{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T02:36:15Z","timestamp":1648953375115},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":10238,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1986,3]]},"abstract":"<jats:p>This paper is a continuation of the authors' paper [7]; in particular, we give a sharper and more useful criterion for the approximate elementary equivalence of <jats:italic>C<\/jats:italic><jats:sub><jats:italic>\u03c3<\/jats:italic><\/jats:sub>(<jats:italic>K<\/jats:italic>) spaces, where <jats:italic>K<\/jats:italic> is a totally disconnected compact Hausdorff space. (See Theorem 2 below.) As an application, we obtain a complete description of the Banach spaces <jats:italic>X<\/jats:italic> which are approximately equivalent to <jats:italic>c<\/jats:italic><jats:sub>0<\/jats:sub>. Namely, <jats:italic>X<\/jats:italic> \u2261 <jats:sub>A<\/jats:sub><jats:italic>c<\/jats:italic><jats:sub>0<\/jats:sub> iff <jats:italic>X<\/jats:italic> = <jats:italic>C<\/jats:italic><jats:sub><jats:italic>\u03c3<\/jats:italic><\/jats:sub>(<jats:italic>K<\/jats:italic>) where <jats:italic>K<\/jats:italic> is a totally disconnected compact Hausdorff space which has a dense set of isolated points and <jats:italic>\u03c3<\/jats:italic> is an involutory homeomorphism on <jats:italic>K<\/jats:italic> which has a unique fixed point <jats:italic>t<\/jats:italic>, and <jats:italic>t<\/jats:italic> is not an isolated point. (See Theorem 7.)<\/jats:p><jats:p>These results are derived from an analysis which we give of the elementary theories of structures (<jats:italic>B<\/jats:italic>, <jats:italic>\u03c3<\/jats:italic>), where <jats:italic>B<\/jats:italic> is a Boolean algebra and <jats:italic>\u03c3<\/jats:italic> is an involutory automorphism of <jats:italic>B<\/jats:italic> which leaves at most one nontrivial ultrafilter invariant. Let <jats:italic>U<\/jats:italic>(<jats:italic>\u03c3<\/jats:italic>) denote this ultrafilter, if it exists; let <jats:italic>U<\/jats:italic>(<jats:italic>\u03c3<\/jats:italic>) = <jats:italic>B<\/jats:italic> in case <jats:italic>\u03c3<\/jats:italic> leaves no nontrivial ultrafilter invariant. We show that the elementary theory of (<jats:italic>B<\/jats:italic>, <jats:italic>\u03c3<\/jats:italic>) is completely determined by the theory of (<jats:italic>B, U<\/jats:italic>(<jats:italic>\u03c3<\/jats:italic>)) (and conversely, because <jats:italic>U<\/jats:italic>(<jats:italic>\u03c3<\/jats:italic>) is definable in (<jats:italic>B<\/jats:italic>, <jats:italic>\u03c3<\/jats:italic>)). This makes possible the use of the explicit invariants given by \u00c9r\u0161ov [4] for structures (<jats:italic>B, U<\/jats:italic>) where <jats:italic>U<\/jats:italic> is an ultrafilter on <jats:italic>B<\/jats:italic>. (These generalize the Tarski invariants for Boolean algebras [15].) We also <jats:italic>use<\/jats:italic> the \u00c9r\u0161ov invariants in the proof of our main result.<\/jats:p>","DOI":"10.2307\/2273950","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:16:32Z","timestamp":1146953792000},"page":"135-146","source":"Crossref","is-referenced-by-count":0,"title":["Elementary equivalence of <i>C<\/i><sub><i>\u03c3<\/i><\/sub>(<i>K<\/i>) spaces for totally disconnected, compact Hausdorff K"],"prefix":"10.1017","volume":"51","author":[{"given":"S.","family":"Heinrich","sequence":"first","affiliation":[]},{"given":"C. Ward","family":"Henson","sequence":"additional","affiliation":[]},{"suffix":"Jr.","given":"L. 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