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

摘要

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 for some special families of θ-invariant codes such as prefix (bifix) codes, codes with a finite (two-way) deciphering delay, uniformly synchronous 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.