{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:55Z","timestamp":1753893835890,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>For a finite loop $Q$, let $P(Q)$ be the set of elements that can be represented as a product containing each element of $Q$ precisely once. Motivated by the recent proof of the Hall-Paige conjecture, we prove several universal implications between the following conditions: (A) $Q$ has a complete mapping, i.e. the multiplication table of $Q$ has a transversal,  (B) there is no $N \\trianglelefteq Q$ such that $|N|$ is odd and $Q\/N \\cong {\\Bbb Z}_{2^m}$ for $m \\geq 1$, and  (C) $P(Q)$ intersects the associator subloop of $Q$.  We prove $(A) \\Longrightarrow (C)$ and $(B) \\Longleftrightarrow (C)$ and show that when $Q$ is a group, these conditions reduce to familiar statements related to the Hall-Paige conjecture (which essentially says that in groups $(B) \\Longrightarrow (A))$. We also establish properties of $P(Q)$, prove a generalization of the D\u00e9nes-Hermann theorem, and present an elementary proof of a weak form of the Hall-Paige conjecture.<\/jats:p>","DOI":"10.37236\/146","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T23:31:48Z","timestamp":1578699108000},"source":"Crossref","is-referenced-by-count":1,"title":["Products of All Elements in a Loop and a Framework for Non-Associative Analogues of the Hall-Paige Conjecture"],"prefix":"10.37236","volume":"16","author":[{"given":"Kyle","family":"Pula","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2009,5,11]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r57\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r57\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T21:55:35Z","timestamp":1579298135000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1r57"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,5,11]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/146","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,5,11]]},"article-number":"R57"}}