Distributive and Anti-distributive Mendelsohn Triple Systems

Donovan, Diane; Griggs, T. S.; McCourt, Thomas A.; Opršal, Jakub; Stanovský, David

doi:10.4153/cmb-2015-053-2

Cited by 7 publications

(20 citation statements)

References 11 publications

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“…Table 2 displays these numbers for every nonassociative commutative Moufang loop of order 81 and 243. The entries for order 81 can be found already in [9,16,17] and have been independently verified by our calculations. The entries in the last row can be found in [3] and have also been independently verified.…”

Section: 1supporting

confidence: 82%

Distributive and trimedial quasigroups of order 243

Jedlička

Stanovský

Vojtźchovský

2017

Discrete Mathematics

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We enumerate three classes of non-medial quasigroups of order 243 = 3 5 up to isomorphism. There are 17004 non-medial trimedial quasigroups of order 243 (extending the work of Kepka, Bénéteau and Lacaze), 92 non-medial distributive quasigroups of order 243 (extending the work of Kepka and Němec), and 6 non-medial distributive Mendelsohn quasigroups of order 243 (extending the work of Donovan, Griggs, McCourt, Opršal and Stanovský).The enumeration technique is based on affine representations over commutative Moufang loops, on properties of automorphism groups of commutative Moufang loops, and on computer calculations with the LOOPS package in GAP.2000 Mathematics Subject Classification. Primary: 20N05. Secondary: 05B15, 05B07.

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Section: 1supporting

confidence: 82%

Distributive and trimedial quasigroups of order 243

Jedlička

Stanovský

Vojtźchovský

2017

Discrete Mathematics

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“…Proposition 4.2, equivalent to Theorem 2.12. (iii) in [13], clarifies the case of r i = 2. is the companion matrix of f (X).…”

Section: The Ramified Casementioning

confidence: 85%

“…The left self-distributive (LD) rule (1.1) x(yz) = (xy)(xz) is a well-studied phenomenon appearing in the theory of knots [21], Hopf algebras [1], symmetric spaces [27], term rewriting systems [10, Ch. V], quasigroups [35], and combinatorial designs [13]. The results of this paper lie at the intersection of the latter two subjects, and what is more, bring algebraic number theory into the fold of disciplines connected by self-distributivity.…”

Section: Introductionmentioning

confidence: 84%

“…Multiply both sides of (2.15) by (2 − ζ). Then The following lemma and its proof appear under Lemma 2.2 in [13]. Next, we generalize the argument of [13, Th.…”

Section: Introductionmentioning

confidence: 90%

“…In [13], the authors classify distributive Mendelsohn quasigroups of orders p and p 2 , for any prime p. They also enumerate, using GAP [16], isomorphism classes for prime powers less than 1000. The following theorem is the key to our improvement of these results.…”

Section: Introductionmentioning

confidence: 99%

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Distributive Mendelsohn triple systems and the Eisenstein integers

Nowak¹

2019

Preprint

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We define a Mendelsohn triple system (MTS) with selfdistributive quasigroup multiplication and order coprime with 3 to be distributive, non-ramified (DNR). We classify, up to isomorphism, all DNR MTS and enumerate isomorphism classes (extending the work of Donovan, Griggs, McCourt, Opršal, and Stanovský). The classification is accomplished via the representation theory of the Eisenstein integers,Containing the class of DNR MTS is that of MTS with an entropic (linear over an abelian group) quasigroup operation. Partial results on the classification of entropic MTS with order divisible by 3 are given, and a complete classification is conjectured. We also prove that for any entropic MTS, the qualities of being non-ramified, pure, and self-orthogonal are equivalent. We introduce the varieties RE and LE of (resp. right and left) Eisenstein quasigroups, whose respective linear representation theories correspond to the alternative presentation Z[X]/(X 2 + X + 1) of the Eisenstein integers.

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Affine Mendelsohn triple systems and the Eisenstein integers

Nowak

2020

J of Combinatorial Designs

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We define a Mendelsohn triple system (MTS) of order coprime with 3, and having multiplication affine over an abelian group, to be affine, nonramified. By exhibiting a one‐to‐one correspondence between isomorphism classes of affine MTS and those of modules over the Eisenstein integers, we solve the isomorphism problem for affine, nonramified MTS and enumerate these isomorphism classes (extending the work of Donovan, Griggs, McCourt, Opršal, and Stanovský). As a consequence, all entropic MTSs of order coprime with 3 and distributive MTS of order coprime with 3 are classified. Partial results on the isomorphism problem for affine MTS with order divisible by 3 are given, and a complete classification is conjectured. We also prove that for any affine MTS, the qualities of being nonramified, pure, and self‐orthogonal are equivalent.

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Distributive and Anti-distributive Mendelsohn Triple Systems

Cited by 7 publications

References 11 publications

Distributive and trimedial quasigroups of order 243

Distributive and trimedial quasigroups of order 243

Distributive Mendelsohn triple systems and the Eisenstein integers

Affine Mendelsohn triple systems and the Eisenstein integers

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