| Title:
|
Linear operator identities in quasigroups (English) |
| Author:
|
Akhtar, Reza |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
63 |
| Issue:
|
1 |
| Year:
|
2022 |
| Pages:
|
1-9 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study identities of the form $$ L_{x_0} \varphi_1 \cdots \varphi_n R_{x_{n+1}} = R_{x_{n+1}} \varphi_{\sigma(1)} \cdots \varphi_{\sigma(n)} L_{x_0} $$ in quasigroups, where $n \geq 1$, $\sigma$ is a permutation of $\{1, \ldots, n\}$, and for each $i$, $\varphi_i$ is either $L_{x_i}$ or $R_{x_i}$. We prove that in a quasigroup, every such identity implies commutativity. Moreover, if $\sigma$ is chosen randomly and uniformly, it also satisfies associativity with probability approaching $1$ as $n \rightarrow \infty$. (English) |
| Keyword:
|
quasigroup |
| Keyword:
|
linear identity |
| Keyword:
|
associativity |
| Keyword:
|
commutativity |
| MSC:
|
05C78 |
| idZBL:
|
Zbl 07584109 |
| idMR:
|
MR4445733 |
| DOI:
|
10.14712/1213-7243.2022.010 |
| . |
| Date available:
|
2022-07-18T11:45:29Z |
| Last updated:
|
2024-04-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/150427 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| . |
