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- W3026692196 abstract "Abstract Let q be an odd prime power and suppose that $a,bin mathbb {F}_q$ are such that $ab$ and $(1{-}a)(1{-}b)$ are nonzero squares. Let $Q_{a,b} = (mathbb {F}_q,*)$ be the quasigroup in which the operation is defined by $u*v=u+a(v{-}u)$ if $v-u$ is a square, and $u*v=u+b(v{-}u)$ if $v-u$ is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies $x*(y*z) = (x*y)*z Leftrightarrow x=y=z$ . Denote by $sigma (q)$ the number of $(a,b)$ for which $Q_{a,b}$ is maximally nonassociative. We show that there exist constants $alpha approx 0.029,08$ and $beta approx 0.012,59$ such that if $qequiv 1 bmod 4$ , then $lim sigma (q)/q^2 = alpha $ , and if $q equiv 3 bmod 4$ , then $lim sigma (q)/q^2 = beta $ ." @default.
- W3026692196 created "2020-05-29" @default.
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- W3026692196 date "2023-02-20" @default.
- W3026692196 modified "2023-09-26" @default.
- W3026692196 title "ON THE NUMBER OF QUADRATIC ORTHOMORPHISMS THAT PRODUCE MAXIMALLY NONASSOCIATIVE QUASIGROUPS" @default.
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- W3026692196 doi "https://doi.org/10.1017/s1446788722000386" @default.
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