{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,1,5]],"date-time":"2024-01-05T09:03:07Z","timestamp":1704445387386},"reference-count":15,"publisher":"Wiley","issue":"6","license":[{"start":{"date-parts":[[2012,10,19]],"date-time":"2012-10-19T00:00:00Z","timestamp":1350604800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2012,11]]},"abstract":"Abstract<\/jats:title>Let \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathbb {N}^\\mathbb {N}$\\end{document}<\/jats:styled-content> be the semigroup of all mappings on the natural numbers \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathbb {N}$\\end{document}<\/jats:styled-content>, and let U<\/jats:italic> and V<\/jats:italic> be subsets of \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathbb {N}^\\mathbb {N}$\\end{document}<\/jats:styled-content>. We write U<\/jats:italic>\u227cV<\/jats:italic> if there exists a countable subset C<\/jats:italic> of \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathbb {N}^\\mathbb {N}$\\end{document}<\/jats:styled-content> such that U<\/jats:italic> is contained in the subsemigroup generated by V<\/jats:italic> and C<\/jats:italic>. We give several results about the structure of the preorder \u227c. In particular, we show that a certain statement about this preorder is equivalent to the Continuum Hypothesis.<\/jats:p>The preorder \u227c is analogous to one introduced by Bergman and Shelah on subgroups of the symmetric group on \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathbb {N}$\\end{document}<\/jats:styled-content>. The results in this paper suggest that the preorder on subsemigroups of \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathbb {N}^\\mathbb {N}$\\end{document}<\/jats:styled-content> is much more complicated than that on subgroups of the symmetric group.<\/jats:p>","DOI":"10.1002\/malq.201200002","type":"journal-article","created":{"date-parts":[[2012,10,19]],"date-time":"2012-10-19T09:32:11Z","timestamp":1350639131000},"page":"424-433","source":"Crossref","is-referenced-by-count":3,"title":["The Bergman\u2010Shelah preorder on transformation semigroups"],"prefix":"10.1002","volume":"58","author":[{"given":"Zak","family":"Mesyan","sequence":"first","affiliation":[]},{"given":"James D.","family":"Mitchell","sequence":"additional","affiliation":[]},{"given":"Micha\u0142","family":"Morayne","sequence":"additional","affiliation":[]},{"given":"Yann H.","family":"P\u00e9resse","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,10,19]]},"reference":[{"key":"e_1_2_8_2_1","doi-asserted-by":"publisher","DOI":"10.4064\/fm-25-1-5-6"},{"key":"e_1_2_8_3_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00012-006-1959-z"},{"key":"e_1_2_8_4_1","volume-title":"Permutation groups, London Mathematical Society Student Texts Vol. 45","author":"Cameron P. J.","year":"1999"},{"key":"e_1_2_8_5_1","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.50.6.1143"},{"key":"e_1_2_8_6_1","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.51.1.105"},{"key":"e_1_2_8_7_1","doi-asserted-by":"publisher","DOI":"10.1307\/mmj\/1028999028"},{"key":"e_1_2_8_8_1","doi-asserted-by":"crossref","DOI":"10.1515\/9781400881635","volume-title":"The Consistency of the Continuum Hypothesis, Annals of Mathematics Studies Vol. 3","author":"G\u00f6del K.","year":"1940"},{"key":"e_1_2_8_9_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4190-4"},{"key":"e_1_2_8_10_1","volume-title":"Set theory, Studies in Logic and the Foundations of Mathematics Vol. 102","author":"Kunen K.","year":"1983"},{"key":"e_1_2_8_11_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00233-007-0731-9"},{"key":"e_1_2_8_12_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2010.05.001"},{"key":"e_1_2_8_13_1","doi-asserted-by":"crossref","first-page":"129","DOI":"10.4064\/cm-53-1-129-132","article-title":"On differentiability of Peano type functions","volume":"53","author":"Morayne M.","year":"1987","journal-title":"Colloq. Math."},{"key":"e_1_2_8_14_1","volume-title":"Hypoth\u00e8se du continu, Monografie Matematyczne Vol. 4","author":"Sierpi\u0144ski W.","year":"1934"},{"key":"e_1_2_8_15_1","doi-asserted-by":"publisher","DOI":"10.4064\/fm-24-1-209-212"},{"key":"e_1_2_8_16_1","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-444-51621-3.50002-5"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.201200002","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.201200002","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,30]],"date-time":"2023-10-30T22:09:17Z","timestamp":1698703757000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.201200002"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,10,19]]},"references-count":15,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2012,11]]}},"alternative-id":["10.1002\/malq.201200002"],"URL":"https:\/\/doi.org\/10.1002\/malq.201200002","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,10,19]]}}}