## Aleš Drápal, Jan Hora

*Nonassociative triples in involutory loops and in loops of small order*

Comment.Math.Univ.Carolin. 61,4 (2020) 459-479.**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

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