{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T02:31:18Z","timestamp":1775097078790,"version":"3.50.1"},"reference-count":19,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":4338,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[2002,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In 1941, Tarski published an abstract, finitely axiomatized version of the theory of binary relations, called the theory of <jats:italic>relation algebras<\/jats:italic>. He asked whether every model of his abstract theory could be represented as a concrete algebra of binary relations. He and J\u00f3nsson obtained some initial, positive results for special classes of abstract relation algebras. But Lyndon showed, in 1950, that in general the answer to Tarski's question is negative. Monk proved later that the answer remains negative even if one adjoins finitely many new axioms to Tarski's system. In this paper we describe a far-reaching generalization of the positive results of J\u00f3nsson and Tarski, as well as of some later, related results of Maddux. We construct a class of concrete models of Tarski's axioms\u2014called <jats:italic>coset relation algebras<\/jats:italic>\u2014that are very close in spirit to algebras of binary relations, but are built using systems of groups and cosets instead of elements of a base set. The models include all algebras of binary relations, and many non-representable relation algebras as well. We prove that every atomic relation algebra satisfying a certain measurability condition\u2014a condition generalizing the conditions imposed by J\u00f3nsson and Tarski\u2014is essentially isomorphic to a coset relation algebra. The theorem raises the possibility of providing a positive solution to Tarski's problem by using coset relation algebras instead of the standard algebras of binary relations.<\/jats:p>","DOI":"10.2178\/bsl\/1182353852","type":"journal-article","created":{"date-parts":[[2008,4,3]],"date-time":"2008-04-03T17:08:52Z","timestamp":1207242532000},"page":"38-64","source":"Crossref","is-referenced-by-count":12,"title":["Groups and Algebras of Binary Relations"],"prefix":"10.1017","volume":"8","author":[{"given":"Steven","family":"Givant","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hajnal","family":"Andr\u00e9ka","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600005217_ref014","volume":"III","author":"Schr\u00f6der","journal-title":"Vorlesungen \u00fcber die Algebra der Logik (exakte Logik)"},{"key":"S1079898600005217_ref002","first-page":"82","volume-title":"The mathematical analysis of logic, being an essay towards a calculus of deductive reasoning","author":"Boole","year":"1847"},{"key":"S1079898600005217_ref017","article-title":"Contributions to the theory of models. II","volume":"57","author":"Tarski","year":"1954","journal-title":"Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A"},{"key":"S1079898600005217_ref007","doi-asserted-by":"publisher","DOI":"10.2307\/2372074"},{"key":"S1079898600005217_ref010","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1991-1049616-1"},{"key":"S1079898600005217_ref015","doi-asserted-by":"publisher","DOI":"10.2307\/2268577"},{"key":"S1079898600005217_ref016","article-title":"Contributions to the theory of models. I","volume":"57","author":"Tarski","year":"1954","journal-title":"Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A"},{"key":"S1079898600005217_ref008","doi-asserted-by":"publisher","DOI":"10.2307\/2269177"},{"key":"S1079898600005217_ref009","doi-asserted-by":"publisher","DOI":"10.2307\/1969375"},{"key":"S1079898600005217_ref018","article-title":"Contributions to the theory of models. 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IV, and on the logic of relations","volume":"10","author":"de Morgan","year":"1864","journal-title":"Transactions of the Cambridge Philosophical Society"}],"container-title":["Bulletin of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1079898600005217","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,7]],"date-time":"2019-05-07T01:33:14Z","timestamp":1557192794000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1079898600005217\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,3]]},"references-count":19,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2002,3]]}},"alternative-id":["S1079898600005217"],"URL":"https:\/\/doi.org\/10.2178\/bsl\/1182353852","relation":{},"ISSN":["1079-8986","1943-5894"],"issn-type":[{"value":"1079-8986","type":"print"},{"value":"1943-5894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,3]]}}}