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.

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.

Equations

$$x\theta (y)={\theta }^2(y)\theta (x).$$

$$y=ax\overline{a},z={\overline{a}}^{|y|}y{a}^{|y|}={\overline{a}}^{|y|}ax\overline{a}{a}^{|y|}.$$

$${\theta }^i(z)={\overline{b}}^{|y|}{\theta }^i(y)b^{|y|}={\overline{b}}^{|y|}b{x}^{\prime }\overline{b}{b}^{|y|}.$$

$$z_2=u{c}^h,z_1=c^{h}s,|u|=|s|\ge 2|y|,h\le |y|,$$

$$z_1=t{d}^k,z_3=d^{k}v,|v|=|t|\ge 2|y|,k\le |y|.$$

$$x_1{z}_1{x}_2{z}_2\cdots x_{n-1}{z}_{n-1}{x}_n=x_1^{\prime }{z}_1^{\prime }{x}_2^{\prime }{z}_1^{\prime }\cdots x_{p-1}^{\prime }{z}_{p-1}^{\prime }{x}_p^{\prime }.$$

$$y_1{y}_2\cdots y_n=s{y}_2^{\prime }{y}_3^{\prime }\cdots y_m^{\prime }p\mathrm{and}{y}_1^{\prime }=ps.$$

$$a\ne b,b\ne c,\theta (a)=b,\theta (b)=a.$$

Full text

1 Introduction

During the last decade, in the free monoid theory, due to their powerful applications, in particular in DNA-computing, one-to-one morphic or antimorphic correspondences play a particularly important part. Given a finite or countable alphabet, say $A$, any such mapping is a substitution which is fully determined by extending a unique permutation of $A$, to a mapping onto $A^{*}$ (the free monoid that is generated by $A$). The resulting mapping is commonly referred to as literal (or letter-to-letter) moreover, in the case of a finite alphabet, it is well known that, with respect to the composition, some power of such a correspondence is the identity (classically, in the case where this power corresponds to the square, we say that the correspondence is involutive).

In that special case of involutive morphisms or antimorphisms -we write (anti)morphisms for short, lots of successful investigations have been done for extending the now classical combinatorical properties on words: we mention the study of the so-called pseudo-palindromes [13, 4], or that of pseudo-repetitions [3, 8, 12]. The framework of some peculiar families of codes [11] and equations in words [5, 6] have been also concerned. Moreover, in the larger family of one-to-one (anti)morphisms, a nice generalization of the famous theorem of Fine and Wilf [14, Proposition 1.3.5] has been recently established in [7].

Equations in words are also the starting point of the study in the present paper, where we adopt the point of view from [14, Chap. 9]. Let $A$ be a finite or countable alphabet; a one-to-one literal (anti)morphism onto $A^{*}$, namely $\theta$, being fixed, consider a finite collection of unknown words, say $Z$. In view of making the present foreword more readable, in the first instance we take $\theta$ as an involutive literal substitution (that is $\theta^{2}=id_{A^{*}}$). We assign that the words in $Z$ and their images by $\theta$ to satisfy a given equation, and we are interested in the cardinality of any set $T$, whose elements allow by concatenation to compute all the words in $Z$. Actually, such a question might be more complex than in the classical configuration, where $\theta$ does not interfer: it is well known that in that classical case, according to the famous defect theorem [14, Theorem 1.2.5], the words in $Z$ may be computed as the concatenation of at most $|Z|-1$ words that don’t satisfy any non-trivial equation. With the terminology of [14, 9], $T$, the set of such words is a code, or equivalently $T^{*}$, the submonoid that it generates, is free: more precisely, with respect to the inclusion of sets it is the smallest free submonoid of $A^{*}$ that contains $Z$.

Along the way, for solving our problem, applying the defect theorem to the set $X=Z\cup\theta(Z)$ might appear as natural. Such a methodology garantees the existence of a code $T$, with $|T|\leq|X|-1$, and such that $T^{*}$ is the smallest free submonoid of $A^{*}$ that contains $X$. Unfortunately, since both the words in $Z$ and $\theta(Z)$ are expressed as concatenations of words in $T$, among the elements of $T\cup\theta(T)$ non-trivial equations can remain; in other words, by applying that methodology, the initial problem would be transferred among the words in $T\cup\theta(T)$. This situation is particularly illustrated by [12, Proposition 3], where the authors prove that, given an involutive antimorphism $\theta$, the solutions of the equation $xy=\theta(y)x$ are $x=(uv)^{i}u,y=vu$, where the elements $u,v$ of $T$ satisfy the non-trivial equation $vu=\theta(u)\theta(v)$.

In the general case where $\theta$ is a literal one-to-one (anti)morphism, we note that the union, say $Y$, of the sets $\theta^{i}(T)$, for all i\in{\mathchoice{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\scriptstyle Z\kern-2.10002ptZ}}{\hbox{\sf\scriptscriptstyle Z\kern-0.99998ptZ}}}, is itself $\theta$-invariant, therefore an alternative methodology will consist in asking for some code $Y$ which is invariant under $\theta$, and such that $Y^{*}$ is the smallest free submonoid of $A^{*}$ that contains X=\bigcup_{i\in{\mathchoice{\hbox{\sf\textstyle Z\kern-2.79996ptZ}}{\hbox{\sf\textstyle Z\kern-2.79996ptZ}}{\hbox{\sf\scriptstyle Z\kern-1.47002ptZ}}{\hbox{\sf\scriptscriptstyle Z\kern-0.69998ptZ}}}}\theta^{i}(Z). By the way, it is straightforward to prove that the intersection of an arbitrary family of $\theta$-invariant free submonoids is itself a $\theta$-invariant free submonoid. In the present paper we prove the following result:

Theorem 1. *Let $A$ be a finite or countable alphabet, let $\theta$ be a literal (anti)morphism onto $A^{*}$, and let $X$ be a finite $\theta$-invariant set. If $X$ it is not a code, then the smallest $\theta$-invariant free submonoid of $A^{*}$ that contains $X$ is generated by a $\theta$-invariant code $Y$ which satisfies $|Y|\leq|X|-1$. *

For illustrating this result in term of equations, we refer to [5, 6], where the authors considered generalizations of the famous equation in three unknowns of Lyndon-Shützenberger [14, Sect. 9.2]. They proved that, an involutive (anti)morphism $\theta$ being fixed, given such an equation with sufficiently long members, a word $t$ exists such that any 3-uple of “solutions” can be expressed as a concatenation of words in $\{t\}\cup\{\theta(t)\}$. With the notation of Theorem 1, the elements of the $\theta$-invariant set $X$ are $x,y,z,\theta(x),\theta(y),\theta(z)$ and those of $Y$ are $t$ and $\theta(t)$: we verify that $Y$ is a $\theta$-invariant code with $|Y|\leq|X|-1$.

In the sequel, we will continue our investigation by studying the properties of complete $\theta$-invariant codes: a subset $X$ of $A^{*}$ is complete if any word of $A^{*}$ is a factor of some words in $X^{*}$. From this point of view, a famous result from Schützenberger states that, for the wide family of the so-called thin codes (which contains regular codes) [9, Sect. 2.5], maximality and completeness are two equivalent notions. In the framework of invariant codes, we prove the following result:

Theorem 2. *Let $A$ be a finite or countable alphabet. Given a thin $\theta$-invariant code $X\subseteq A^{*}$, the three following conditions are equivalent:

(i) $X$ is complete

(ii) $X$ is a maximal code

(iii) $X$ is maximal in the family of the $\theta$-invariant codes. *

In the proof, the main feature consists in establishing that a non-complete $\theta$-invariant code $X$ cannot be maximal in the family of $\theta$-invariant codes: actually, the most delicate step lays upon the construction of a convenient $\theta$-invariant set $Z\subseteq A^{*}$, with $X\cap Z=\emptyset$ and such that $X\cup Z$ remains itself a $\theta$-invariant code.

It is well known that the preceding result from Schützenberger has been successfully extended to some famous families of thin codes, such as prefix (bifix, uniformly synchronous, circular) codes (cf [9, Proposition 3.3.8], [9, Proposition 6.2.1], [9, Theorem 10.2.11], [15, Proposition 3.6] and [10, Theorem 3.5]) and codes with a finite deciphering delay (f.d.d. codes, for short) [9, Theorem 5.2.2]. From this point of view, we will examine the behavior of corresponding families of $\theta$-invariant codes. Actually we establish a result similar to the preceding theorem 2 in the framework of the family of prefix (bifix, f.d.d., two-way f.d.d, uniformly synchronized, circular codes). In the proof, a construction very similar to the previous one may be used in the case of prefix, bifix, f.d.d., two-way f.d.d codes. At the contrary, investigating the behavior of circular codes with regards to the question necessitates the computation of a more sofisticated set; moreover the family of uniformly synchronized codes itself impose to make use of a significantly different methodology.

In the last part of our study, we address to the problem of embedding a non-complete $\theta$-invariant code into a complete one. For the first time, this question was stated in [2], where the author asked whether any finite code can be imbedded into a regular one. A positive answer was provided in [1], where was established a formula for embedding any regular code into a complete one. From the point of view of $\theta$-invariant codes, we obtain a positive answer only in the case where $\theta$ is an involutive antimorphism which is different of the so-called miror image; actually the general question remains open.

We now describe the contents of the paper. Section 2 contains the preliminaries: the terminology of the free monoid is settled, and the definitions of some classical families of codes are recalled. Theorem 1 is established in Section 3, where an original example of equation is studied. The proof of Theorem 2 is done in Section 3, and extensions for special familes of $\theta$-invariant codes are studied in Section 4. The question of embedding a regular $\theta$-invariant code into a complete one is examined in Section 5.

2 Preliminaries

We adopt the notation of the free monoid theory: given an alphabet $A$, we denote by $A^{*}$ the free monoid that it generates. Given a word $w$, we denote by $|w|$ its length, the empty word, that we denote by $\varepsilon$, being the word with length [math]. We denote by $w_{i}$ the letter of position $i$ in $w$: with this notation we have $w=w_{1}\cdots w_{|w|}$. We set $A^{+}=A^{*}\setminus\{\varepsilon\}$. Given $x\in A^{*}$ and $w\in A^{+}$, we say that $x$ is a prefix (suffix) of $w$ if a word $u$ exists such that $w=xu$ ($w=ux$). Similarly, $x$ is a factor of $w$ if a pair of words $u,v$ exists such that $w=uxv$. Given a non-empty set $X\subseteq A^{*}$, we denote by $P(X)$ ($S(X),F(X)$) the set of the words that are prefix (suffix, factor) of some word in $X$. Clearly, we have $X\subseteq P(X)$ ($S(X),F(X)$). A set $X\subseteq A^{*}$ is complete iff $F(X^{*})=A^{*}$. Given a pair of words $w,w^{\prime}$, we say that it overlaps if words $u,v$ exist such that $uw^{\prime}=wv$ or $w^{\prime}u=vw$, with $1\leq|u|<|w|$ and $1\leq|v|<|w^{\prime}|$; otherwise, the pair is overlapping-free (in such a case, if $w=w^{\prime}$, we simply say that $w$ is overlapping-free).

It is assumed that the reader has a fundamental understanding with the main concepts of the theory of variable length codes: we only recall some of the main definitions and we suggest, if necessary, that he (she) report to [9]. A set $X$ is a variable length code (a code for short) iff any equation among the words of $X$ is trivial, that is, for any pair of sequences of words in $X$, namely $(x_{i})_{1\leq i\leq m}$, $(y_{j})_{1\leq i\leq n}$, the equation $x_{1}\cdots x_{m}=y_{1}\cdots y_{n}$ implies $m=n$ and $x_{i}=y_{i}$ for each integer $i\in[1,m]$. By definition $X^{*}$, the submonoid of $A^{*}$ which is generated by $X$, is free. Equivalently, $X^{*}$ satisfies the property of equidivisibility, that is $(X^{*})^{-1}X^{*}\cap X^{*}(X^{*})^{-1}=X^{*}$.

Some famous families of codes that have been studied in the literature: $X$ is a *prefix * (suffix, bifix) code iff $X\neq\{\varepsilon\}$ and $X\cap XA^{+}=\emptyset$ $(X\cap A^{+}X=\emptyset,X\cap XA^{+}=X\cap A^{+}X=\emptyset$). $X$ is a code with a finite deciphering delay (f.d.d. code for short) if it is a code and if a non-negative integer $d$ exists such that $X^{-1}X^{*}\cap X^{d}A^{+}\subseteq X^{+}$. With this condition, if another integer $d^{\prime}$ exists such that we have $X^{*}X^{-1}\cap A^{+}X^{d^{\prime}}\subseteq X^{+}$, we say that $X$ is a two-way f.d.d. code. $X$ is a uniformly synchronized code if it is a code and if a positive integer $k$ exists such that, for all $x,y\in X^{k},u,v\in A^{+}$: $uxyv\in X^{*}\Longrightarrow ux,xv\in X^{*}.$ $X$ is a circular code if for any pair of sequences of words in $X$, namely $(x_{i})_{1\leq i\leq m},(y_{j})_{1\leq j\leq n}$, and any pair of words $s,p$, with $s\neq\varepsilon$, the equation $x_{1}\cdots x_{m}=sy_{2}\cdots y_{n}p$, with $y_{1}=ps$, implies $m=n$, $p=\varepsilon$ and $x_{i}=y_{i}$ for each $i\in[1,m]$.

In the whole paper, we consider a finite or countable alphabet $A$ and a mapping $\theta$ which satisfies each of the three following conditions:

(a) $\theta$ is a one-to-one correspondence onto $A^{*}$

(b) $\theta$ is literal, that is $\theta(A)\subseteq A$

(c) either $\theta$ is a morphism or it is an antimorphism (it is an antimophism if $\theta(\varepsilon)=\varepsilon$ and $\theta(xy)=\theta(y)\theta(x)$, for any pair of words $x,y$); for short in any case we write that $\theta$ is an (anti)morphism.

In the case where $A$ is a finite set, it is well known that a positive integer $n$ exists such that $\theta^{n}=id_{A^{*}}$. In the whole paper, we are interested in the family of sets $X\subseteq A^{*}$ that are invariant under the mapping $\theta$ ($\theta$-invariant for short), that is $\theta(X)=X$.

3 A Defect Effect for Invariant Sets

Informally, the famous defect theorem says that if some words of a set $X$ satisfy a non-trivial equation, then these words may be written upon an alphabet of smaller size. In this section, we examine whether a corresponding result may be stated in the frameword of $\theta$-invariant sets. The following property comes from the definition:

Proposition 1

Let $M$ be a submonoid of $A^{*}$ and let $S\subseteq A^{*}$ be such that $M=S^{*}$. Then $M$ is $\theta$-invariant if and only if $S$ is $\theta$-invariant.

Clearly the intersection of a non-empty family of $\theta$-invariant free submonoids of $A^{*}$ is itself a $\theta$-invariant free submonoid. Given a submonoid $M$ of $A^{*}$, recall that its minimal generating set is $(M\setminus\{\varepsilon\})\setminus(M\setminus\{\varepsilon\})^{2}$.

Theorem 3.1

Let $A$ be a finite or countable alphabet, let $X\subseteq A^{*}$ be a $\theta$-invariant set and let $Y$ be the minimal generating set of the smallest $\theta$-invariant free submonoid of $A^{*}$ which contains $X$. If $X$ is not a code, then we have $|Y|\leq|X|-1$.

Proof

With the notation of Theorem 3.1, since $Y$ is a code, each word $x\in X$ has a unique factorization upon the words of $Y$, namely $x=y_{1}\cdots y_{n}$, with $y_{i}\in Y$ ($1\leq i\leq n$). In a classical way, we say that $y_{1}$ ($y_{n}$) is the initial (terminal) factor of $x$ (with respect to such a factorization). At first, we shall establish the following lemma:

Lemma 1

With the preceding notation, each word in $Y$ is the initial (terminal) factor of a word in $X$.

Proof

By contradiction, assume that a word $y\in Y$ that is never initial of any word in $X$ exists. Set $Y_{0}=(Y\setminus\{y\})\{y\}^{*}$ and $Y_{i}=\theta^{i}(Y_{0})$, for each integer i\in{\mathchoice{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\scriptstyle Z\kern-2.10002ptZ}}{\hbox{\sf\scriptscriptstyle Z\kern-0.99998ptZ}}}. In a classical way (cf e.g. [14, p. 7]), since $Y$ is a code, $Y_{0}$ itself is a code. Since $\theta^{i}$ is a one-to-one correspondence, for each integer i\in{\mathchoice{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\scriptstyle Z\kern-2.10002ptZ}}{\hbox{\sf\scriptscriptstyle Z\kern-0.99998ptZ}}}, $Y_{i}$ is a code, that is $Y_{i}^{*}$ is a free submonoid of $A^{*}$. Consequently, the intersection, namely $M$, of the family (Y_{i}^{*})_{i\in{\mathchoice{\hbox{\sf\textstyle Z\kern-2.79996ptZ}}{\hbox{\sf\textstyle Z\kern-2.79996ptZ}}{\hbox{\sf\scriptstyle Z\kern-1.47002ptZ}}{\hbox{\sf\scriptscriptstyle Z\kern-0.69998ptZ}}}} is itself a free submonoid of $A^{*}$. Moreover we have $\theta(M)\subseteq M$ (indeed, given a word $w\in M$, $\theta(w)\not\in Y_{i}$ implies $w\not\in Y_{i-1}$) therefore, since $\theta$ is onto, we obtain $\theta(M)=M$. Let $x$ be an arbitrary word in $X$. Since $X\subseteq Y^{*}$, and according to the definition of $y$, we have $x=(y_{1}y^{k_{1}})(y_{2}y^{k_{2}})\cdots(y_{n}y^{k_{n}})$, with $y_{1},\cdots y_{n}\in Y\setminus\{y\}$ and $k_{1},\cdots k_{n}\geq 0$. Consequently $x$ belongs to $Y_{0}^{*}$, therefore we have $X\subseteq Y_{0}^{*}$. Since $X$ is $\theta$-invariant, this implies $X=\theta(X)\subseteq Y_{i}^{*}$ for each i\in{\mathchoice{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\scriptstyle Z\kern-2.10002ptZ}}{\hbox{\sf\scriptscriptstyle Z\kern-0.99998ptZ}}}, thus $X\subseteq M$.

But the word $y$ belongs to $Y^{*}$ and doesn’t belong to $Y_{0}^{*}$ thus it doesn’t belong to $M$. This implies $X\subseteq M\subsetneq Y^{*}$: a contradiction with the minimality of $Y^{*}$.

Proof of Theorem 3.1. Let $\alpha$ be the mapping from $X$ onto $Y$ which, with every word $x\in X$, associates the initial factor of $x$ in its (unique) factorization over $Y^{*}$. According to Lemma 1, $\alpha$ is onto. We will prove that it is not one-to-one. Classically, since $X$ is not a code, a non-trivial equation may be written among its words, say: $x_{1}\cdots x_{n}=x^{\prime}_{1}\cdots x^{\prime}_{m},~{}~{}{\rm with}~{}~{}x_{i},x^{\prime}_{j}\in X~{}~{}x_{1}\neq x^{\prime}_{1}~{}~{}(1\leq i\leq n,1\leq j\leq m).$ Since $Y$ is a code, a unique sequence of words in $Y$, namely $y_{1},\cdots,y_{p}$ exists such that: $x_{1}\cdots x_{n}=x^{\prime}_{1}\cdots x^{\prime}_{m}=y_{1}\cdots y_{p}.$ This implies $y_{1}=\alpha(x_{1})=\alpha(x^{\prime}_{1})$ and completes the proof.

In what follows we discuss some interpretation of Theorem 3.1 with regards to equations in words. For this purpose, we assume that $A$ is finite, thus a positive integer $n$ exists such that $\theta^{n}=id_{A^{*}}$. Consider a finite set of words, say $Z$, and denote by $X$ the union of the sets $\theta^{i}(Z)$, for $i\in[1,n]$; assume that a non-trivial equation holds among the words of $X$, namely $x_{1}\cdots x_{m}=y_{1}\cdots y_{p}$. By construction $X$ is $\theta$-invariant therefore, according to Theorem 3.1, a $\theta$-invariant code $Y$ exists such that $X\subseteq Y^{*}$, with $|Y|\leq|X|-1$. This means that each of the words in $X$ can be expressed by making use of at most $|X|-1$ words of type $\theta^{i}(u)$, with $u\in Y$ and $1\leq i\leq n$. It will be easily verified that the examples from [5, 12, 6] corroborate this fact, moreover below we mention an original one:

Example 1

Let $\theta$ be a literal antimorphism such that $\theta^{3}=id_{A^{*}}$. Consider two different words $x,y$, with $|x|>|y|$, which satisfy the following equation:

[TABLE]

With these conditions, a pair of words $u,v$ exists such that $x=uv$, $\theta^{2}(y)=u$, thus $y=\theta(u)$, moreover we have $v=\theta(v)$ and $u=\theta(u)=\theta^{2}(u)$. With the preceding notation, we have $Z=\{x,y\}$, $X=Z\cup\theta(Z)\cup\theta^{2}(Z)$, $Y=\{u\}\cup\{v\}\cup\{\theta(u)\}\cup\{\theta(v)\}\cup\{\theta^{2}(u)\}\cup\{\theta^{2}(v)\}$. It follows from $y=\theta(y)=\theta^{2}(y)$ that $X=\{x\}\cup\{\theta(x)\}\cup\{\theta^{2}(x)\}\cup\{y\}$.

• At first, assume that no word $t$ exists such that $u,v\in t^{+}$. In a classical way, we have $uv\neq vu$, thus $X=\{x,\theta(x),\theta^{2}(x),y\}$ and $Y=\{u,v\}$. We verify that $|Y|\leq|X|-1$.

• Now, assume that we have $u,v\in t^{+}$. We obtain $X=Z=\{x,y\}$ and $Y=\{t\}$. Once more we have $|Y|\leq|X|-1$.

4 Maximal $\theta$-Invariant Codes

Given set $X\subseteq A^{*}$, we say that it is thin if $A^{*}\neq F(X)$. Regular codes are well known examples of thin codes. From the point of view of maximal codes, below we recall one of the famous result stated by Schützenberger:

Theorem 4.1

[9, Theorem 2.5.16]*

Let $X$ be an thin code. Then the following conditions are equivalent:

(i) $X$ is complete

(ii) $X$ is a maximal code.*

The aim of this section is to examine whether a corresponding result may be stated in the family of thin $\theta$-invariant codes.

In the case where $|A|=1$, we have $\theta=id_{A^{*}}$, moreover the codes are all the singletons in $A^{+}$. Therefore any code is $\theta$-invariant, maximal and complete. In the rest of the paper, we assume that $|A|\geq 2$.

Some notations. Let $X$ be a non-complete $\theta$-invariant code, and let $y\not\in F(X^{*})$. Without loss of generality, we may assume that the initial and the terminal letters of $y$ are different (otherwise, substitute to $y$ the word $ay{\overline{a}}$, with $a,{\overline{a}}\in A$ and $a\neq{\overline{a}}$), we may also assume that $|y|\geq 2$. Set:

[TABLE]

Since $\theta$ is a literal (anti)morphism, for each integer i\in{\mathchoice{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\scriptstyle Z\kern-2.10002ptZ}}{\hbox{\sf\scriptscriptstyle Z\kern-0.99998ptZ}}}, a pair of different letters $b,{\overline{b}}$ and a word $x^{\prime}$ exist such that $|x^{\prime}|=|x|=|y|-2$, and:

[TABLE]

Given two (not necessarily different) integers i,j\in{\mathchoice{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\scriptstyle Z\kern-2.10002ptZ}}{\hbox{\sf\scriptscriptstyle Z\kern-0.99998ptZ}}}, we will accurately study how the two words $\theta^{i}(z),\theta^{j}(z)$ may overlap.

Lemma 2

With the notation in (2), let $u,v\in A^{+}$ and i,j\in{\mathchoice{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\scriptstyle Z\kern-2.10002ptZ}}{\hbox{\sf\scriptscriptstyle Z\kern-0.99998ptZ}}} such that $|u|\leq|z|-1$ and $\theta^{i}(z)v=u\theta^{j}(z)$. Then we have $|u|=|v|\geq 2|y|$, moreover a letter $b$ and a unique positive integer $k$ (depending of $|u|$) exist such that we have $\theta^{i}(z)=ub^{k}$, $\theta^{j}(z)=b^{k}v$, with $k\leq|y|$.

Proof

According to (2), we set $\theta^{i}(z)={\overline{b}^{|y|}}bx^{\prime}{\overline{b}}b^{|y|}$ and $\theta^{j}(z)={\overline{c}^{|y|}}cx^{\prime\prime}{\overline{c}}c^{|y|}$, with $b,{\overline{b}},c,{\overline{c}}\in A$ and $b\neq{\overline{b}},c\neq{\overline{c}}$. Since $\theta$ is a literal (anti)morphism, we have $|\theta^{i}(z)|=|\theta^{j}(z)|$ thus $|u|=|v|$; since we have $1\leq|u|\leq 3|y|-1$, exactly one of the following cases occurs:

Case 1: $1\leq|u|\leq|y|-1$. With this condition, we have ($\theta^{i}(z))_{|u|+1}={\overline{b}}={\overline{c}}=(u\theta^{j}(z))_{|u|+1}$ and $(\theta^{i}(z))_{|y|+1}=b={\overline{c}}=(u\theta^{j}(z))_{|y|+1}$, which contradicts $b\neq{\overline{b}}$.

Case 2: $|u|=|y|$. This condition implies $(\theta^{i}(z))_{|u|+1}=b={\overline{c}}=(u\theta^{j}(z))_{|u|+1}$ and $(\theta^{i}(z))_{2|y|}={\overline{b}}={\overline{c}}=(u\theta^{j}(z))_{2|y|}$, which contradicts $b\neq{\overline{b}}$.

Case 3: $|y|+1\leq|u|\leq 2|y|-1$. We obtain $(\theta^{i}(z))_{2|y|}={\overline{b}}={\overline{c}}=(u\theta^{j}(z))_{2|y|}$ and $(\theta^{i}(z))_{2|y|+1}=b={\overline{c}}=(u\theta^{j}(z))_{2|y|+1}$ which contradicts $b\neq{\overline{b}}$.

Case 4: $2|y|\leq|u|\leq 3|y|-1$. With this condition, necessarily we have $b={\overline{c}}$, therefore an integer $k\in[1,|y|]$ exists such that $\theta^{i}(z)=ub^{k}$ and $\theta^{j}(z)=b^{k}v$.

Set Z=\{\theta^{i}(z)|i\in{\mathchoice{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\scriptstyle Z\kern-2.10002ptZ}}{\hbox{\sf\scriptscriptstyle Z\kern-0.99998ptZ}}}\}. Since $y\notin F(X^{*})$ and since $X$ is $\theta$-invariant, for any integer i\in{\mathchoice{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\scriptstyle Z\kern-2.10002ptZ}}{\hbox{\sf\scriptscriptstyle Z\kern-0.99998ptZ}}} we have $\theta^{i}(z)\not\in F(X^{*})$, hence we obtain $Z\cap F(X^{*})=\emptyset$. By construction, all the words in $Z$ have length $|z|$ moreover, as a consequence of Lemma 2:

Lemma 3

With the preceding notation, we have $A^{+}ZA^{+}\cap ZX^{*}Z=\emptyset$.

Proof

By contradiction, assume that $z_{1},z_{2},z_{3}\in Z$ , $x\in X^{*}$ and $u,v\in A^{+}$ exist such that $uz_{1}v=z_{2}xz_{3}$. By comparing the lengths of the words $u$ and $v$ with $|z|$, exactly one of the three following cases occurs:

Case 1: $|z|\leq|u|$ and $|z|\leq|v|$. With this condition, we have $z_{2}\in P(u)$ and $z_{3}\in S(v)$, therefore the word $z_{1}$ is a factor of $x$: this contradicts $Z\cap F(X^{*})=\emptyset$.

Case 2: $|u|<|z|\leq|v|$. We have in fact $u\in P(z_{2})$ and $z_{3}\in S(v)$. We are in the condition of Lemma 2: the words $z_{2}$, $z_{1}$ overlap. Consequently, $u\in A^{+}$ and $b\in A$ exist such that $z_{2}=ub^{k}$ and $z_{1}=b^{k}z^{\prime}_{1}$, with $1\leq k\leq|y|$. But by construction we have $|uz_{1}|=|z_{2}xz_{3}|-|v|$: since we assume $|v|\geq|z|$, this implies $|uz_{1}|\leq|z_{2}xz_{3}|-|z|=|z_{2}x|$, therefore we obtain $uz_{1}=ub^{k}z^{\prime}_{1}\in P(z_{2}x)$. It follows from $z_{2}=ub^{k}$ that $z^{\prime}_{1}\in P(x)$. Since $z_{1}\in Z$ and according to (2), i\in{\mathchoice{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\scriptstyle Z\kern-2.10002ptZ}}{\hbox{\sf\scriptscriptstyle Z\kern-0.99998ptZ}}} and ${\overline{b}}\in A$ exist such that we have $z_{1}=b^{k}z^{\prime}_{1}=b^{|y|}\theta^{i}(y){\overline{b}}^{|y|}$. Since by Lemma 2 we have $|z_{1}^{\prime}|=|u|\geq 2|y|$, we obtain $\theta^{i}(y)\in F(z^{\prime}_{1})$, which contradicts $y\notin F(X^{*})$.

Case 3: $|v|<|z|\leq|u|$. Same arguments on the reversed words lead to a conclusion similar to that of Case 2.

Case 4: $|z|>|u|$ and $|z|>|v|$. With this condition, both the pairs of words $z_{2},z_{1}$ and $z_{1},z_{3}$ overlap. Once more we are in the condition of Lemma 2: letters $c,d$, words $u,v,s,t$, and integers $h,k$ exist such that the two following properties hold:

[TABLE]

It follows from $uz_{1}v=z_{2}xz_{3}$ that $uz_{1}v=(uc^{h})x(d^{k}v)$, thus $z_{1}=c^{h}xd^{k}$. Once more according to (2), i\in{\mathchoice{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\scriptstyle Z\kern-2.10002ptZ}}{\hbox{\sf\scriptscriptstyle Z\kern-0.99998ptZ}}} and ${\overline{c}}\in A$ exist such that we have $z_{1}=c^{|y|}\theta^{i}(y){\overline{c}}^{|y|}$. Since we have $h,k\leq|y|$, this implies $d={\overline{c}}$ moreover $\theta^{i}(y)$ is a factor of $x$. Once more, this contradicts $y\notin F(X^{*})$.

Thanks to Lemma 3 we will prove some meaningful results in Section 5. Presently, we will apply it in a special context:

Corollary 1

With the preceding notation, $X^{*}Z$ is a prefix code.

Proof

Let $z_{1},z_{2}\in Z$, $x_{1},x_{2}\in X^{*},u\in A^{+}$, such that $x_{1}z_{1}u=x_{2}z_{2}$. For any word $z_{3}\in Z$, we have $(z_{3}x_{1})z_{1}(u)=z_{3}x_{2}z_{1}$, a contradiction with Lemma 3.

We are now ready to prove the main result of the section:

Theorem 4.2

*Let $A$ be a finite or countable alphabet and let $X\subseteq A^{*}$ be a thin $\theta$-invariant code. Then the following conditions are equivalent:

(i) $X$ is complete

(ii) $X$ is a maximal code

(iii) $X$ is maximal in the family $\theta$-invariant codes.*

Proof

Let $X$ be a $\theta$-invariant code. According to Theorem 4.1, if $X$ is thin and complete, then it is a maximal code, therefore $X$ is maximal in the family of $\theta$-invariant codes. For proving the converse, we consider a set $X$ which is maximal in the family of $\theta$-invariant codes.

Assume that $X$ is not complete and let $y\not\in F(X^{*})$. Define the word $z$ as in (1) and consider the set Z=\{\theta^{i}(z)|i\in{\mathchoice{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\scriptstyle Z\kern-2.10002ptZ}}{\hbox{\sf\scriptscriptstyle Z\kern-0.99998ptZ}}}\}. At first, we will prove that $X\cup Z$ remains a code. In view of that, we consider an arbitrary equation between the words in $X\cup Z$. Since $X$ is a code, without loss of generality, we may assume that at least one element of $Z$ has at least one occurrence in one of the two sides of this equation. As a matter of fact, with such a condition and since $Z\cap F(X^{*})=\emptyset$, two sequences of words in $X^{*}$, namely $(x_{i})_{1\leq i\leq n},(x^{\prime}_{j})_{1\leq j\leq p}$ and two sequences of words in $Z$, namely $(z_{i})_{1\leq i\leq n-1},(z^{\prime}_{j})_{1\leq j\leq p-1}$ exist such that the equation takes the following form:

[TABLE]

Without loss of generality, we assume $n\geq p$. At first, according to Corollary 1, necessarily, we have $x_{1}=x^{\prime}_{1}$, therefore Equation (5) is equivalent to: $z_{1}x_{2}z_{2}\cdots x_{n-1}z_{n-1}x_{n}=z^{\prime}_{1}x^{\prime}_{2}z^{\prime}_{2}\cdots x^{\prime}_{p-1}z^{\prime}_{p-1}x^{\prime}_{p},$ however, since all the words in $Z$ have a common length, we have $z_{1}=z^{\prime}_{1}$ hence our equation is equivalent to $x_{2}z_{2}\cdots x_{n-1}z_{n-1}x_{n}=x^{\prime}_{2}z^{\prime}_{2}\cdots x^{\prime}_{p-1}z^{\prime}_{p-1}x^{\prime}_{p}.$ Consequently, by applying iteratively the result of Corollary 1, we obtain: $x_{2}=x^{\prime}_{2},\cdots,x_{p}=x^{\prime}_{p}$, which implies $x_{p+1}z_{p+1}\cdots z_{n-1}x_{n}=\varepsilon$, thus $n=p$. In other words Equation (5) is trivial, thus $X\cup Z$ is a code.

Next, since $\theta$ is one-to-one and since we have $\theta(X\cup Z)\subseteq\theta(X)\cup\theta(Z)=X\cup Z$, the code $X\cup Z$ is $\theta$-invariant. It follows from $z\in Z\setminus X$ that $X$ is strictly included in $X\cup Z$: this contradicts the maximality of $X$ in the whole family of $\theta$-invariant codes, and completes the proof of Theorem 4.2.

Example 2

Let $A=\{a,b,c\}$. Consider the antimorphism $\theta$ which is generated by the permutation $\sigma(a)=b,\sigma(b)=c,\sigma(c)=a$ and let $X=\{ab,cb,ca,ba,bc,ac\}$; it can be easily verified that $X$ is a $\theta$-invariant code. Since we have $a^{3}\not\in F(X^{*})$, by setting $y=a^{3}b$ and $z=b^{4}\cdot a^{3}b\cdot a^{4}$ we are in Condition (1). The corresponding set $Z$ is \{\theta^{i}(z)|i\in{\mathchoice{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\scriptstyle Z\kern-2.10002ptZ}}{\hbox{\sf\scriptscriptstyle Z\kern-0.99998ptZ}}}\} $=\{b^{4}cb^{3}c^{4},a^{4}c^{3}ac^{4},a^{4}ba^{3}b^{4},c^{4}b^{3}cb^{4},c^{4}ac^{3}a^{4},b^{4}a^{3}ba^{4}\}$. Since $X\cup Z$ is a prefix set, this guarantees that $X\cup Z$ remains a $\theta$-invariant code.

5 Maximality in Some Families of $\theta$-Invariant Codes

In the literature, statements similar to Theorem 4.1 were established in the framework of some special families of thin codes. In this section we will draw similar investigations with regards to $\theta$-invariant codes. We will establish the following result:

Theorem 5.1

*Let $A$ be a finite or countable alphabet and let $X\subseteq A^{*}$ be a thin $\theta$-invariant prefix (resp. bifix, f.d.d., two-way f.d.d, uniformly synchronized, circular) code. Then the following conditions are equivalent:

(i) $X$ is complete

(ii) $X$ is a maximal code

(iii) $X$ is maximal in the family of prefix (bifix, f.d.d., two-way f.d.d, uniformly synchronized, circular) codes

(iv) $X$ is maximal in the family $\theta$-invariant codes

(v) $X$ is maximal in the family of $\theta$ invariant prefix (bifix, f.d.d., two-way f.d.d, uniformly synchronized, circular) codes.*

*Sketch proof. *According to to Theorem 4.2, and thanks to [9, Proposition 3.3.8], [9, Proposition 6.2.1], [9, Theorem 5.2.2], [15, Proposition 3.6] and [10, Theorem 3.5], if $X$ is complete then it is maximal in the family of $\theta$-invariant codes and maximal in the family of $\theta$-invariant prefix (bifix, f.d.d., two-way f.d.d, uniformly synchronized, circular) codes. Consequently, the proof of Proposition 5.1 comes down to establish that if $X$ is not complete, then it cannot be maximal in the family of $\theta$-invariant prefix (bifix, f.d.d., wo-way f.d.d, uniformly synchronized, circular) codes.

1. We begin by $\theta$-invariant prefix codes. At first, we assume that $\theta$ is an antimorphism. Since $X\cap XA^{+}=\emptyset$, and since $\theta$ is injective, we have $\theta(X)\cap\theta(XA^{+})=\emptyset$, thus $X\cap A^{+}X=\emptyset$, hence $X$ is also a suffix code. Assume that $X$ is not complete. According to [9, Proposition 3.3.8], it is non-maximal in both the families of prefix codes and suffix codes. Therefore a pair of words $y,y^{\prime}\in A^{+}\setminus X$ exists such $X\cup\{y\}$ ($X\cup\{y^{\prime}\}$) remains a prefix (suffix) code. By construction $X\cup\{yy^{\prime}\}$ remains a code which is both prefix and suffix.

Set Y=\{\theta^{i}(yy^{\prime})|i\in{\mathchoice{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\scriptstyle Z\kern-2.10002ptZ}}{\hbox{\sf\scriptscriptstyle Z\kern-0.99998ptZ}}}\}: since all the words in $Y$ have same positive length, $Y$ is a prefix code. From the fact that $\theta$ is one-to-one, for any integer i\in{\mathchoice{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\scriptstyle Z\kern-2.10002ptZ}}{\hbox{\sf\scriptscriptstyle Z\kern-0.99998ptZ}}} we obtain $\theta^{i}(\{yy^{\prime}\})\cap\theta^{i}(P(X))=\theta^{i}(X)\cap P(\theta^{i}(yy^{\prime}))=\emptyset$, consequently $X\cup Y$ remains a prefix code. By construction, $Y$ is $\theta$-invariant and it is not included in $X$, thus $X$ is not a maximal prefix code.

In the case where $\theta$ is a morphism, the preceding arguments may be simplified. Actually, a word $y\in A^{+}\setminus X$ exists such that $X\cup\{y\}$ remains a prefix code, thereferore by setting Y=\{\theta^{i}(y)|i\in{\mathchoice{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\scriptstyle Z\kern-2.10002ptZ}}{\hbox{\sf\scriptscriptstyle Z\kern-0.99998ptZ}}}\}, $X\cup Y$ remains a prefix code.

2. (sketch) The preceding arguments may be applied for proving that in any case, if $X$ is a non-complete bifix code, then it is maximal.

3,4) (sketch) In the case where $X$ is a (two-way) f.d.d.-code, according to [9, Proposition 5.2.1], similar arguments leads to a similar conclusion.

5. In the case where $X$ is a $\theta$-invariant uniformly synchronized code with verbal delay $k$ ([9, Section 10.2]), we must make use of different arguments. Actually, according to [15, Theorem 3.10], a complete uniformly synchronized code $X^{\prime}$ exists, with synchronizing delay $k$, and such that $X\subsetneq X^{\prime}$. More precisely, $X^{\prime}$ is the minimal generating set of the submonoid $M$ of $A^{*}$ which is defined by $M=(X^{2k}A^{*}\cap A^{*}X^{2k})\cup X^{*}$. According to Proposition 1 in the present paper, $X^{\prime}$ is $\theta$-invariant. Since $X$ is stictly included in $X^{\prime}$, it cannot be maximal in the family of $\theta$-invariant uniformly synchronized codes with delay $k$.

6. It remains to study the case where $X$ is a non-complete $\theta$-invariant circular code. Let $y\not\in F(X^{*})$ and let $z$ and $Z$ be computed as in Section 3: this guarantees that $X\cup Z$ is a $\theta$-invariant set. For proving that $X\cup Z$ is a circular code, by contradiction we assume that some words $y_{1},\cdots y_{n},y^{\prime}_{1},\cdots,y^{\prime}_{m}\in X\cup Z$ (with $m+n$ minimal), $p\in A^{*}$ and $s\in A^{+}$, exist such that the following equation holds:

[TABLE]

Once more since $X$ is a code, and since $Z\cap F(X^{*})=\emptyset$, without loss of generality we assume that at least one integer i\in{\mathchoice{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\textstyle Z\kern-3.99994ptZ}}{\hbox{\sf\scriptstyle Z\kern-2.10002ptZ}}{\hbox{\sf\scriptscriptstyle Z\kern-0.99998ptZ}}} exists such that $y_{i}\in Z$; similarly, at least one integer $j\in[1,m]$ exists such that $y^{\prime}_{j}\in Z$. By construction, we have $y_{i}\in F(y^{\prime}_{j}\cdots y^{\prime}_{m}y^{\prime}_{1}\cdots y^{\prime}_{j}\cdots y^{\prime}_{m}y^{\prime}_{1}\cdots y^{\prime}_{j})$; consequently, since all the words in $Z$ have the same length, a pair of integers $h,k\in[1,m]$ and a pair of words $u,v$ exist such that $uy_{i}v\in y^{\prime}_{h}X^{*}y^{\prime}_{k}$. According to Lemma 3, necessarily we have either $u=\varepsilon$ or $v=\varepsilon$; this implies $y_{i}=y^{\prime}_{h}$ or $y_{i}=y^{\prime}_{k}$, which contradicts the minimality of $m+n$, therefore $X\cup Z$ is a circular code.

6 Embedding a Regular Invariant Code into a Complete One

In this section, we consider a non-complete regular $\theta$-invariant code $X$ and we are interested in the problem of computing a complete one, namely $Y$, such that $X\subseteq Y$. Historically, such a question appears for the first time in [2], where the author asked for the possibility of embedding a finite code into a regular complete one. With regards to $\theta$-invariant codes, it seems natural to generalize the formula from [1] by making use of the code $Z$ that was introduced in Section 4. More precisely we would consider the set $X^{\prime}=X\cup(ZU)^{*}Z$, with $U=A^{*}\setminus(X^{*}\cup A^{*}ZA^{*})$. Unfortunately, with such a construction we observe that some pairs of words in $Z$ may overlap, therefore a non-trivial equation could hold among the words of $X^{\prime}$.

Nevertheless, we shall see that in the very special case where $\theta$ is an involutive antimorphism, convenient invariant overlapping-free words can be computed. Denote by $\theta_{0}$ the antimorphism which is generated by the identity onto $A$; in other words, with every word $w=w_{1}\cdots w_{n}\in A^{*}$ (with $w_{i}\in A$, for $1\leq i\leq n$), it associates $\theta_{0}(w)=w_{n}\cdots w_{1}$.

Proposition 2

Let $A$ be a finite alphabet and let $\theta$ be an antimorphism onto $A^{*}$, with $\theta\neq\theta_{0}$. If $\theta$ is involutive, then any non-complete regular $\theta$-invariant code can be embedded into a complete one.

Proof

Let $X$ be such that $\theta(X)=X$. Assume that $X$ is not complete. We will construct an overlapping-free word $t\notin F(X^{*})$ such that $\theta(t)=t$. At first, we consider a word $x$ such that $x\not\in F(X^{*})$ and $|x|\geq 2$. Without loss of generality, we assume that $x$ is overlapping-free (otherwise, as in [9, Proposition 1.3.6], a word $s$ exists such that $xs$ is overlapping-free). If $\theta(x)=x$, then we set $t=x$, otherwise let $y=cx$, where $c$ stands for the initial letter of $x$. Once more, without loss of generality we assume that $y$ is overlapping-free. By construction we have $y\in ccA^{+}$, thus $|y|\geq 3$ and $y_{1}=y_{2}=c$. If $\theta(y)=y$, then we set $t=y$. Now assume $\theta(y)\neq y$; according to the condition of Proposition 2, we have $\theta|_{A}\neq id_{A}$, therefore a pair of letters $a,b$ exists such that the following property holds:

[TABLE]

Set $t=a^{|y|}b\theta(y)yab^{|y|}$. By construction, we have $\theta(t)=t$, moreover the following property holds:

Claim

$t$ is an overlapping-free word.

Proof

Let $u,v\in A^{*}$ such that $ut=tv$, with $1\leq|u|\leq|t|-1$. According to the length of $u$, exactly one of the following cases occurs:

Case 1: $1\leq|u|\leq|y|$. With this condition, we obtain $t_{|y|+1}=b=(ut)_{|y|+1}=a$: a contradiction with $a\neq b$.

Case 2: $|y|+1\leq|u|\leq 2|y|$. This condition implies $\theta(y_{1})=t_{2|y|+1}=a$, therefore we obtain $c=y_{1}=\theta(a)=b$: a contradiction with (7).

Case 3: $|u|=2|y|+1$. We have $y=a^{|y|}$: since we have $|y|\geq 3$, this contradicts the fact that $y$ is overlaping-free.

Case 4: $|u|=2|y|+2$. We have $t_{2|y|+3}=y_{2}=c=(ut)_{2|y|+3}=a$. It follows from $y_{1}=y_{2}=c$ that $y=a^{|y|}$: once more this contradicts the fact that $y$ is overlapping-free.

Case 5: $2|y|+3\leq|u|\leq 3|y|+2$. By construction, we have $t_{|u|+|y|}=b=(ut)_{|uy|}=a$, a contradiction with (7).

Case 6: $3|y|+3\leq|u|\leq|t|-1=4|y|+1$. We obtain $t_{|u|+1}=b=(ut)_{|u|+1}=a$: once more this contradicts (7).

In any case we obtain a contradiction: this establishes the claim.

Since we have $t\not\in F(X^{*})$, and since $t$ is overlapping-free, the classical method from [1] may be applied without any modification to ensure that $X$ may embedded into a complete code, say $X^{\prime}$. Recall that it computes in fact a code $X^{\prime}$ as $X\cup V$, with $V=t(Ut)^{*}$ and $U=A^{*}\setminus(X^{*}\cup A^{*}tA^{*})$. Moreover, since $\theta(t)=t$, it is straightforward to verify that $\theta(X^{\prime})=X^{\prime}$.

With regards to the antimorphism $\theta_{0}$, necessarily the words $w$, $\theta_{0}(w)$ overlap, therefore the preceding methodology seems to be unreliable in the most general case. We finish our paper by stating the following open problem:

Problem. Let $A$ be a finite alphabet and let $\theta$ be an (anti)morphism onto $A^{*}$. Given a non-complete regular $\theta$-invariant code $X\subset A^{*}$, can we compute a complete regular $\theta$-invariant code $Y$ such that $X\subseteq Y$?