Estimates of a structure of piece-wise periodicity in Shirshov's height theorem / *M. I. Kharitonov.* // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2013. № 1. P. 10-16
[Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 0].

The Gelfand-Kirillov dimension of *l*-generated general matrixes is (*l* -1)*n^{2} + 1. The minimal degree of the identity of this algebra is 2n as a corollary of Amitzur-Levitsky theorem. That is why the essential height of A being an l-generated PI-algebra of degree n over every set of words can be greater than (l -1)n^{2}/4 + 1. We prove that if A has a finite GK-dimension, then the number of lexicographically comparable subwords with the period (n -1) in each monoid of A is not greater than (l -2)(n -1). The case of the subwords with the period 2 is generalized to the proof of Shirshov's Height theorem.*

*Key words*:
essential height, Shirshov's Height theorem, combinatorics of words, *n*-divisibility, Dilworth's theorem.