{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,29]],"date-time":"2023-09-29T08:15:35Z","timestamp":1695975335546},"reference-count":19,"publisher":"Wiley","issue":"6","license":[{"start":{"date-parts":[[2017,12,18]],"date-time":"2017-12-18T00:00:00Z","timestamp":1513555200000},"content-version":"vor","delay-in-days":17,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"funder":[{"DOI":"10.13039\/501100004945","name":"Nankai University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004945","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100008973","name":"University of North Texas","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100008973","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2017,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We study Schauder equivalence relations, which are Borel equivalence relations induced by actions of Banach spaces with Schauder bases. Firstly, we show that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201600074-math-0001.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201600074:malq201600074-math-0001\" \/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201600074-math-0002.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201600074:malq201600074-math-0002\" \/> are minimal Schauder equivalence relations. Then, we prove that neither of them is Borel reducible to the quotient <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201600074-math-0003.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201600074:malq201600074-math-0003\" \/> where <jats:italic>T<\/jats:italic> is the Tsirelson space. This implies that they cannot form a basis for the Schauder equivalence relations. In addition, we apply an argument of Farah to show that every basis for the Schauder equivalence relations, if such exist, has to be of cardinality <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/malq201600074-math-0004.png\" xlink:title=\"urn:x-wiley:09425616:media:malq201600074:malq201600074-math-0004\" \/>.<\/jats:p>","DOI":"10.1002\/malq.201600074","type":"journal-article","created":{"date-parts":[[2017,12,18]],"date-time":"2017-12-18T16:18:52Z","timestamp":1513613932000},"page":"491-500","source":"Crossref","is-referenced-by-count":1,"title":["On Schauder equivalence relations"],"prefix":"10.1002","volume":"63","author":[{"given":"Xin","family":"Ma","sequence":"first","affiliation":[{"name":"Department of Mathematics Texas A&amp;M University College Station TX 77843 United States of America"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2017,12,18]]},"reference":[{"key":"e_1_2_8_2_1","volume-title":"Topics in Banach Space Theory","author":"Albiac F.","year":"2006"},{"key":"e_1_2_8_3_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02764600"},{"key":"e_1_2_8_4_1","doi-asserted-by":"publisher","DOI":"10.4153\/CMB-1975-025-8"},{"key":"e_1_2_8_5_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760508"},{"key":"e_1_2_8_6_1","volume-title":"Tsirelson's Space","author":"Casazza P. 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