{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T21:59:53Z","timestamp":1771019993518,"version":"3.50.1"},"reference-count":23,"publisher":"World Scientific Pub Co Pte Lt","issue":"07","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2015,11]]},"abstract":"<jats:p>It was shown by van Rees [Subsquares and transversals in latin squares, Ars Combin. 29B (1990) 193\u2013204] that a latin square of order [Formula: see text] has at most [Formula: see text] latin subsquares of order [Formula: see text]. He conjectured that this bound is only achieved if [Formula: see text] is a power of [Formula: see text]. We show that it can only be achieved if [Formula: see text]. We also state several conditions that are equivalent to achieving the van Rees bound. One of these is that the Cayley table of a loop achieves the van Rees bound if and only if every loop isotope has exponent [Formula: see text]. We call such loops van Rees loops and show that they form an equationally defined variety. We also show that: (1) In a van Rees loop, any subloop of index 3 is normal. (2) There are exactly six nonassociative van Rees loops of order [Formula: see text] with a nontrivial nucleus and at least 1 with all nuclei trivial. (3) Every commutative van Rees loop has the weak inverse property. (4) For each van Rees loop there is an associated family of Steiner quasigroups.<\/jats:p>","DOI":"10.1142\/s0218196715500356","type":"journal-article","created":{"date-parts":[[2015,10,7]],"date-time":"2015-10-07T02:42:54Z","timestamp":1444185774000},"page":"1159-1177","source":"Crossref","is-referenced-by-count":3,"title":["Loops with exponent three in all isotopes"],"prefix":"10.1142","volume":"25","author":[{"given":"Michael","family":"Kinyon","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Denver, 2280 S. Vine St. Denver, CO 80208, USA"}]},{"given":"Ian M.","family":"Wanless","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences Monash University, Vic 3800, Australia"}]}],"member":"219","published-online":{"date-parts":[[2015,12,8]]},"reference":[{"key":"S0218196715500356BIB001","doi-asserted-by":"publisher","DOI":"10.1002\/jcd.21355"},{"key":"S0218196715500356BIB002","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2014.01.002"},{"key":"S0218196715500356BIB003","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-013-2809-1"},{"key":"S0218196715500356BIB004","first-page":"175","volume":"51","author":"Browning J.","year":"2010","journal-title":"Comment. Math. Univ. 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