algebra of an infinite finitary symmetric group over a field of positive characteristic (for zero charasteristic this was established by A. Zalessky). To solve the Latyshev problem, it is necessary to translate properties of $T$-ideals to the language of polylinear words. In [@BBL97; @Belov1] an attempt was made to realize a project of translation of structure properties of algebras to the language of word combinatorics. Translation to the language of polylinear words is simpler and enables to get some information on words of a general form.

In this paper we transfer V. N. Latyshev’s technique to the non-polylinear case, and this enables us to obtain a subexponential estimate in Shirshov height theorem. G. R. Chelnokov suggested the idea of this transfer in 1996.

Estimates for height
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The original A. I. Shirshov’s proof, being purely combinatorial (it was based on the technique of elimination developed by him for Lie algebras, in particular in the proof of the theorem on freeness), nevertheless implied only primitively recursive estimates. Later A. T. Kolotov [@Kolotov] obtained an estimate for $\Ht(A)\leqslant l^{l^n}$($n=\deg(A)$, $l$ is the number of generators). A. Ya. Belov in [@Bel92] has shown that $\Ht(n,l)<2nl^{n+1}$. The exponential estimate in Shirshov height theorem was 