Abstract
Let X be a two-element set of words over a finite alphabet. If a bi-infinite word possesses two X-factorizations which are not shiftequivalent, then the primitive roots of the words in X are conjugates. Note, that this is a strict sharpening of a defect theorem for bi-infinite words stated in \emphKMP. Moreover, we prove that there is at most one bi-infinite word possessing two different X-factorizations and give a necessary and sufficient conditions on X for the existence of such a word. Finally, we prove that the family of sets X for which such a word exists is parameterizable.
Highlights
Defect theorem is one of the fundamental results on words, cf [Lo]. It states that if n words satisfy a nontrivial relation, these words can be expressed as products of at most n − 1 words
The goal of this note is to look for defect theorems for bi-infinite words
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
Summary
Defect theorem is one of the fundamental results on words, cf [Lo]. Intuitively it states that if n words satisfy a nontrivial relation, these words can be expressed as products of at most n − 1 words. In terms of factorizations of words defect theorem can be stated as follows: Let X ⊆ Σ+ be a finite set of words. The following result was shown in [KMP2]: If a nonperiodic bi-infinite word w has two different X-factorizations, the combinatorial rank rc (X) of X is at most card(X) − 1. To prove this seems to be quite complicated. The extended abstract of this paper and paper [KMP2] has appeared in [KMP1]
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