the number of abstractly distinct complete atomic proper finite relation algebras of 2<jats:sup><jats:italic>m<\/jats:italic><\/jats:sup> elements is equal to the number of distinct partitions of the positive integer <jats:italic>m<\/jats:italic> into summands which are perfect squares.<\/jats:p><jats:p>A relation algebra is called complete atomic if its Boolean algebra is complete atomic. A proper relation algebra is a relation algebra whose elements are relations. Lyndon has recently proved, by constructing an appropriate finite relation algebra, that not every complete atomic relation algebra is isomorphic to a proper relation algebra.<\/jats:p><jats:p>Henceforth, let us consider a complete atomic proper relation algebra. The atoms {\u3008<jats:italic>a, b<\/jats:italic>\u3009}, {\u3008<jats:italic>c, d<\/jats:italic>\u3009} will be called <jats:italic>connected<\/jats:italic> if and only if {<jats:italic>a, b<\/jats:italic>} \u22c2 {<jats:italic>c, d<\/jats:italic>} is not empty. Let the relation of connectedness over the set <jats:italic>B<\/jats:italic> of all atoms be denoted by ~. For brevity let <jats:italic>B<\/jats:italic> = {<jats:italic>b<\/jats:italic><jats:sub>1<\/jats:sub>,<jats:italic>b<\/jats:italic><jats:sub>2<\/jats:sub>, \u2026}. The relation ~ is obviously reflexive and symmetric. But ~ is not transitive, since <jats:italic>a, b, c, d<\/jats:italic> distinct implies {\u3008<jats:italic>a<\/jats:italic>, 0\u3009} ~ {\u3008<jats:italic>b, c<\/jats:italic>\u3009}, {\u3008<jats:italic>b, c<\/jats:italic>\u3009} ~ {\u3008<jats:italic>c, d<\/jats:italic>\u3009}, and {\u3008<jats:italic>a, b<\/jats:italic>\u3009} \u2241 {\u3008<jats:italic>c, d<\/jats:italic>\u3009}.<\/jats:p>","DOI":"10.2307\/2266785","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T15:12:14Z","timestamp":1146928334000},"page":"197-198","source":"Crossref","is-referenced-by-count":1,"title":["On complete atomic proper relation algebras"],"prefix":"10.1017","volume":"15","author":[{"given":"Frank","family":"Harary","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200103160_ref002","first-page":"513","article-title":"The representation of algebras of relations","volume":"55","author":"Lyndon","year":"1949","journal-title":"Bulletin of the American Mathematical Society"},{"key":"S0022481200103160_ref001","first-page":"80","article-title":"Representation problems for relation algebras","volume":"54","author":"Jonsson","year":"1948","journal-title":"Bulletin of the American Mathematical Society"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200103160","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,8]],"date-time":"2019-06-08T06:13:56Z","timestamp":1559974436000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200103160\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1950,9]]},"references-count":2,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1950,9]]}},"alternative-id":["S0022481200103160"],"URL":"https:\/\/doi.org\/10.2307\/2266785","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1950,9]]}}}