{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T02:39:03Z","timestamp":1774406343084,"version":"3.50.1"},"reference-count":12,"publisher":"World Scientific Pub Co Pte Ltd","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2021,6]]},"abstract":"<jats:p> We investigate finitary functions from [Formula: see text] to [Formula: see text] for a square-free number [Formula: see text]. We show that the lattice of all clones on the square-free set [Formula: see text] which contain the addition of [Formula: see text] is finite. We provide an upper bound for the cardinality of this lattice through an injective function to the direct product of the lattices of all [Formula: see text]-linearly closed clonoids, [Formula: see text], to the [Formula: see text] power, where [Formula: see text]. These lattices are studied in [S.\u00a0Fioravanti, Closed sets of finitary functions between products of finite fields of pair-wise coprime order, preprint (2020), arXiv:2009.02237] and there we can find an upper bound for their cardinality. Furthermore, we prove that these clones can be generated by a set of functions of arity at most [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s0218196721500302","type":"journal-article","created":{"date-parts":[[2021,4,12]],"date-time":"2021-04-12T14:53:07Z","timestamp":1618239187000},"page":"623-638","source":"Crossref","is-referenced-by-count":8,"title":["Expansions of abelian square-free groups"],"prefix":"10.1142","volume":"31","author":[{"given":"Stefano","family":"Fioravanti","sequence":"first","affiliation":[{"name":"Stefano Fioravanti, Institut f\u00fcr Algebra, Johannes Kepler Universit\u00e4t Linz, 4040 Linz, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2021,4,10]]},"reference":[{"key":"S0218196721500302BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/s10474-006-0530-x"},{"key":"S0218196721500302BIB002","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196715500344"},{"key":"S0218196721500302BIB003","first-page":"193","volume-title":"Multiple-valued Logic","volume":"8","author":"Bulatov A. A.","year":"2002"},{"key":"S0218196721500302BIB004","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5350-1"},{"key":"S0218196721500302BIB007","doi-asserted-by":"publisher","DOI":"10.1142\/S021819679900014X"},{"issue":"1","key":"S0218196721500302BIB008","first-page":"51","volume":"33","author":"Kreinecker S.","year":"2019","journal-title":"J. Mult.Val. Logic Soft Comput."},{"key":"S0218196721500302BIB009","doi-asserted-by":"publisher","DOI":"10.1007\/1-4020-3817-8_8"},{"key":"S0218196721500302BIB010","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196708004597"},{"key":"S0218196721500302BIB011","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-5547-1"},{"key":"S0218196721500302BIB012","unstructured":"E. L. Post ,  The Two-valued Iterative Systems of Mathematical Logic, Annals of Mathematics Studies,  No. 5  (Princeton University Press,  Princeton, N. J.,  1941),  viii+122 pp."},{"key":"S0218196721500302BIB013","doi-asserted-by":"publisher","DOI":"10.1007\/BF02849497"},{"key":"S0218196721500302BIB014","series-title":"S\u00e9minaire de Math\u00e9matiques Sup\u00e9rieures [Seminar on Higher Mathematics]","volume-title":"Clones in Universal Algebra","volume":"99","author":"Szendrei \u00c1.","year":"1986"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196721500302","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,6,24]],"date-time":"2021-06-24T04:21:09Z","timestamp":1624508469000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196721500302"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,10]]},"references-count":12,"journal-issue":{"issue":"04","published-print":{"date-parts":[[2021,6]]}},"alternative-id":["10.1142\/S0218196721500302"],"URL":"https:\/\/doi.org\/10.1142\/s0218196721500302","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,10]]}}}