{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,10]],"date-time":"2024-09-10T16:31:50Z","timestamp":1725985910159},"reference-count":48,"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":5799,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[1998,3]]},"abstract":"<jats:p>The purpose of this communication is to survey a theory of liftings, as developed in author's thesis ([8]). The first result in this area was Shelah's construction of a model of set theory in which every automorphism of<jats:italic>P<\/jats:italic>(\u2115)\/ Fin, where Fin is the ideal of finite sets, is<jats:italic>trivial<\/jats:italic>, or inother words, it is induced by a function mapping integers into integers ([33]). (It is a classical result of W. Rudin [31] that under the Continuum Hypothesis there are automorphisms other than trivial ones.) Soon afterwards, Velickovic ([47]), was able to extract from Shelah's argument the fact that every automorphism of<jats:italic>P<\/jats:italic>(\u2115)\/ Fin with a Baire-measurable lifting has to be trivial. This, for instance, implies that in Solovay's model ([36]) all automorphisms are trivial. Later on, an axiomatic approach was adopted and Shelah's conclusion was drawn first from the Proper Forcing Axiom (PFA) ([34]) and then from the milder Open Coloring Axiom (OCA) and Martin's Axiom (MA) ([48], see \u00a75 for definitions). Both shifts from the quotient<jats:italic>P<\/jats:italic>(\u2115)\/ Fin to quotients over more general ideals<jats:italic>P<\/jats:italic>(\u2115)\/<jats:italic>I<\/jats:italic>and from automorphisms to arbitrary ho-momorphisms were made by Just in a series of papers ([14]-[17]), motivated by some problems in algebra ([7, pp. 38\u201339], [43, I.12.11], [45, Q48]) and topology ([46, p. 537]).<\/jats:p>","DOI":"10.2307\/421005","type":"journal-article","created":{"date-parts":[[2006,5,7]],"date-time":"2006-05-07T07:11:08Z","timestamp":1146985868000},"page":"37-54","source":"Crossref","is-referenced-by-count":7,"title":["Completely Additive Liftings"],"prefix":"10.1017","volume":"4","author":[{"given":"Ilijas","family":"Farah","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S107989860000737X_ref037","doi-asserted-by":"publisher","DOI":"10.4064\/sm-67-1-13-43"},{"key":"S107989860000737X_ref018","first-page":"411","article-title":"On certain Boolean algebras P(\u03c9)\/I","volume":"285","author":"Just","year":"1984","journal-title":"Transactions of the American Mathematical Society"},{"key":"S107989860000737X_ref024","first-page":"1","article-title":"Progres recents sur le probleme de Maharam d'apres N. 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