Defect Effect of Bi-infinite Words in the Two-element Case

Defect theorem is one of the fundamental results on words, cf [Lo]. Intuitively it states that if n words satisfy a nontrivial relation, then these words can be expressed as products of at most n−1 words. Actually, as discussed in [CK], for example, there does not exist just one defect theorem but several ones depending on restrictions put on the required n−1 words. It is also well-known that the nontrivial relation above can be replaced by a weaker condition, namely by the nontrivial one-way infinite relation, cf. [Br] and [HK]. The goal of this note is to look for defect theorems for bi-infinite words. In a strict sense such results do not exist: the set X = {ab,ba} of words satisfies a bi-infinite nontrivial relation since (ab)Z = (ba)Z, but there exists no word ρ such that X ⊆ ρ+. However, in [KMP2] there was proved one result and we are going to prove another one in a special case which both can be viewed as defect theorems for bi-infinite words. In terms of factorizations of words defect theorem can be stated as follows: Let X ⊆ Σ+ be a finite set of words. If there exists a word w∈ Σ+ having two different X-factorizations, then the rank of X is at most card(X)−1. Here the rank of X can be defined in different ways, cf again [CK]. For example, it can be defined as a combinatorial rank rc (X) denoting the smallest number k such that X ⊆Y + with card(Y ) = k. To describe our results let w be a bi-infinite word, i.e., an element of ΣZ, and X a finite subset of Σ+. We say that w has an X-factorization if w∈ XZ, and that w has two different X-factorizations, if it has two †Supported by Academy of Finland under Grant No. 14047.