Notes on the number of associative triples. (English) Zbl 0786.20042

Some questions concerning the number of associative triples in a quasigroup are discussed. The authors divide the paper into three parts. In part 1 some definitions and examples are given. For instance, they present two examples of a quasigroup \(H\) of order 6 with \(a(H) = 16\) and a quasigroup \(Q\) of order 6 with \(a(Q) = 189\) (\(a(Q)\) is the associative spectrum of \(n\), for every \(n\geq 1\), where \(Q\) runs over the quasigroups of order \(n\)). In part 2 are given three examples, and in part 3 they discuss the group distance and the numbers \(b_{\min}(n)\), where \(b_{\min}(n)\) denote the minimum of the numbers \(b(Q)\), for \(Q\) running over all the nonassociative quasigroups of order \(n\geq 3\).


20N05 Loops, quasigroups
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