| Title:
|
Quasigroup covers of division groupoids (English) |
| Author:
|
Ježek, Jaroslav J. |
| Author:
|
Kepka, Tomáš J. |
| Author:
|
Němec, Petr C. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
64 |
| Issue:
|
3 |
| Year:
|
2023 |
| Pages:
|
265-278 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $G$ be a division groupoid that is not a quasigroup. For each regular cardinal $\alpha>|G|$ we construct a quasigroup $Q$ on $G\times\alpha$ that is a quasigroup cover of $G$ (i.e., $G$ is a homomorphic image of $Q$ and $G$ is not an image of any quasigroup that is a proper factor of $Q$). We also show how to easily obtain quasigroup covers from free quasigroups. (English) |
| Keyword:
|
groupoid |
| Keyword:
|
division |
| Keyword:
|
quasigroup |
| Keyword:
|
cover |
| MSC:
|
20N05 |
| idZBL:
|
Zbl 07830508 |
| idMR:
|
MR4717501 |
| DOI:
|
10.14712/1213-7243.2024.002 |
| . |
| Date available:
|
2024-03-18T10:34:32Z |
| Last updated:
|
2025-10-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152297 |
| . |
| Reference:
|
[1] Albert A. A.: Qusigroups. I.Trans. Amer. Math. Soc. 54 (1943), 507–519. MR 0009962, 10.1090/S0002-9947-1943-0009962-7 |
| Reference:
|
[2] Baer R.: The homomorphism theorems for loops.Amer. J. Math. 67 (1945), 458–460. Zbl 0063.00166, MR 0012302, 10.2307/2371960 |
| Reference:
|
[3] Bates G. E., Kiokemeister F.: A note on homomorphic mapping of quasigroups into multiplicative systems.Bull. Amer. Math. Soc. 54 (1948), 1180–1185. MR 0027768, 10.1090/S0002-9904-1948-09146-7 |
| Reference:
|
[4] Bruck R. H.: Simple quasigroups.Bull. Amer. Math. Soc. 50 (1944), 769–781. MR 0011311, 10.1090/S0002-9904-1944-08236-0 |
| Reference:
|
[5] Bruck R. H.: Some results in the theory of linear non-associative algebras.Trans. Amer. Math. Soc. 56 (1944), 141–199. MR 0011083, 10.1090/S0002-9947-1944-0011083-5 |
| Reference:
|
[6] Bruck R. H.: A Survey of Binary Systems.Gruppentheorie, Ergebnisse der Mathematik und ihre Grenzgebiete, (N.F.), 20, Springer, Berlin, 1958. Zbl 0141.01401, MR 0093552 |
| Reference:
|
[7] Evans T.: On multiplicative systems defined by generators and relations. I. Normal form theorems.Proc. Cambridge Philos. Soc. 47 (1951), 637–649. MR 0043764 |
| Reference:
|
[8] Garrison G. N.: Quasi-groups.Ann. of Math. (2) 41 (1940), 474–487. MR 0002150 |
| Reference:
|
[9] Hausmann B. A., Ore O.: Theory of quasi-groups.Amer. J. Math. 59 (1937), no. 4, 983–1004. MR 1507296, 10.2307/2371362 |
| Reference:
|
[10] Kepka T., Němec P.: In memory of Jaroslav Ježek.Acta Univ. Carolin. Math. Phys. 53 (2012), no. 2, 3–4. MR 3099636 |
| Reference:
|
[11] Kiokemeister F.: A theory of normality for quasigroups.Amer. J. Math. 70 (1948), 99–106. MR 0023252, 10.2307/2371934 |
| . |
