Classification of quasigroups according to directions of translations II

Fedir Sokhatsky; Alla Lutsenko

Commentationes Mathematicae Universitatis Carolinae (2021)

  • Volume: 62, Issue: 3, page 309-323
  • ISSN: 0010-2628

Abstract

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In each quasigroup there are defined six types of translations: the left, right and middle translations and their inverses. Two translations may coincide as permutations of , and yet be different when considered upon the web of the quasigroup. We shall call each of the translation types a direction and will associate it with one of the elements and , i.e., the elements of a symmetric group . Properties of the directions are considered in part 1 of “Classification of quasigroups according to directions of translations I” by F. M. Sokhatsky and A. V. Lutsenko. Let denote the set of all translations of a direction . The conditions , where and , define nine quasigroup varieties. Four of them are well known: , , and . The remaining five quasigroup varieties are relatively new because they are left and right inverses of variety and generalization of commutative, left and right symmetric quasigroups.

How to cite

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Sokhatsky, Fedir, and Lutsenko, Alla. "Classification of quasigroups according to directions of translations II." Commentationes Mathematicae Universitatis Carolinae 62.3 (2021): 309-323. <http://eudml.org/doc/297756>.

@article{Sokhatsky2021,
abstract = {In each quasigroup $Q$ there are defined six types of translations: the left, right and middle translations and their inverses. Two translations may coincide as permutations of $Q$, and yet be different when considered upon the web of the quasigroup. We shall call each of the translation types a direction and will associate it with one of the elements $\iota , l, r, s, ls $ and $rs$, i.e., the elements of a symmetric group $S_3$. Properties of the directions are considered in part 1 of “Classification of quasigroups according to directions of translations I” by F. M. Sokhatsky and A. V. Lutsenko. Let $\{^\{\sigma \}\mathcal \{M\}\}$ denote the set of all translations of a direction $\sigma \in S_\{3\}$. The conditions $\{^\{\sigma \}\mathcal \{M\}\}=\{^\{\kappa \}\mathcal \{M\}\}$, where $\sigma ,\kappa \in S_\{3\}$ and $\sigma \ne \kappa $, define nine quasigroup varieties. Four of them are well known: $LIP$, $RIP$, $MIP$ and $CIP$. The remaining five quasigroup varieties are relatively new because they are left and right inverses of $ CIP$ variety and generalization of commutative, left and right symmetric quasigroups.},
author = {Sokhatsky, Fedir, Lutsenko, Alla},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {quasigroup; parastrophe; parastrophic symmetry; parastrophic orbit; translation; direction; matrix quasigroup},
language = {eng},
number = {3},
pages = {309-323},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Classification of quasigroups according to directions of translations II},
url = {http://eudml.org/doc/297756},
volume = {62},
year = {2021},
}

TY - JOUR
AU - Sokhatsky, Fedir
AU - Lutsenko, Alla
TI - Classification of quasigroups according to directions of translations II
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2021
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62
IS - 3
SP - 309
EP - 323
AB - In each quasigroup $Q$ there are defined six types of translations: the left, right and middle translations and their inverses. Two translations may coincide as permutations of $Q$, and yet be different when considered upon the web of the quasigroup. We shall call each of the translation types a direction and will associate it with one of the elements $\iota , l, r, s, ls $ and $rs$, i.e., the elements of a symmetric group $S_3$. Properties of the directions are considered in part 1 of “Classification of quasigroups according to directions of translations I” by F. M. Sokhatsky and A. V. Lutsenko. Let ${^{\sigma }\mathcal {M}}$ denote the set of all translations of a direction $\sigma \in S_{3}$. The conditions ${^{\sigma }\mathcal {M}}={^{\kappa }\mathcal {M}}$, where $\sigma ,\kappa \in S_{3}$ and $\sigma \ne \kappa $, define nine quasigroup varieties. Four of them are well known: $LIP$, $RIP$, $MIP$ and $CIP$. The remaining five quasigroup varieties are relatively new because they are left and right inverses of $ CIP$ variety and generalization of commutative, left and right symmetric quasigroups.
LA - eng
KW - quasigroup; parastrophe; parastrophic symmetry; parastrophic orbit; translation; direction; matrix quasigroup
UR - http://eudml.org/doc/297756
ER -

References

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