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Positional operatives with invertible elements
L. M. Gluskin, L. N. Èl'kin
Abstract:
The main result of this paper is a proof of the fact that if $S$ is a $\Pi$-operative (i.e. an $n$-ary operation on a set $S$ satisfying the identities
\begin{multline*}
x_1\dots x_{k-1}(y_1\dots y_n)x_{k+1}\dots x_n=\\
=(x_{\sigma_k1}\dots x_{\sigma_k(k-1)}y_{\pi_k1}\dots y_{\pi_k(n-k+1)})\dots y_{\pi_kn}x_{\sigma_k(k+1)}\dots x_{\sigma_kn},
\end{multline*}
where $\sigma_k$ and $\pi_k$ are permutations, $k=1,\dots,n$, $\sigma_1=\pi_1=\varepsilon$, and $\sigma_kk=k$), and if $S$ contains a two-sided invertible element $\alpha$ (i.e. $S=\alpha S\dots S = S\dots S\alpha$), then a semigroup operation $*$ can be defined on $S$ such that
$$
x_1x_2\dots x_n=x_1*\psi_2x_2*\dots*\psi_{n-1}x_{n-1}*u*\psi_nx_n
$$
for some invertible element $u$ of the semigroup $S(*)$ and certain of its automorphisms or inverse automorphisms $\psi_2,\dots,\psi_n$ for which $\psi_ku=u$.
Bibliography: 13 titles.
Received: 23.01.1973
Citation:
L. M. Gluskin, L. N. Èl'kin, “Positional operatives with invertible elements”, Mat. Sb. (N.S.), 92(134):3(11) (1973), 420–429; Math. USSR-Sb., 21:3 (1973), 412–422
Linking options:
https://www.mathnet.ru/eng/sm3356https://doi.org/10.1070/SM1973v021n03ABEH002024 https://www.mathnet.ru/eng/sm/v134/i3/p420
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Abstract page: | 235 | Russian version PDF: | 71 | English version PDF: | 9 | References: | 45 |
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