it generates $A$ as an algebra. In the representable case the converse is also true.

Suppose $A$ is a finitely generated representable algebra and $H_{Ess}{}_Y(A)<\infty$. Then $H_{Ess}{}_Y(A)=\GK(A)$.

The Gelfand — Kirillov dimension of a finitely generated representable algebra is an integer.

If $H_{Ess}{}_Y(A)<\infty$ and $A$ is representable then $H_{Ess}{}_Y(A)$ is independent of choice of the $s$-basis $Y$.

In this case the Gelfand — Kirillov dimension also is equal to the essential height by virtue of local representability of relatively free algebras.

Although in the representable case the Gelfand — Kirillov dimension and the essential height behave well, even in this case the set of degree vectors may have a bad structure, namely, it can be the complement to the set of solutions of a system of exponential-polynomial Diophantine equations [@BBL97]. That is why there exists an instance of a representable algebra with the trascendent Hilbert series. However for a relatively free algebra, the Hilbert series is rational [@Belov501].

$n$-divisibility and Dilworth theorem
-------------------------------------

The significance of the notion of [*$n$-divisibility*]{} exceeds the limits of Burnside-type problems. This notion is also actual in investigation of polylinear words and estimation of their number; a word is [*polylinear*]{} if each letter occurs in 