Invariance: a Theoretical Approach for Coding Sets of Words Modulo Literal (Anti)Morphisms

Let $A$ be a finite or countable alphabet and let $θ$ 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 $θ$ ($θ$-invariant for short).We establish an extension of the famous defect theorem. Moreover, we prove that for the so-called thin $θ$-invariant codes, maximality and completeness are two equivalent notions. We prove that a similar property holds in the framework of some special families of $θ$-invariant codes such as prefix (bifix) codes, codes with a finite deciphering delay, uniformly synchronized codes and circular codes. For a special class of involutive antimorphisms, we prove that any regular $θ$-invariant code may be embedded into a complete one.