Invariance: a Theoretical Approach for Coding Sets of Words Modulo Literal (Anti)Morphisms
Résumé
Let $A$ be a finite or countable alphabet and let $\theta$ be literal (anti)morphism onto $A^*$ (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under $\theta$ ($\theta$-invariant for short).
We establish an extension of the famous defect theorem. Moreover, we prove that for the
so-called thin $\theta$-invariant codes, maximality and completeness are two equivalent notions.
We prove that a similar property holds for some special families of $\theta$-invariant codes such as prefix (bifix) codes, codes with a finite deciphering delay, uniformly synchronous codes and circular codes. For a special class of involutive antimorphisms, we prove that any regular $\theta$-invariant code may be embedded into a complete one.
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