Classification of quasigroups according to directions of translations I

Fedir Sokhatsky; Alla Lutsenko

Commentationes Mathematicae Universitatis Carolinae (2020)

  • Volume: 61, Issue: 4, page 567-579
  • ISSN: 0010-2628

Abstract

top
It is proved that every translation in a quasigroup has two independent parameters. One of them is a bijection of the carrier set. The second parameter is called a direction here. Properties of directions in a quasigroup are considered in the first part of the work. In particular, totally symmetric, semisymmetric, commutative, left and right symmetric and also asymmetric quasigroups are characterized within these concepts. The sets of translations of the same direction are under consideration in the second part of the work. Coincidence of these sets defines nine varieties, among them are varieties of L I P , R I P , M I P and C I P quasigroups. Quasigroups in other five varieties also have some invertibility properties.

How to cite

top

Sokhatsky, Fedir, and Lutsenko, Alla. "Classification of quasigroups according to directions of translations I." Commentationes Mathematicae Universitatis Carolinae 61.4 (2020): 567-579. <http://eudml.org/doc/296965>.

@article{Sokhatsky2020,
abstract = {It is proved that every translation in a quasigroup has two independent parameters. One of them is a bijection of the carrier set. The second parameter is called a direction here. Properties of directions in a quasigroup are considered in the first part of the work. In particular, totally symmetric, semisymmetric, commutative, left and right symmetric and also asymmetric quasigroups are characterized within these concepts. The sets of translations of the same direction are under consideration in the second part of the work. Coincidence of these sets defines nine varieties, among them are varieties of $LIP$, $RIP$, $MIP$ and $CIP$ quasigroups. Quasigroups in other five varieties also have some invertibility properties.},
author = {Sokhatsky, Fedir, Lutsenko, Alla},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {quasigroup; parastrophe; parastrophic symmetry; parastrophic orbit; translation; direction},
language = {eng},
number = {4},
pages = {567-579},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Classification of quasigroups according to directions of translations I},
url = {http://eudml.org/doc/296965},
volume = {61},
year = {2020},
}

TY - JOUR
AU - Sokhatsky, Fedir
AU - Lutsenko, Alla
TI - Classification of quasigroups according to directions of translations I
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2020
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 61
IS - 4
SP - 567
EP - 579
AB - It is proved that every translation in a quasigroup has two independent parameters. One of them is a bijection of the carrier set. The second parameter is called a direction here. Properties of directions in a quasigroup are considered in the first part of the work. In particular, totally symmetric, semisymmetric, commutative, left and right symmetric and also asymmetric quasigroups are characterized within these concepts. The sets of translations of the same direction are under consideration in the second part of the work. Coincidence of these sets defines nine varieties, among them are varieties of $LIP$, $RIP$, $MIP$ and $CIP$ quasigroups. Quasigroups in other five varieties also have some invertibility properties.
LA - eng
KW - quasigroup; parastrophe; parastrophic symmetry; parastrophic orbit; translation; direction
UR - http://eudml.org/doc/296965
ER -

References

top
  1. Belousov V. D., The group associated with a quasigroup, Mat. Issled. 4 (1969), no. 3, 21–39 (Russian). MR0262407
  2. Dénes J., Keedwell A. D., Latin Squares and Their Applications, Academic Press, New York, 1974. MR0351850
  3. Duplák J., On quasi-identities of transitive quasigroups, Math. Slovaca 34 (1984), no. 3, 281–294. MR0756985
  4. Duplák J., 10.21136/CMJ.1986.102119, Czechoslovak Math. J. 36(111) (1986), no. 4, 599–616. MR0863190DOI10.21136/CMJ.1986.102119
  5. Duplák J., A parastrophic equivalence in quasigroups, Quasigroups Related Systems 7 (2000), 7–14. MR1848538
  6. Krainichuk H., Tarkovska O., Semi-symmetric isotopic closure of some group varieties and the corresponding identities, Bul. Acad. Ştiinţe Repub. Mold. Mat. Number 3(85) (2017), 3–22. MR3760535
  7. Smith J. D. H., An introduction to quasigroups and their representations, Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, 2007. MR2268350
  8. Sokhatsky F. M., On pseudoisomorphy and distributivity of quasigroups, Bul. Acad. Ştiinţe Repub. Mold. Mat. 2(81) (2016), 125–142. MR3570801
  9. Sokhatsky F. M., Parastrophic symmetry in quasigroup theory, Visnik DonNU. Ser. A Natural Sciences 1–2 (2016), 70–83. 
  10. 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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.