Introduction

The notion of a dimension, when available, is a powerful mathematical tool in proving finiteness conditions in combinatorics. An example of this is Eilenberg's equality theorem, which provides an optimal criterion for the equality of two rational series over a (skew) field. In this example a problem on words, i.e., on free semigroups, is first transformed into a problem on vector spaces, and then it is solved using the dimension property of those algebraic structures. One can raise the natural question: do sets of words possess dimension properties of some kind?

We approach this problem through systems of equations in semigroups. As a starting point we recall the well-known defect theorem (see Chapter 6), which states that if a set of n words satisfies a nontrivial relation, then these words can be expressed simultaneously as products of at most n — 1 words. The defect effect can be seen as a weak dimension property of words. In order to analyze it further one can examine what happens when n words satisfy several independent relations, where independence is formalized as follows: a set E of relations on n words is independent, if E, viewed as a system of equations, does not contain a proper subset having the same solutions as E.

It is not difficult to see that a set of n words can satisfy two or more equations even in the case where the words cannot be expressed as products of fewer than n—1 words.