| Title:
|
Classification of quasigroups according to directions of translations II (English) |
| Author:
|
Sokhatsky, Fedir |
| Author:
|
Lutsenko, Alla |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
62 |
| Issue:
|
3 |
| Year:
|
2021 |
| Pages:
|
309-323 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
quasigroup |
| Keyword:
|
parastrophe |
| Keyword:
|
parastrophic symmetry |
| Keyword:
|
parastrophic orbit |
| Keyword:
|
translation |
| Keyword:
|
direction |
| Keyword:
|
matrix quasigroup |
| MSC:
|
20N05 |
| idMR:
|
MR4331285 |
| DOI:
|
10.14712/1213-7243.2021.021 |
| . |
| Date available:
|
2021-10-20T08:10:54Z |
| Last updated:
|
2023-10-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149151 |
| . |
| Reference:
|
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