| Title:
|
Medial quasigroups of type $(n,k)$ (English) |
| Author:
|
Vanžurová, Alena |
| Language:
|
English |
| Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
| ISSN:
|
0231-9721 |
| Volume:
|
49 |
| Issue:
|
2 |
| Year:
|
2010 |
| Pages:
|
107-122 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Our aim is to demonstrate how the apparatus of groupoid terms (on two variables) might be employed for studying properties of parallelism in the so called $(n,k)$-quasigroups. We show that an incidence structure associated with a medial quasigroup of type $(n,k)$, $n>k\ge 3$, is either an affine space of dimension at least three, or a desarguesian plane. Conversely, if we start either with an affine space of order $k>2$ and dimension $m$, or with a desarguesian affine plane of order $k>2$ then there is a medial quasigroup of type $(k^m,k)$, $m>2$ such that the incidence structure naturally associated to a quasigroup is isomorphic with the starting one (the simplest case $k=2$ can be examined separately but is of little interest). The proofs are mostly based on properties of groupoid term functions, applied to idempotent medial quasigroups (idempotency means that $x\cdot x=x$ holds, and mediality means that the identity $(xy)(uv)=(xu)(yv)$ is satisfied). (English) |
| Keyword:
|
Quasigroup |
| Keyword:
|
idempotent groupoid term |
| Keyword:
|
mediality |
| Keyword:
|
incidence structure |
| Keyword:
|
parallelism |
| Keyword:
|
affine space |
| Keyword:
|
desarguesian affine plane |
| MSC:
|
05B25 |
| MSC:
|
20N05 |
| idZBL:
|
Zbl 1236.20066 |
| idMR:
|
MR2796951 |
| . |
| Date available:
|
2011-02-18T07:41:56Z |
| Last updated:
|
2013-09-18 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141421 |
| . |
| Reference:
|
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