Positional operatives with invertible elements

L. M. Gluskin ; L. N. Èl'kin

Sbornik. Mathematics, Tome 21 (1973) no. 3, pp. 412-422

Résumé

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.