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. 68, N 1, 2013.].
The Gelfand-Kirillov dimension of l-generated general matrixes is (l -1)n2 + 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)n2/4 + 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.