On the Number of Quadratic Orthomorphisms That Produce Maximally Nonassociative Quasigroups

Drápal, Aleš; Wanless, Ian M.

doi:10.1017/s1446788722000386

Cited by 3 publications

(6 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Such squares are called quadratic Latin squares. Quadratic Latin squares have previously been used to construct perfect 1-factorisations [34,15,1], mutually orthogonal Latin squares [14,13], atomic Latin squares [34], Falconer varieties [1], and maximally non-associative quasigroups [9,10]. Quadratic Latin squares are the main focus of this paper.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Cycles of quadratic Latin squares and anti-perfect $1$-factorisations

Jack¹

2023

Preprint

View full text Add to dashboard Cite

A Latin square of order n is an n × n matrix of n symbols, such that each symbol occurs exactly once in each row and column. For an odd prime power q let F q denote the finite field of order q. A quadratic Latin square is a Latin square L[a, b] defined by,for some {a, b} ⊆ F q such that ab and (a − 1)(b − 1) are quadratic residues in F q . Quadratic Latin squares have previously been used to construct perfect 1-factorisations, mutually orthogonal Latin squares and atomic Latin squares. We first characterise quadratic Latin squares which are devoid of 2 × 2 Latin subsquares. Let G be a graph and F a 1-factorisation of G. If the union of every pair of 1-factors in F induces a Hamiltonian cycle in G then F is called perfect, and if there is no pair of 1-factors in F which induce a Hamiltonian cycle in G then F is called anti-perfect. We use quadratic Latin squares to construct new examples of anti-perfect 1-factorisations of complete graphs and complete bipartite graphs. We also demonstrate that for each odd prime p, there are only finitely many orders q, which are powers of p, such that quadratic Latin squares of order q could be used to construct perfect 1-factorisations of complete graphs or complete bipartite graphs.

show abstract

Section: Introductionmentioning

confidence: 99%

“…10. A permutation α ∈ Γ contains a Type One c-cycle if and only if it satisfies a sequence in X c,α / ∼.…”

mentioning

confidence: 99%

Cycles of quadratic Latin squares and anti-perfect $1$-factorisations

Jack¹

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…The condition

\{ab,(a-1)(b-1)\}\subseteq {{\rm{ {\mathcal R} }}}_{q}

ensures that

{\rm{ {\mathcal L} }}[a,b]

is a Latin square [16]. Quadratic Latin squares have previously been used to construct perfect 1‐factorisations [1, 17, 38], mutually orthogonal Latin squares [15, 16], atomic Latin squares [38], Falconer varieties [1] and maximally nonassociative quasigroups [10, 11]. Quadratic Latin squares are the main focus of this paper.…”

Section: Introductionmentioning

confidence: 99%

“…[ , ]  is a Latin square [16]. Quadratic Latin squares have previously been used to construct perfect 1-factorisations [1,17,38], mutually orthogonal Latin squares [15,16], atomic Latin squares [38], Falconer varieties [1] and maximally nonassociative quasigroups [10,11]. Quadratic Latin squares are the main focus of this paper.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Cycles of quadratic Latin squares and antiperfect 1‐factorisations

Jack

2023

J of Combinatorial Designs

View full text Add to dashboard Cite

A Latin square of order is an matrix of symbols, such that each symbol occurs exactly once in each row and column. For an odd prime power let denote the finite field of order . A quadratic Latin square is a Latin square defined by for some such that and are quadratic residues in . Quadratic Latin squares have previously been used to construct perfect 1‐factorisations, mutually orthogonal Latin squares and atomic Latin squares. We first characterise quadratic Latin squares which are devoid of Latin subsquares. Let be a graph and a 1‐factorisation of . If the union of every pair of 1‐factors in induces a Hamiltonian cycle in then is called perfect, and if there is no pair of 1‐factors in which induce a Hamiltonian cycle in then is called antiperfect. We use quadratic Latin squares to construct new examples of antiperfect 1‐factorisations of complete graphs and complete bipartite graphs. We also demonstrate that for each odd prime , there are only finitely many orders , which are powers of , such that quadratic Latin squares of order could be used to construct perfect 1‐factorisations of complete graphs or complete bipartite graphs.

show abstract

Isomorphisms of quadratic quasigroups

Drápal,

Wanless

2023

Proceedings of the Edinburgh Mathematical Society

Self Cite

View full text Add to dashboard Cite

Let $\mathbb F$ be a finite field of odd order and $a,b\in\mathbb F\setminus\{0,1\}$ be such that $\chi(a) = \chi(b)$ and $\chi(1-a)=\chi(1-b)$, where χ is the extended quadratic character on $\mathbb F$. Let $Q_{a,b}$ be the quasigroup over $\mathbb F$ defined by $(x,y)\mapsto x+a(y-x)$ if $\chi(y-x) \geqslant 0$, and $(x,y) \mapsto x+b(y-x)$ if $\chi(y-x) = -1$. We show that $Q_{a,b} \cong Q_{c,d}$ if and only if $\{a,b\} = \{\alpha(c),\alpha(d)\}$ for some $\alpha\in \operatorname{Aut}(\mathbb F)$. We also characterize $\operatorname{Aut}(Q_{a,b})$ and exhibit further properties, including establishing when $Q_{a,b}$ is a Steiner quasigroup or is commutative, entropic, left or right distributive, flexible or semisymmetric. In proving our results, we also characterize the minimal subquasigroups of $Q_{a,b}$.

show abstract

On the Number of Quadratic Orthomorphisms That Produce Maximally Nonassociative Quasigroups

Cited by 3 publications

References 14 publications

Cycles of quadratic Latin squares and anti-perfect $1$-factorisations

Cycles of quadratic Latin squares and anti-perfect $1$-factorisations

Cycles of quadratic Latin squares and antiperfect 1‐factorisations

Isomorphisms of quadratic quasigroups

Contact Info

Product

Resources

About