{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,2,4]],"date-time":"2024-02-04T08:10:21Z","timestamp":1707034221809},"reference-count":11,"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":7132,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1994,9]]},"abstract":"<jats:title>Abstract.<\/jats:title><jats:p>One of the basic theorems in universal algebra is Birkhoff's variety theorem: the smallest equationally axiomatizable class containing a class<jats:bold>K<\/jats:bold>of algebras coincides with the class obtained by taking homomorphic images of subalgebras of direct products of elements of<jats:bold>K<\/jats:bold>. G. Gr\u00e4tzer asked whether the variety theorem is equivalent to the Axiom of Choice. In 1980, two of the present authors proved that Birkhoff's theorem can already be derived in<jats:italic>ZF<\/jats:italic>. Surprisingly, the Axiom of Foundation plays a crucial role here: we show that Birkhoff's theorem cannot be derived in<jats:italic>ZF<\/jats:italic>+<jats:italic>AC<\/jats:italic>\\{Foundation}, even if we add Foundation for Finite Sets. We also prove that the variety theorem is equivalent to a purely set-theoretical statement, the Collection Principle. This principle is independent of<jats:italic>ZF<\/jats:italic>\\{Foundation}. The second part of the paper deals with further connections between axioms of<jats:italic>ZF<\/jats:italic>-set theory and theorems of universal algebra.<\/jats:p>","DOI":"10.2307\/2275917","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:52:43Z","timestamp":1146955963000},"page":"912-923","source":"Crossref","is-referenced-by-count":0,"title":["Connections between axioms of set theory and basic theorems of universal algebra"],"prefix":"10.1017","volume":"59","author":[{"given":"H.","family":"Andr\u00e9ka","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"\u00c1.","family":"Kurucz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"I.","family":"N\u00e9meti","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200019605_ref011","first-page":"783\u2013802","volume":"49","author":"Simpson","year":"1984","journal-title":"Which set existence axioms are needed to prove the Cauchy\/Peano theorem?"},{"key":"S0022481200019605_ref009","volume-title":"Cylindric algebras, Parts I, II","author":"Henkin","year":"1971"},{"key":"S0022481200019605_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/BF01190911"},{"key":"S0022481200019605_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF02483831"},{"key":"S0022481200019605_ref007","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-77487-9","volume-title":"Universal algebra","author":"Gr\u00e4tzer","year":"1979"},{"key":"S0022481200019605_ref002","first-page":"699\u2013706","article-title":"DoesSP K \u2287 PS Kimply the axiom of choice?","volume":"21","author":"Andr\u00e9ka","year":"1980","journal-title":"Commentationes Mathematicae Universitatis Carolinae"},{"key":"S0022481200019605_ref001","first-page":"325\u2013372","article-title":"Quasirarieties of partial algebras\u2014a unifying approach towards a two-valued model theory for partial algebras","volume":"16","author":"Andr\u00e9ka","year":"1981","journal-title":"Studia Scientarium Mathematicarum Hungarica"},{"key":"S0022481200019605_ref010","volume-title":"Set theory","author":"Jech","year":"1978"},{"key":"S0022481200019605_ref006","first-page":"217","article-title":"A statement equivalent to the axiom of choice","volume":"12","author":"Gr\u00e4tzer","year":"1965","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200019605_ref005","first-page":"141\u2013181","article-title":"Countable algebra and set existence axioms","volume":"25","author":"Friedman","year":"1983","journal-title":"Annals of Pure and Applied Logic"},{"key":"S0022481200019605_ref004","volume-title":"Lecture Notes in Mathematics","volume":"223","author":"Felgner","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\/S0022481200019605","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,4]],"date-time":"2024-02-04T07:28:01Z","timestamp":1707031681000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200019605\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1994,9]]},"references-count":11,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1994,9]]}},"alternative-id":["S0022481200019605"],"URL":"https:\/\/doi.org\/10.2307\/2275917","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1994,9]]}}}