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\iteman{io-port 06010613}
\itemau{Chen, Yuqun; Li, Yu}
\itemti{Some remarks on the Akivis algebras and the pre-Lie algebras.}
\itemso{Czech. Math. J. 61, No. 3, 707-720 (2011).}
\itemab
Summary: In this paper, by using the composition-diamond lemma for nonassociative algebras introduced by {\it A. I. Shirshov} in [Sib. Mat. Zh. 3, 132--137 (1962; Zbl 0143.25602)] and [Sib. Mat. Zh. 3, 292--296 (1962; Zbl 0104.26004)], we give Gr\"obner-Shirshov bases for free pre-Lie algebras and the universal enveloping nonassociative algebra of an Akivis algebra, respectively. As applications, we show Shestakov's result that any Akivis algebra is linear and Segal's result that the set of all good words in $X^{**}$ forms a linear basis of the free pre-Lie algebra $\text{PLie}(X)$ generated by the set $X$. For completeness, we give the details of the proof of Shirshov's composition-diamond lemma for nonassociative algebras.
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\itemcc{}
\itemut{nonassociative algebra; Akivis algebra; universal enveloping algebra; pre-Lie algebra; Gr\"obner-Shirshov basis}
\itemli{doi:10.1007/s10587-011-0041-y}
\end