Maximal nonassociativity via fields
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Maximal nonassociativity via fields Petr Lisonek ˇ 1 Received: 24 November 2019 / Revised: 25 July 2020 / Accepted: 27 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We say that (x, y, z) ∈ Q 3 is an associative triple in a quasigroup (Q, ∗) if (x ∗ y) ∗ z = x ∗ (y ∗ z). Let a(Q) denote the number of associative triples in Q. It is easy to show that a(Q) ≥ |Q|, and we call the quasigroup maximally nonassociative if a(Q) = |Q|. It was conjectured that maximally nonassociative quasigroups do not exist when |Q| > 1. Drápal and Lisonˇek recently refuted this conjecture by proving the existence of maximally nonassociative quasigroups for a certain infinite set of orders |Q|. In this paper we prove the existence of maximally nonassociative quasigroups for a much larger set of orders |Q|. Our main tools are finite fields and the Weil bound on quadratic character sums. Unlike in the previous work, our results are to a large extent constructive. Keywords Nonassociativity · Latin square · Finite field · Weil bound
1 Maximally nonassociative quasigroups A quasigroup (Q, ∗) is a set Q with a binary operation ∗ such that for all a, b ∈ Q there exist unique x, y ∈ Q such that a ∗ x = b and y ∗ a = b. Hence a binary operation on a finite set yields a quasigroup if and only if its multiplication table is a Latin square. In this paper we only deal with finite quasigroups. We call a triple (x, y, z) ∈ Q 3 associative if (x ∗ y) ∗ z = x ∗ (y ∗ z), and we denote the number of associative triples in Q by a(Q). For each c ∈ Q there exist x, y such that c∗x = c and y ∗ c = c. Then (y, c, x) is an associative triple and it follows that a(Q) ≥ |Q|. We call Q maximally nonassociative when a(Q) = |Q|. The existence of such quasigroups has been investigated for at least four decades [10,11]. It was shown [9] that quasigroups with few associative triples can be used in the design of hash functions in cryptography. Grošek and Horák conjectured that maximally nonassociative quasigroups do not exist when |Q| > 1
Communicated by D. Panario. Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).
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Petr Lisonˇek [email protected] Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada
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P. Lisonˇek
[9, Conjecture 1.2]. An example of a maximally nonassociative quasigroup of order 9 was found recently [4]. No example of order greater than 1 and less than 9 exists, and no example of order 10 exists [3,4]. It is known [9, Theorem 1.1] that any maximally nonassociative quasigroup must be idempotent, that is, x ∗ x = x for all x ∈ Q. Very recently Drápal and Lisonˇek [2] used Dickson’s quadratic nearfields to prove existence of an infinite set of maximally nonassociative quasigroups. Specifically they proved:
Theorem 1.1 [2, Corollary 5.8] Let m = 23k r where k ≥ 0 is an integer and r is odd. There exists a maximally nonassociative quasigroup of order m 2 .
In this paper we greatly extend the set of orders for which the exist
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