  Let A be a finite or countable alphabet and let $\theta$ be a literal
(anti-)automorphism 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) that is,
languages L such that $\theta$ (L) is a subset of L.We establish an extension
of the famous defect theorem. With regards to the so-called notion of
completeness, we provide a series of examples of finite complete
$\theta$-invariant codes. Moreover, we establish a formula which allows to
embed any non-complete $\theta$-invariant code into a complete one. As a
consequence, in the family of the so-called thin $\theta$--invariant codes,
maximality and completeness are two equivalent notions.
