over the set of words of degree not exceeding $n-1$.*]{}

The Burnside-type problems related to height theorem are considered in [@Zelmanov]. The authors believe that Shirshov theorem on height is a fundamental fact in word combinatorics independently of its applications to $\PI$-theory. (All our proofs are elementary and fit in the framework of word combinatorics.) Unfortunately the experts in combinatorics haven’t sufficiently appraised this fact yet. As regards the notion of [*$n$-divisibility*]{} itself, it seems to be fundamental as well. V. N. Latyshev’s estimates on $\xi_n(k)$, the number of non-$n$-divisible polylinear words in $k$ symbols, have lead to fundamental results in $\PI$-theory. At the same time, this number is nothing but the number of arrangements of integers from $1$ to $k$, such that no $n$ integers (not necessarily consecutive) are placed in decreasing order. Furthermore it is the number of permutationally ordered sets of diameter $n$ consisting of $k$ elements. (A set is called [*permutationally ordered*]{} if its ordering is the intersection of two linear orderings, [*the diameter*]{} of an ordered set is the length of its maximal antichain.)

Height theorem implies the solution of a set of problems in ring theory. Suppose an associative algebra over a field satisfies 