Maximal nonassociativity via fields

Lisonĕk, P

doi:10.1007/s10623-020-00800-4

Cited by 3 publications

(10 citation statements)

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“…These quasigroups will play a central role in this paper. Petr Lisoněk [10] has independently and concurrently obtained some of the results in our paper, also by using quadratic orthomorphisms. The following basic properties of quadratic orthomorphisms are known, see [6,13].…”

Section: Introductionmentioning

confidence: 60%

“…For (a, b) ∈ Σ and i, j, r, s ∈ {0, 1} denote by E rs ij (a, b) the set of all nontrivial solutions (x, y) to the Associativity Equation (6) that fulfil (10) i = η(x), j = η(−y), r = η(ψ(x) − y) and s = η(x − y − ψ(−y)).…”

Section: Quadratic Orthomorphismsmentioning

confidence: 99%

“…Since (1, ρ) is a solution of (11), then every solution to (11) is equal to (x, ρx) for some x ∈ F. However, if x is a nonsquare, then this solution does not fulfil (10). Hence The equations (11) have been obtained by combining the row (r, s) = (1, 0) of ( 9) with the row (i, j) = (0, 1) of ( 8).…”

Section: Quadratic Orthomorphismsmentioning

confidence: 99%

“…This work was supported in part by Australian Research Council grant DP150100506. We thank Petr Lisoněk for keeping us informed regarding his paper [10] and Gabriel Verret for posing the question to us as to whether a maximally nonassociative quasigroup can have trivial automorphism group.…”

mentioning

confidence: 99%

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Maximally nonassociative quasigroups via quadratic orthomorphisms

Drápal¹,

Wanless²

2021

Algebraic Combinatorics

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Section: Introductionmentioning

confidence: 60%

Section: Quadratic Orthomorphismsmentioning

confidence: 99%

Section: Quadratic Orthomorphismsmentioning

confidence: 99%

mentioning

confidence: 99%

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Maximally nonassociative quasigroups via quadratic orthomorphisms

Drápal¹,

Wanless²

2021

Algebraic Combinatorics

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“…If a = b ∈ F \ {0, 1}, then Equation (1-2) defines a quasigroup in which u * (v * u) = (u * v) * u for all u, v ∈ F. This means that such a quasigroup is never maximally nonassociative. If q 13, then there always exists (a, b) ∈ Σ(F q ) such that Q a,b is maximally nonassociative [6,13]. This paper is concerned with the density of such (a, b).…”

Section: Introductionmentioning

confidence: 99%

On the Number of Quadratic Orthomorphisms That Produce Maximally Nonassociative Quasigroups

Drápal¹,

Wanless

2023

J. Aust. Math. Soc.

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Let q be an odd prime power and suppose that $a,b\in \mathbb {F}_q$ 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 $q\equiv 1 \bmod 4$ , then $\lim \sigma (q)/q^2 = \alpha $ , and if $q \equiv 3 \bmod 4$ , then $\lim \sigma (q)/q^2 = \beta $ .

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