{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:29:42Z","timestamp":1760236182935,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,10,27]],"date-time":"2021-10-27T00:00:00Z","timestamp":1635292800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In the framework of finitely supported atomic sets, by using the notion of atomic cardinality and the T-finite support principle (a closure property for supports in some higher-order constructions), we present some finiteness properties of the finitely supported binary relations between infinite atomic sets. Of particular interest are finitely supported Dedekind-finite sets because they do not contain finitely supported, countably infinite subsets. We prove that the infinite sets \u2118fs(Ak\u00d7Al), \u2118fs(Ak\u00d7\u2118m(A)), \u2118fs(\u2118n(A)\u00d7Ak) and \u2118fs(\u2118n(A)\u00d7\u2118m(A)) do not contain uniformly supported infinite subsets. Moreover, the functions space ZAm does not contain a uniformly supported infinite subset whenever Z does not contain a uniformly supported infinite subset. All these sets are Dedekind-finite in the framework of finitely supported structures.<\/jats:p>","DOI":"10.3390\/sym13112028","type":"journal-article","created":{"date-parts":[[2021,10,27]],"date-time":"2021-10-27T23:24:42Z","timestamp":1635377082000},"page":"2028","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Finitely Supported Binary Relations between Infinite Atomic Sets"],"prefix":"10.3390","volume":"13","author":[{"given":"Andrei","family":"Alexandru","sequence":"first","affiliation":[{"name":"Institute of Computer Science, Romanian Academy, 700505 Ia\u015fi, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8166-9456","authenticated-orcid":false,"given":"Gabriel","family":"Ciobanu","sequence":"additional","affiliation":[{"name":"Faculty of Computer Science, Alexandru Ioan Cuza University, 700506 Ia\u015fi, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,27]]},"reference":[{"key":"ref_1","unstructured":"Barwise, J. 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