{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T12:04:33Z","timestamp":1753877073939,"version":"3.41.2"},"reference-count":16,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2025,1,24]],"date-time":"2025-01-24T00:00:00Z","timestamp":1737676800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"publisher","award":["12101466"],"award-info":[{"award-number":["12101466"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,5,23]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>A function is boundedly finite-to-one if there is a natural number $k$ such that each point has at most $k$ inverse images. In this paper, we prove in $\\textsf{ZF}$ (i.e. the Zermelo\u2013Fraenkel set theory without the axiom of choice) several results concerning this notion, among which are the following:<\/jats:p>\n               <jats:p>(1)For each infinite set $A$ and natural number $n$, there is no boundedly finite-to-one function from $\\mathcal{S}(A)$ to $\\mathcal{S}_{\\leq n}(A)$, where $\\mathcal{S}(A)$ is the set of all permutations of $A$ and $\\mathcal{S}_{\\leq n}(A)$ is the set of all permutations of $A$ moving at most $n$ points.<\/jats:p>\n               <jats:p>(2) For each infinite set $A$, there is no boundedly finite-to-one function from $\\mathcal{B}(A)$ to $\\textrm{fin}(A)$, where $\\mathcal{B}(A)$ is the set of all partitions of $A$ such that every block is finite and $\\textrm{fin}(A)$ is the set of all finite subsets of $A$.<\/jats:p>","DOI":"10.1093\/jigpal\/jzae130","type":"journal-article","created":{"date-parts":[[2025,1,24]],"date-time":"2025-01-24T04:19:59Z","timestamp":1737692399000},"source":"Crossref","is-referenced-by-count":0,"title":["Boundedly finite-to-one functions"],"prefix":"10.1093","volume":"33","author":[{"given":"Xiao","family":"Hu","sequence":"first","affiliation":[{"name":"School of Philosophy , Wuhan University, No. 299 Bayi Road, Wuhan, Hubei Province 430072, People\u2019s Republic of China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guozhen","family":"Shen","sequence":"additional","affiliation":[{"name":"School of Philosophy , Wuhan University, No. 299 Bayi Road, Wuhan, Hubei Province 430072, People\u2019s Republic of China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2025,1,24]]},"reference":[{"key":"2025052605514100900_ref1","first-page":"75","article-title":"\u00dcber eine elementare Frage der Mannigfaltigkeitslehre","volume":"1","author":"Cantor","year":"1891","journal-title":"Jahresber Dtsch Math-Ver"},{"key":"2025052605514100900_ref2","doi-asserted-by":"crossref","first-page":"185","DOI":"10.4064\/fm-93-3-185-195","article-title":"Factorials of infinite cardinals","volume":"93","author":"Dawson Jr","year":"1976","journal-title":"Fund Math"},{"key":"2025052605514100900_ref3","doi-asserted-by":"crossref","first-page":"1251","DOI":"10.2178\/jsl\/1067620184","article-title":"Finite-to-one maps","volume":"68","author":"Forster","year":"2003","journal-title":"J Symb Log"},{"key":"2025052605514100900_ref4","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-60231-8","volume-title":"Combinatorial Set Theory: With a Gentle Introduction to Forcing","author":"Halbeisen","year":"2017"},{"key":"2025052605514100900_ref5","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-0348-0224-6","volume-title":"Proofs of the Cantor\u2013Bernstein Theorem: A Mathematical 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