Distributive and trimedial quasigroups of order 243

Jedlička, Přemysl; Stanovský, David; Vojtźchovský, Petr

doi:10.1016/j.disc.2016.08.022

Cited by 5 publications

(2 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Propositions 3.3 and 3.5 specify the possible quasigroup isomoprhism classes for the factors, and these agree with (3.4). ) is part of [20]. Theorem 3.7 allows us to calculate d(p n ) for arbitrary powers, as long as p = 3.…”

Section: Introductionmentioning

confidence: 99%

Distributive Mendelsohn triple systems and the Eisenstein integers

Nowak¹

2019

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

Distributive Mendelsohn triple systems and the Eisenstein integers

Nowak¹

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…This identity is also referred to as the medial law (cf. [13,28,53]). Hence, Problem 4.2.1 is equivalent to the isomorphism problem for entropic Mendelsohn quasigroups.…”

Section: Affine Mts Entropicity and Distributivitymentioning

confidence: 99%

Linear aspects of equational triality in quasigroups

Nowak¹

View full text Add to dashboard Cite

show abstract

Affine Mendelsohn triple systems and the Eisenstein integers

Nowak

2020

J of Combinatorial Designs

View full text Add to dashboard Cite

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.

show abstract

Distributive and trimedial quasigroups of order 243

Cited by 5 publications

References 18 publications

Distributive Mendelsohn triple systems and the Eisenstein integers

Distributive Mendelsohn triple systems and the Eisenstein integers

Linear aspects of equational triality in quasigroups

Affine Mendelsohn triple systems and the Eisenstein integers

Contact Info

Product

Resources

About