Precircular Codes and Periodic Biinfinite Words

Finitary codes for biinfinite words

“…Barbin and Le Rest have studied the two-element codes [1]. As an application, Devolder shows that [u,v) is a precircular code if and only if the primitive roots of u and v are not conjugate [6], It is now easy to show the same result for bioe-codes: [u,v} is a bico-code whenever it is a precircular code: PROPOSITION…”

“…The notion of precircular codes has been defined to study the factorizations of periodic biinfinite words [6]: DÉFINITION: A language C a A + is said to be precircular if Vw, p^l, VM 0 , . .…”

“…Let C 0 = {u 0 ,v 0 }. In ^C™ there is at most one word whose représentatives have two distinct C 0 -decompositions [15,1,6]. It is the word "w 0 where w is the single imprimitive word in u 0 v$ U UQ V 0 (if exists).…”

“…So u = v and tC*sC\ C* <= w*. As C is a code, ?C*5P|C* is monogeneous [6], So C is an almost circular code. Let us give a définition useful in the sequel of the proof:…”

Codes générateurs minimaux de langages de mots bi-infinis

“…Un langage C Ç A+ est un code précirculaire [7] si V n,p > 1 Vito,. • • ,Un-ii ^o, •• • )Vp_i G C V t,s G A* tels que uo = (u 0 .…”