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 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 ι , l , r , s , l s and r s , 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 σ ℳ denote the set of all translations of a direction σ ∈ S 3 . The conditions σ ℳ = κ ℳ , where σ , κ ∈ S 3 and σ ≠ κ , define nine quasigroup varieties. Four of them are well known: L I P , R I P , M I P and C I P . The remaining five quasigroup varieties are relatively new because they are left and right inverses of C I P 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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  1. Belousov V. D., Foundations of the Theory of Quasigroups and Loops, Nauka, Moskva, 1967 (Russian). 
  2. Belousov V. D., Curkan B. V., Crossed-inverse quasigroups (CI-quasigroups), Izv. Vysš. Učebn. Zaved. Math. 1969 (1969), no. 3(82), 21–27 (Russian). 
  3. Duplák J., 10.21136/CMJ.1986.102119, Czechoslovak Math. J. 36(111) (1986), no. 4, 599–616. DOI10.21136/CMJ.1986.102119
  4. Issa A. N., On quasigroups with the left loop property, Comment. Math. Univ. Carolin. 41 (2000), no. 4, 663–669. 
  5. Izbash V., Labo N., Crossed-inverse-property groupoids, Bul. Acad. Ştiinţe Repub. Mold. Mat. (2007), no. 2, 101–106. 
  6. Keedwell A. D., Shcherbacov V. A., On m -inverse loops and quasigroups with a long inverse cycle, Australas. J. Combin. 26 (2002), 99–119. Zbl1020.20041
  7. Keedwell A. D., Shcherbacov V. A., 10.1016/S0012-365X(02)00814-2, The 18th British Combinatorial Conf., Brighton, 2001, Discrete Math. 266 (2003), no. 1–3, 275–291. DOI10.1016/S0012-365X(02)00814-2
  8. Keedwell A. D., Shcherbacov V. A., 10.1016/j.disc.2004.06.020, Discrete Math. 288 (2004), no. 1–3, 61–71. DOI10.1016/j.disc.2004.06.020
  9. Krainichuk H., Tarkovska O., Semi-symmetric isotopic closure of some group varieties and the corresponding identities, Bul. Acad. Ştiinţe Repub. Mold. Mat. 3(85) (2017), no. 3, 3–22. 
  10. Krapež A., Generalized quadratic quasigroup equations with three variables, Quasigroups Related Systems 17 (2009), no. 2, 253–270. 
  11. Lindner C. C., 10.1007/BF02018044, Period. Math. Hungar. 8 (1977), no. 1, 33–39. DOI10.1007/BF02018044
  12. Smith J. D. H., An Introduction to Quasigroups and Their Representations, Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, 2007. Zbl1122.20035
  13. Sokhatsky F. M., On pseudoisomorphy and distributivity of quasigroups, Bul. Acad. Ştiinţe Repub. Mold. Mat. 2(81) (2016), 125–142. 
  14. Sokhatsky F. M., Parastrophic symmetry in quasigroup theory, Visnik DonNU. Ser. A: Natural Sciences 1–2 (2016), 70–83. 
  15. Sokhatsky F. M., Lutsenko A. V., The bunch of varieties of inverse property quasigroups, Visnik DonNU. Ser. A: natural Sciences 1–2 (2018), 56–69. 
  16. Sokhatsky F. M., Lutsenko A. V., Classification of quasigroups according to directions of translations I, Comment. Math. Univ. Carolin. 4 (2020), no. 4, 567–579. 

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