Linear operator identities in quasigroups

Akhtar, Reza. "Linear operator identities in quasigroups." Commentationes Mathematicae Universitatis Carolinae 62 63.1 (2022): 1-9. <http://eudml.org/doc/299278>.

@article{Akhtar2022,
abstract = {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 $n \ge 1$, $\sigma $ is a permutation of $\lbrace 1, \ldots , n\rbrace $, 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 $.},
author = {Akhtar, Reza},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {quasigroup; linear identity; associativity; commutativity},
language = {eng},
number = {1},
pages = {1-9},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Linear operator identities in quasigroups},
url = {http://eudml.org/doc/299278},
volume = {62 63},
year = {2022},
}

TY - JOUR
AU - Akhtar, Reza
TI - Linear operator identities in quasigroups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2022
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62 63
IS - 1
SP - 1
EP - 9
AB - 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 $n \ge 1$, $\sigma $ is a permutation of $\lbrace 1, \ldots , n\rbrace $, 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 $.
LA - eng
KW - quasigroup; linear identity; associativity; commutativity
UR - http://eudml.org/doc/299278
ER -