| Title:
|
Quasigroups arisen by right nuclear extension (English) |
| Author:
|
Nagy, Péter T. |
| Author:
|
Stuhl, Izabella |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
53 |
| Issue:
|
3 |
| Year:
|
2012 |
| Pages:
|
391-395 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The aim of this paper is to prove that a quasigroup $Q$ with right unit is isomorphic to an $f$-extension of a right nuclear normal subgroup $G$ by the factor quasigroup $Q/G$ if and only if there exists a normalized left transversal $\Sigma \subset Q$ to $G$ in $Q$ such that the right translations by elements of $\Sigma$ commute with all right translations by elements of the subgroup $G$. Moreover, a loop $Q$ is isomorphic to an $f$-extension of a right nuclear normal subgroup $G$ by a loop if and only if $G$ is middle-nuclear, and there exists a normalized left transversal to $G$ in $Q$ contained in the commutant of $G$. (English) |
| Keyword:
|
extension of quasigroups |
| Keyword:
|
right nucleus |
| Keyword:
|
quasigroup with right unit |
| Keyword:
|
transversal |
| MSC:
|
20N05 |
| idZBL:
|
Zbl 1257.20069 |
| idMR:
|
MR3017838 |
| . |
| Date available:
|
2012-08-31T11:38:20Z |
| Last updated:
|
2014-10-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142932 |
| . |
| Reference:
|
[1] Nagy P.T., Strambach K.: Schreier loops.Czechoslovak Math. J. 58 (133) (2008), 759–786. Zbl 1166.20058, MR 2455937, 10.1007/s10587-008-0050-7 |
| Reference:
|
[2] Nagy P.T., Stuhl I.: Right nuclei of quasigroup extensions.Comm. Alg. 40 (2012), 1893-1900. 10.1080/00927872.2011.575676 |
| Reference:
|
[3] Smith J.D.H., Romanowska A.B.: Post-modern algebra.Wiley, New York, 1999. Zbl 0946.00001, MR 1673047 |
| . |
