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Codes et motifs. (Codes and motifs). (French) Zbl 0689.68102
We prove, for two codes X and Y, the equivalence of the equality $$XY=YX$$ and the existence of two positive integers i and j such that $$X^ i=Y^ j$$. Moreover, if one of them is singular, these two conditions are equivalent to the existence of a code Z such that X and Y are powers of Z. Then, we show for each non empty language L commuting with a prefix or circular code X, the existence of a code Y, an integer j and $$I\subset {\mathbb{N}}$$ such that $$L=\cup_{i\in I}Y^ i$$ and $$X=Y^ j.$$
In particular, if X is a circular code then $$j=1$$ and if L is a code, L is a power of X.

##### MSC:
 68Q45 Formal languages and automata
prefix codes
Full Text:
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