Abstract:A loop of order $n$ possesses at least $3n^2-3n+1$ associative triples. However, no loop of order $n>1$ that achieves this bound seems to be known. If the loop is involutory, then it possesses at least $3n^2-2n$ associative triples. Involutory loops with $3n^2-2n$ associative triples can be obtained by prolongation of certain maximally nonassociative quasigroups whenever $n-1$ is a prime greater than or equal to $13$ or $n-1=p^{2k}$, $p$ an odd prime. For orders $n\le 9$ the minimum number of associative triples is reported for both general and involutory loops, and the structure of the corresponding loops is described.
Keywords: quasigroup; loop; prolongation; involutory loop; associative triple; maximally nonassociative
DOI: DOI 10.14712/1213-7243.2020.037
AMS Subject Classification: 20N05 05B15