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Nonassociativity in two operations
Lehká, Martina ; Drápal, Aleš (advisor) ; Patáková, Zuzana (referee)
This thesis follows up mainly on the research of Drápal and Valent, who studied the nonassociativity of one quasigroup operation. Its central objective is to examine the number of triples (x, y, z) ∈ Q3 such that (x ∗ y) ◦ z = x ∗ (y ◦ z), where (Q, ∗) and (Q, ◦) are two quasigroups, |Q| = n. Let a2(C) be the number of such triples in a quasigroup couple C. Call it the associativity index. Denote by a2(n) the minimal a2(C), where C is a couple of order n. By averaging the associativity index over all the principal isotopes of a quasigroup couple, we prove that a2(n) ≤ n2 (1+1/(n−1)), n > 2. We then characterize the couples C that, on average, attain a2(C) = n2 and we prove that this value is an improved upper bound on a2(n), n > 2. Furthermore, we begin research on couples of quasigroups isotopic to groups. Lastly, we present computational results with examples, including a2(4) = 8 and a2(5) = 9. 1
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