Few associative triples, isotopisms and groups

Drápal, Aleš; Valent, Viliam

doi:10.1007/s10623-017-0341-9

Cited by 5 publications

(30 citation statements)

References 3 publications

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“…The first eight elements of the sequence

bolda (n)

are 1, 4, 9, 16, 15, 16, 17, and 16 (cf. Drápal & Valent ).…”

Section: Discussionmentioning

confidence: 96%

“…Isotopic quasigroups may have a completely different number of associative triples. In fact, there exists a result that may be regarded as a demonstration that there is no direct casual relationship between isotopy and associativity: Consider all isotopes of a quasigroup

Q

. Represent each isotope by a pair of permutations of

Q

and compute the average associativity index.…”

Section: Introductionmentioning

confidence: 99%

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High nonassociativity in order 8 and an associative index estimate

Drápal

Valent

2018

J of Combinatorial Designs

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Let Q be a quasigroup. Put bolda ( Q ) = ∣ { ( x , y , z ) ∈ Q 3 ; x ( y z ) = ( x y ) z } ∣ and assume that ∣ Q ∣ = n. Let δ L and δ R be the number of left and right translations of Q that are fixed point free. Put δ ( Q ) = δ normalL + δ normalR. Denote by i ( Q ) the number of idempotents of Q. It is shown that bolda ( Q ) ≥ 2 n − i ( Q ) + δ ( Q ). Call Q extremely nonassociative if bolda ( Q ) = 2 n − i ( Q ). The paper reports what seems to be the first known example of such a quasigroup, with n = 8, bolda ( Q ) = 16, and i ( Q ) = 0. It also provides supporting theory for a search that verified bolda ( Q ) ≥ 16 for all quasigroups of order 8.

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“…The first eight elements of the sequence

bolda (n)

are 1, 4, 9, 16, 15, 16, 17, and 16 (cf. Drápal & Valent ).…”

Section: Discussionmentioning

confidence: 96%

Q

. Represent each isotope by a pair of permutations of

Q

and compute the average associativity index.…”

Section: Introductionmentioning

confidence: 99%

High nonassociativity in order 8 and an associative index estimate

Drápal

Valent

2018

J of Combinatorial Designs

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“…Both the pseudocoboundary and the pseudococyclic frameworks over Latin rectangles described in this paper are based on the existence of non-associative triples within a Latin rectangle. As was already indicated in the introductory section and in Section 3, the study of this type of triple in the case of dealing with Latin squares has received particular attention in the recent literature [32][33][34][35][36] because of its possible application in different areas as cryptography [30,31]. A comprehensive study of non-associative triples in the case of dealing with Latin rectangles instead of Latin squares is established, therefore, as natural further work.…”

Section: Conclusion and Further Workmentioning

confidence: 98%

“…Of particular interest in our study, the relevant role that quasigroups with few associative triples play in cryptography [30,31] is remarkable. It is so that quasigroups with a high amount of non-associative triples are receiving particular attention [32][33][34][35][36].…”

Section: Introductionmentioning

confidence: 99%

Pseudococyclic Partial Hadamard Matrices over Latin Rectangles

Ganfornina¹,

Álvarez²,

Frau³

et al. 2021

Mathematics

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The classical design of cocyclic Hadamard matrices has recently been generalized by means of both the notions of the cocycle of Hadamard matrices over Latin rectangles and the pseudococycle of Hadamard matrices over quasigroups. This paper delves into this topic by introducing the concept of the pseudococycle of a partial Hadamard matrix over a Latin rectangle, whose fundamentals are comprehensively studied and illustrated.

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“…New results on isotopisms [134], autotopisms [135], automorphisms [136] and parastrophisms [137] of quasigroups have continued to progress until the present day. Furthermore, different applications of autotopisms of quasigroups in Cryptography have been developed [138,139].…”

Section: Quasigroups Latin Squares and Related Structuresmentioning

confidence: 99%

A Historical Perspective of the Theory of Isotopisms

2018

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In the middle of the twentieth century, Albert and Bruck introduced the theory of isotopisms of non-associative algebras and quasigroups as a generalization of the classical theory of isomorphisms in order to study and classify such structures according to more general symmetries. Since then, a wide range of applications have arisen in the literature concerning the classification and enumeration of different algebraic and combinatorial structures according to their isotopism classes. In spite of that, there does not exist any contribution dealing with the origin and development of such a theory. This paper is a first approach in this regard.

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Few associative triples, isotopisms and groups

Cited by 5 publications

References 3 publications

High nonassociativity in order 8 and an associative index estimate

High nonassociativity in order 8 and an associative index estimate

Pseudococyclic Partial Hadamard Matrices over Latin Rectangles

A Historical Perspective of the Theory of Isotopisms

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