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

## TL;DR

This paper develops a theoretical framework for analyzing sets of words invariant under literal (anti)morphisms, extending the defect theorem and exploring properties of various classes of $ heta$-invariant codes.

## Contribution

It extends the defect theorem to $ heta$-invariant sets and establishes the equivalence of maximality and completeness for thin $ heta$-invariant codes.

## Key findings

- Extended the defect theorem for $ heta$-invariant sets.
- Proved equivalence of maximality and completeness for thin $ heta$-invariant codes.
- Embedded regular $ heta$-invariant codes into complete ones for certain antimorphisms.

## Abstract

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 in the framework of some special families of $\theta$-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 $\theta$-invariant code may be embedded into a complete one.