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Let us recall the well-known Shirshov's Height Theorem. "Let A be a finitelygenerated algebra of degree d. Then there exists a finite set Y which is thesubset of A that A has and an integer h' = h(A) such that A has Shirshov'sheight h' over set Y. For Y we may take the set of words of length <d. Such Yis called a Shirshov's base of algebra A." Shirshov's original proof was purelycombinatorical, but did not provide a reasonable upper estimate for the height.Kolotov obtained an estimate for h(A) < s^(s^m) (m = deg(A), and s is thenumber of generators) in 1981. I. Zelmanov asked for an exponential bound inthe Dniester Notebook. In 2011 Belov and Kharitonov get the estimate 2^87 m^(12log_3(m)+48) l. This result answered Zelmanov question that in fact Shirshov'sheight has subexponential growth. As a corollary from Amitzur-Levitsky theoremwe have the lower estimate of h(A) which is ((m/2)^2 - 1)(s - 1) + 1 . Weconsider essential height over the sets of small words to obtain exactestimations of Shirshov's height. Define by n-divisible such word w that may beconstructed as w = v u_1 u_2...u_n, where for every i from 2 to n u_(i-1) islexicographically less than u_i. The main results of the paper are thefollowing: Let M be the set of non-n-dividable words with: 1) finite essentialheight over the words degree 2, then the number of lexicographically comparablewords with period 2 is less than (2l-1)(n-1)(n-2)/2; 2) finite essential heightover the words degree 3, then the number of lexicographically comparable wordswith period 3 is less than (2l-1)(n-1)(n-2); 3) finite essential height overthe words degree (n-1), then the number of lexicographically comparable wordswith period (n-1) is less than (l-2)(n-1). The case of the words of length 2 issynthesized to the proof of the exponential estimation in Shirshov's Heighttheorem.

Estimations of the particular periodicity in case of the extremal periods in Shirshov's Height theorem