Watson-Crick strong bi-catenation on words

TL;DR

This paper introduces and analyzes the strong-$$-bi-catenation operation inspired by DNA Watson-Crick complementarity, exploring its properties, effects on language classes, and related conjugacy concepts.

Contribution

It defines the strong-$$-bi-catenation operation, studies its algebraic properties, closure under language classes, and extends the concept to language equations and conjugacy.

Findings

The operation is commutative but not associative.

Iterative application generates words over u, , and their combinations.

Regular, context-free, and context-sensitive languages are closed under this operation.

Abstract

In this paper we define and investigate the binary word operation of strong-$\phi$-bi-catenation (denoted by $\leftrightarrows_\phi$) where $\phi$ is either a morphic or an antimorphic involution. In particular, we concentrate on the mapping $\phi=\theta_{DNA}$, which models the Watson-Crick complementarity of DNA single strands. We show that such an operation is commutative and not associative and when iteratively applied to a word $u$, this operation generates words over $\{u, \theta(u)\}$. We then extend this operation to languages and show that the families of regular, context-free and context-sensitive languages are closed under the operation of strong-$\phi$-bi-catenation. We also define the notion of $\leftrightarrows_\theta$-conjugacy and study conditions on words $u$ and $v$ where $u$ is a $\leftrightarrows_\theta$-conjugate of $v$. We then extend this relation to language equations and provide solutions under some special cases.

Equations

$$\varphi (L)=\{\varphi (w):w\in L\}$$

$$L^R=\{w^R:w\in L\}.$$

$$u^{\circ (0)}=\{\lambda \},u^{\circ (1)}=u\circ \lambda ,u^{\circ (i)}=u^{\circ (i-1)}\circ u,i\ge 2$$

$$u\otimes v=\{uv,u\varphi (v),\varphi (u)v,\varphi (u)\varphi (v)\}.$$

$$u{\leftrightarrows }_{\varphi }v=(u\otimes v)\cup (v\otimes u)=u_{\varphi }{v}_{\varphi }\cup v_{\varphi }{u}_{\varphi }$$

$$u{\leftrightarrows }_{\varphi }v=\{uv,u\varphi (v),\varphi (u)v,\varphi (u)\varphi (v),vu,v\varphi (u),\varphi (v)u,\varphi (v)\varphi (u)\}$$

$$u{\leftrightarrows }_{\theta }v$$

$$\cup \{GCTAATC,GCTAGAT,TAGCATC,TAGCGAT\}$$

$$u{\leftrightarrows }_{\varphi }v=u_1{\leftrightarrows }_{\varphi }{v}_1=v_1{\leftrightarrows }_{\varphi }{u}_1$$

$$(u\circ v)\circ w\supseteq u\circ (v\circ w)$$

$$(u\circ v)\circ w\subseteq u\circ (v\circ w).$$

$$(u\circ v)\circ w=u\circ (v\circ w)$$

$$w\in \{uv,u\varphi (v),\varphi (u)v,\varphi (u)\varphi (v),vu,v\varphi (u),\varphi (v)u,\varphi (v)\varphi (u)\}$$

$$\begin{array}{rl}|w{|}_{a,\varphi (a)}& =|w{|}_a+|w{|}_{\varphi (a)}\\ & =|\varphi (v){|}_{a,\varphi (a)}+|u{|}_{a,\varphi (a)}\\ & =|\varphi (v){|}_{\varphi (a)}+|\varphi (v){|}_a+|u{|}_{a,\varphi (a)}\\ & =|v{|}_a+|v{|}_{\varphi (a)}+|u{|}_{a,\varphi (a)}\\ & =|u{|}_{a,\varphi (a)}+|v{|}_{a,\varphi (a)}\end{array}$$

$$u{\leftrightarrows }_{\varphi }v=u_{\varphi }{v}_{\varphi }\cup v_{\varphi }{u}_{\varphi }=v{\leftrightarrows }_{\varphi }u$$

$$(u{\leftrightarrows }_{\varphi }v){\leftrightarrows }_{\varphi }w=u{\leftrightarrows }_{\varphi }(v{\leftrightarrows }_{\varphi }w).$$

$$(u{\leftrightarrows }_{\varphi }v){\leftrightarrows }_{\varphi }w$$

$$=u_{\varphi }{v}_{\varphi }{w}_{\varphi }\cup v_{\varphi }{u}_{\varphi }{w}_{\varphi }\cup w_{\varphi }{u}_{\varphi }{v}_{\varphi }\cup w_{\varphi }{v}_{\varphi }{u}_{\varphi }$$

$$u{\leftrightarrows }_{\varphi }(v{\leftrightarrows }_{\varphi }w)$$

$$=u_{\varphi }{v}_{\varphi }{w}_{\varphi }\cup u_{\varphi }{w}_{\varphi }{v}_{\varphi }\cup v_{\varphi }{w}_{\varphi }{u}_{\varphi }\cup w_{\varphi }{v}_{\varphi }{u}_{\varphi }$$

$$L_1{\leftrightarrows }_{\varphi }{L}_2=\bigcup _{u\in L_1,v\in L_2}(u{\leftrightarrows }_{\varphi }v)$$

$$L^{{\leftrightarrows }_{\varphi }(0)}=\{\lambda \},L^{{\leftrightarrows }_{\varphi }(1)}=L_{\varphi },L^{{\leftrightarrows }_{\varphi }(i)}=(L{\leftrightarrows }_{\varphi }^{i-1}L),i\ge 1$$

$$L^{{\leftrightarrows }_{\varphi }(+)}=\bigcup _{k\ge 1}{L}^{{\leftrightarrows }_{\varphi }(k)}$$

$$u^{{\leftrightarrows }_{\varphi }(n)}=u^{\otimes (n)}=\{u_1{u}_2\cdots u_n:u_i=ATC\text{or}{u}_i=GAT,1\le i\le n\}$$

$$U{\leftrightarrows }_{\varphi }V=U_{\varphi }{V}_{\varphi }\cup V_{\varphi }{U}_{\varphi }$$

$$\begin{array}{rl}U{\leftrightarrows }_{\varphi }V& =\bigcup _{u_1\in U,u_2\in V}{u}_1{\leftrightarrows }_{\varphi }{u}_2\\ & =\bigcup _{u_1\in U,u_2\in V}((u_1{)}_{\varphi }(u_2{)}_{\varphi }\cup (u_2{)}_{\varphi }(u_1{)}_{\varphi }))\\ & =\bigcup _{u_1\in U,u_2\in V}(u_1\otimes u_2)\cup (u_2\otimes u_1)\\ & =U\otimes V\cup V\otimes U\\ & =U_{\varphi }{V}_{\varphi }\cup V_{\varphi }{U}_{\varphi }\end{array}$$

$$\begin{array}{rl}{L}^{{\leftrightarrows }_{\varphi }(2)}=L{\leftrightarrows }_{\varphi }L& =\bigcup _{u_1,u_2\in L}{u}_1{\leftrightarrows }_{\varphi }{u}_2\\ & =\bigcup _{u_1,u_2\in L}((u_1{)}_{\varphi }(u_2{)}_{\varphi }\cup (u_2{)}_{\varphi }(u_1{)}_{\varphi }))\\ & =\bigcup _{u_1,u_2\in L}(u_1{)}_{\varphi }(u_2{)}_{\varphi }\\ & =\{u_1{u}_2:u_1,u_2\in L_{\varphi }\}\end{array}$$

$$\begin{array}{rl}{L}^{{\leftrightarrows }_{\varphi }(n+1)}=L^{{\leftrightarrows }_{\varphi }(n)}{\leftrightarrows }_{\varphi }L& =\bigcup _{u\in L^{{\leftrightarrows }_{\varphi }(n)},u^{\prime }\in L}u{\leftrightarrows }_{\varphi }{u}^{\prime }\\ & =\bigcup _{u\in L^{{\leftrightarrows }_{\varphi }(n)},u^{\prime }\in L}((u{)}_{\varphi }(u^{\prime }{)}_{\varphi }\cup (u^{\prime }{)}_{\theta }(u{)}_{\theta }))\\ & =\bigcup _{u\in L^{{\leftrightarrows }_{\varphi }(n)},u^{\prime }\in L}(u{)}_{\varphi }(u^{\prime }{)}_{\varphi }\\ & =\{u{u}^{\prime }:u\in L^{{\leftrightarrows }_{\varphi }(n)},u^{\prime }\in L_{\varphi }\}\\ & =\{u_1{u}_2...u_n{u}_{n+1}:u_i\in L_{\varphi }\}\end{array}$$

$$L^{{\leftrightarrows }_{\varphi }(n)}{\leftrightarrows }_{\varphi }{L}^{{\leftrightarrows }_{\varphi }(m)}=L^{{\leftrightarrows }_{\varphi }(n+m)}$$

$$L^{{\leftrightarrows }_{\varphi }(n+1)}$$

$$=\{u_1{u}_2...u_n{u}_{n+1}:u_i\in L_{\varphi }\}$$

$$L^{{\leftrightarrows }_{\varphi }(n+m)}=\{u_1{u}_2...u_{n+m}:u_i\in L_{\varphi }\}=L^{{\leftrightarrows }_{\varphi }(n)}{\leftrightarrows }_{\varphi }{L}^{{\leftrightarrows }_{\varphi }(m)}$$

Full text

**Watson-Crick strong bi-catenation on words ††footnotetext: ∗Corresponding author. Kalpana Mahalingam ††footnotetext: 2020 Mathematics Subject Classification. FILL SUBJECT MSCs HERE.  Keywords and Phrases. Binary Operation, Bi-Catenation, Watson-Crick powers

*Kalpana Mahalingam

Department Of Mathematics,

Indian Institute of Technology,

Chennai-600036. India.

Email : kmahalingam@iitm.ac.in*

In this paper we define and investigate the binary word operation of strong-$\phi$-bi-catenation (denoted by $\leftrightarrows_{\phi}$) where $\phi$ is either a morphic or an antimorphic involution. In particular, we concentrate on the mapping $\phi=\theta_{DNA}$, which models the Watson-Crick complementarity of DNA single strands. We show that such an operation is commutative and not associative and when iteratively applied to a word $u$, this operation generates words over $\{u,\theta(u)\}$. We then extend this operation to languages and show that the families of regular, context-free and context-sensitive languages are closed under the operation of strong-$\phi$-bi-catenation. We also define the notion of $\leftrightarrows_{\theta}$-conjugacy and study conditions on words $u$ and $v$ where $u$ is a $\leftrightarrows_{\theta}$-conjugate of $v$. We then extend this relation to language equations and provide solutions under some special cases.

1. Introduction

Combinatorics on words focuses on the study of words and formal languages([12, 29]). A word is basically formed from alphabets by simply juxtaposing the alphabets. Such an operation is called as concatenation, which is indeed a basic binary operation on words. Some of the well known basic word operations defined and studied in literature are quotient, shuffle([30, 23]), bi-catenation([3]), $k$-catenation([24]), insertion([28]) and deletion([28]) to name a few. These operations were naturally extended to languages and authors in general studied closure properties of the families in the Chomsky hierarchy under the above operations among others.

In [14], the authors used the $k$-catenation operation defined in [24] to define $k$-involution codes. The $k$-involution codes formally denote DNA strands (possibly used in DNA based computations) avoiding certain non-specific hybridizations that pose potential problems for the results of the biocomputation. A DNA strand is basically a word over the alphabet $\{A,G,C,T\}$ and its Watson-Crick complement is mathematically formalized by an antimorphic involution denoted by $\theta_{DNA}$ which is an antimorphism ($\theta_{DNA}(uv)=\theta_{DNA}(v)\theta_{DNA}(u)$) and an involution ($\theta^{2}_{DNA}(u)=u$) that maps $A\mapsto T$, $C\mapsto G$ and vice-versa. Concatenation of DNA strands is a process to combine various DNA strands linearly to form new DNA strands. One such recombination is obtained by repeatedly concatenating a DNA strand $u$ and its Watson-Crick complement $\theta_{DNA}(u)$ in random order. Such a strand is called a $\theta_{DNA}$-power of $u$. The authors in [15, 22] extended the notion of catenation to $\phi$-catenation and strong $\phi$-catenation respectively that generates all possible $\phi$-powers of a given word where $\phi$ is either a morphic or an antimorphic involution.

Observe that, the operation strong-$\phi$-catenation, when applied iteratively to a word $u$, results in all possible $\phi$-powers of $u$ (i.e.) words that belong to the set $\{u,\phi(u)\}^{+}$. However, when this operation is applied between two distinct words, say $u$ and $v$, the resulting set does not provide all possible combinations of words of the set $\{u,v,\phi(u),\phi(v)\}$ as the catenation is one-sided. To fill this gap, we introduce the notion of strong-$\phi$-bi-catenation of words.

In this paper, we combine the notion of strong-$\phi$-catenation and bi-catenation to obtain a new binary operation which we call strong-$\phi$-bi-catenation of words. We define and investigate some basic properties of strong-$\phi$-bi-catenation in Section 3. We also mention its connection to the previously defined notion of strong-$\phi$-catenation. In Section 3.1, we naturally extend the operation to languages and show that the families of regular, context-free and context-sensitive languages are closed under this operation.

Section 3.2 briefly explore closure properties of languages closed under strong-$\phi$-bi-catenation. Section 4 investigates conjugacy with respect to $\leftrightarrows_{\phi}$ and Section 5 studies some language equations with respect to the strong $\phi$-bi-catenation operation. We end the paper with few concluding remarks.

2. Preliminaries

Let $\Sigma$ be a finite alphabet. We denote by $\Sigma^{*}$ the set of all words over $\Sigma$ including the empty word $\lambda$. By $\Sigma^{+}$, we denote the set of all non-empty words over $\Sigma$. The length of a word $w\in\Sigma^{*}$ is the number of letter occurrences in $w$, denoted by $|w|$; i.e. if $w=a_{1}a_{2}...a_{n},~a_{i}\in\Sigma$ then $|w|=n$. $|w|_{a}$ denotes the number of occurrences of $a$ in $w$. The reverse of the word $w=a_{1}a_{2}\cdots a_{n-1}a_{n}$ denoted by $w^{R}$ is the word $a_{n}a_{n-1}\cdots a_{2}a_{1}$ where $a_{i}\in\Sigma$, $1\leq i\leq n$. A word $w$ is called primitive it is not the non-trivial power of another word; i.e. if $w=u^{i}$ then $w=u$ and $i=1$. The primitive root of a word $w$ is the shortest $u$ such that $w=u^{i}$ for some $i$, denoted by $\rho(w)=u$. We denote by $\mathcal{Q}$, the set of all primitive words.

We first recall some results from [25, 22].

Lemma 2.1**.**

[25]** Let $u,\;v,\;w\in\Sigma^{+}$ be such that, $uv=vw$, then for $k\geq 0$, $x\in\Sigma^{+}$ and $y\in\Sigma^{*}$, $u=xy$, $v=(xy)^{k}x$, $w=yx$.

Lemma 2.2**.**

[25]** If $xy=yx$ then $x$ and $y$ are powers of a common word; i.e. $x=u^{i}$ and $y=u^{j}$ for some $u\in\Sigma^{+}$.

Lemma 2.3**.**

[22]** For $x,y\in\Sigma^{+}$, if $yxx=xxy$, then $x=\alpha^{m}$ and $y=\alpha^{n}$ for some $m,n\geq 1$ and $\alpha\in\Sigma^{+}$.

A mapping $\phi:\Sigma^{*}\rightarrow\Sigma^{*}$ is called a morphism on $\Sigma^{*}$ if for all words $u,v\in\Sigma^{*}$ we have that $\phi(uv)=\phi(u)\phi(v)$, an antimorphism on $\Sigma^{*}$ if $\phi(uv)=\phi(v)\phi(u)$ and an involution if $\phi(\phi(a))=a$ for all $a\in\Sigma$.

A mapping $\phi:\Sigma^{*}\rightarrow\Sigma^{*}$ is called a morphic involution on $\Sigma^{*}$ (respectively, an antimorphic involution on $\Sigma^{*}$) if it is an involution on $\Sigma$ extended to a morphism (respectively, to an antimorphism) on $\Sigma^{*}$. For convenience, in the remainder of this paper we use the convention that the letter $\phi$ denotes an involution that is either morphic or antimorphic (such a mapping will be termed (anti)morphic involution), that the letter $\theta$ denotes an antimorphic involution, and that the letter $\mu$ denotes a morphic involution. For $L\subseteq\Sigma^{*}$ and an involution $\phi$, we define,

[TABLE]

[TABLE]

A word $u$ is a conjugate of $v$ if for some $w$, $uw=wv$. Two words $u$ and $v$ are said to commute if $uv=vu$. The concept of conjugacy and commutativity was extended to the notion of an involution map $\theta$ in [18]. Recall that $u$ is said to be a $\theta$-conjugate of $w$ if $uv=\theta(v)w$ for some $v\in\Sigma^{+}$, and $u$ is said to $\theta$-commute with $v$ if $uv=\theta(v)u$. We recall the following result from [18] characterizing $\theta$-conjugacy and $\theta$-commutativity for an antimorphic involution $\theta$ (if $\theta=\theta_{DNA}$, these are called Watson-Crick conjugacy, respectively Watson-Crick commutativity). For an antimorphic involution $\theta$, a word $u$ is called a $\theta$-palindrome if $u=\theta(u)$. The set of all $\theta$-palindromes is denoted by $P_{\theta}$.

Proposition 1**.**

[18]** For $u,v,w\in\Sigma^{+}$ and $\theta$ an antimorphic involution,

1. (1)

If $uv=\theta(v)w$, then either there exists $x\in\Sigma^{+}$ and $y\in\Sigma^{*}$ such that $u=xy$ and $w=y\theta(x)$, or $u=\theta(w)$. 2. (2)

If $uv=\theta(v)u$, then $u=x(yx)^{i}$, $v=yx$, for some $i\geq 0$ and $\theta$-palindromes $x\in\Sigma^{*},y\in\Sigma^{+}$.

We recall the following from [17].

Proposition 2**.**

[17]** Let $x,y\in\Sigma^{+}$ and $\theta$ an antimorphic involution, such that $xy=\theta(y)\theta(x)$ and $yx=\theta(x)\theta(y)$. Then, one of the following holds:

1. (1)

$x=\alpha^{i}$, $y=\alpha^{k}$ for some $\alpha\in P_{\theta}$ 2. (2)

$x=[\theta(s)s]^{i}\theta(s)$, $y=[s\theta(s)]^{k}s$ for some $s\in\Sigma^{+}$, $i,k\geq 0$.

We recall the following from [22].

Definition 1**.**

For a given $u\in\Sigma^{*}$, and an (anti)morphic involution $\phi$, the set $\{u,\phi(u)\}$ is denoted by $u_{\phi}$, and is called a $\phi$-complementary pair, or $\phi$-pair for short. The length of a $\phi$-pair $u_{\phi}$ is defined as $|u_{\phi}|=|u|=|\phi(u)|$.

It was also remarked in [22] that, for $u\in\Sigma^{+}$ and $\phi$, an (anti)morphic involution, $|u_{\phi}|_{a}=|u|_{a}+|\phi(u)|_{a}$, $|\phi(u)|_{a}=|u|_{\phi(a)}$ and $|\phi(u)|_{\phi(a)}=|u|_{a}$. For $L\subseteq\Sigma^{*}$, we denote $L_{\phi}=L\cup\phi(L)$. A word is called $\phi$-power of a word $u$ if it is of the form $u_{1}u_{2}...u_{n}$ where $u_{1}=u$ and $u_{i}\in u_{\phi}$ for $2\leq i\leq n$.

3. Strong $\phi$-bi-catenation

In this section, we define and study a new binary operation called the strong $\phi$-bi-catenation. The basic string operation catenation is a binary operation that maps $(u,v)$ to $uv$. The catenation operation has several generalizations. The first one is the notion of Bi-catenation ([3]), which is a binary operation which maps $(u,v)$ to $\{uv,vu\}$. Motivated by the Watson-crick complemantarity of DNA strands, the authors in [15], defined the concept of $\phi$-catenation which incorporates an (anti)-morphic involution mapping $\phi$. The $\phi$-catenation maps $(u,v)$ to $\{uv,u\phi(v)\}$. This concept was further generalized in [22] to define the strong $\phi$-catenation, which generates all possible $\phi$ powers of a given word $u$, (i.e.) all words in the set $\{u,\phi(u)\}^{+}$. In this section, we introduce the notion of strong $\phi$-bi-catenation operation which is indeed a generalization of bi-catenation defined in [3] as well as the strong $\phi$-catenation operation([22]).

Binary operation $\circ$ on $\Sigma^{*}$ is a map $\circ:\Sigma^{*}\times\Sigma^{*}\to 2^{\Sigma^{*}}$. For a given binary operation $\circ$, the $i$-th $\circ$-power of a word is defined by :

[TABLE]

Note that, depending on the operation $\circ$, the $i$-th power of a word can be a singleton word, or a set of words.

A binary operation called $\phi$-catenation denoted by $\odot$, was defined in [15] which generates some $\phi$ powers of a word $u$ under consideration, when $\odot$ is applied iteratively. However, this concept was extended to the notion of strong-$\phi$-catenation denoted by $\otimes$, that generates all the non-trivial $\phi$-powers of $u$, that is, the union of the sets $\{u,\theta(u)\}^{n}$, $n\geq 2$. We begin the section by recalling the formal definition of strong $\phi$-catenation.

Definition 2**.**

[22]** Given an (anti)morphic involution $\phi$ on $\Sigma^{*}$ and two words $u,v\in\Sigma^{*}$, we define the strong-$\phi$-catenation operation of $u$ and $v$ with respect to $\phi$ as

[TABLE]

We recall the following from [22].

Proposition 3**.**

[22]** For an antimorphic involution $\theta$ and $u,v\in\Sigma^{+}$, $u\otimes v=v\otimes u$ iff (i) $u=v$, or (ii) $u=\theta(v)$, or (iii) $u$ and $v$ are powers of a common $\theta$-palindrome.

We now formally define the notion of strong $\phi$-bi-catenation operation.

Definition 3**.**

We define strong $\phi$-bi-catenation $({\leftrightarrows_{\phi}})$ as

[TABLE]

Writing explicitly all the terms of $u\leftrightarrows_{\phi}v$ we get,

[TABLE]

Example 1**.**

Consider the case of $\theta=\theta_{DNA}$, the Watson-Crick complementary function that maps $A\leftrightarrow T$ and $C\leftrightarrow G$ and the words $u=ATC$, $v=GCTA$. Then,

[TABLE]

which is the set of all bi-catenations that involve words $u$ and $v$ and their images under $\theta_{DNA}$.

We have the following remark which follows directly from definition.

Remark 1**.**

Let $\phi$ be an (anti)morphic involution on $\Sigma^{*}$ and $u,v\in\Sigma^{*}$. Then, for $u_{1}\in u_{\phi}$ and $v_{1}\in v_{\phi}$,

[TABLE]

We first observe the following which is straightforward from the definition.

Lemma 3.1**.**

Let $\phi$ be an (anti)morphic involution on $\Sigma^{*}$ and $u,v\in\Sigma^{*}$. Then, $x\in u\leftrightarrows_{\phi}v$ iff $\phi(x)\in u\leftrightarrows_{\phi}v$.

A bw-operation $\circ$ is called length-increasing if for any $u,v\in\Sigma^{+}$ and $w\in u\circ v$, $|w|>max\{|u|,|v|\}$. A bw-operation $\circ$ is called propagating if for any $u,v\in\Sigma^{*}$, $a\in\Sigma$ and $w\in u\circ v$, $|w|_{a}=|u|_{a}+|v|_{a}$. In [15], these notions were generalized to incorporate an (anti)morphic involution $\phi$, as follows. A bw-operation $\circ$ is called $\phi$-propagating if for any $u,v\in\Sigma^{*}$, $a\in\Sigma$ and $w\in u\circ v$, $|w|_{a,\phi(a)}=|u|_{a,\phi(a)}+|v|_{a,\phi(a)}$. It was shown in [15] that the operation $\phi$-catenation is not propagating but is $\phi$-propagating. The concept of $\phi$-catenation was extended to strong $\phi$-catenation in [22]. It was shown in [22] that the operation strong $\phi$-catenation is also not propagating but is $\phi$-propagating.

A bw-operation $\circ$ is called left-inclusive if for any three words $u,v,w\in\Sigma^{*}$ we have

[TABLE]

and is called right-inclusive if

[TABLE]

A bw-operation $\circ$ is associative if for any three words $u,v,w\in\Sigma^{*}$ we have

[TABLE]

Similar to the properties of the operation $\phi$-catenation and strong $\phi$-catenation investigated in [15, 22], one can easily observe that the strong-$\phi$-bi-catenation operation is length increasing, not propagating and $\phi$-propagating. In [15], it was shown that for a morphic involution the $\phi$-catenation operation is trivially associative, whereas for an antimorphic involution the $\phi$-catenation operation is not associative. In contrast, it was shown in [22], that the strong-$\phi$-catenation operation is right inclusive, left inclusive, as well as associative, when $\phi$ is a morphic as well as an antimorphic involution. We also observe that the operation strong $\phi$-bi-catenation operation is commutative and not associative.

Lemma 3.2**.**

Let $\phi$ be an (anti)morphic involution. The strong $\phi$-bi-catenation operation is length increasing, not propagating, $\phi$-propagating, commutative, not associative, and neither right nor left inclusive.

Proof.

We show that the binary operation $\leftrightarrows_{\phi}$ is length increasing, $\phi$-propagating and commutative.

1. (1)

Let $u,v,w\in\Sigma^{+}$ such that $w\in u\leftrightarrows_{\phi}v$. Then, $|w|=|u|+|v|$ and hence $|w|>max\{|u|,|v|\}$. Thus, the operation $\leftrightarrows_{\phi}$ is length increasing. 2. (2)

Consider the words $u,v$ from Example 1. Note that, for $w=GATGCTA\in u\leftrightarrows_{\phi}v$, $|w|_{G}=2\neq|u|_{G}+|v|_{G}=0+1=1$. Hence, the operation $\leftrightarrows_{\phi}$ is not propagating. 3. (3)

Let $w,u,v\in\Sigma^{+}$ be such that $w\in u\leftrightarrows_{\phi}v$. Then,

[TABLE]

Suppose, $w=\phi(v)u$ then,

[TABLE]

The other cases are similar and we omit them. Hence, the operation $\leftrightarrows_{\phi}$ is $\phi$-propagating. 4. (4)

One can easily observe from the definition that for $u,v\in\Sigma^{*}$,

[TABLE]

Hence, $\leftrightarrows_{\phi}$ is commutative. 5. (5)

Note that, for $u=AG$, $v=CA$ and $w=AC$ and $\theta=\theta_{DNA}$, we have $CACTAC\in v_{\phi}u_{\phi}w_{\phi}\subseteq(u\leftrightarrows_{\phi}v)\leftrightarrows_{\phi}w$ but not in $u\leftrightarrows_{\phi}(v\leftrightarrows_{\phi}w)$ Thus, the operation $\leftrightarrows_{\phi}$ is not associative. 6. (6)

It is evident from the example given in Item 5 that the operation $\leftrightarrows_{\phi}$ is neither right nor left inclusive. ∎

∎

We now give a sufficient condition on words $u$ and $w$ such that $(u\leftrightarrows_{\phi}v)\leftrightarrows_{\phi}w=u\leftrightarrows_{\phi}(v\leftrightarrows_{\phi}w)$.

Lemma 3.3**.**

Given an (anti)morphic involution $\phi$ and $u,v,w\in\Sigma^{+}$ such that $u_{\phi}w_{\phi}=w_{\phi}u_{\phi}$ then,

[TABLE]

Proof.

Let $u,v,w\in\Sigma^{+}$. Then,

[TABLE]

and,

[TABLE]

Thus, if $u_{\phi}w_{\phi}=w_{\phi}u_{\phi}$, then $(u\leftrightarrows_{\phi}v)\leftrightarrows_{\phi}w=u\leftrightarrows_{\phi}(v\leftrightarrows_{\phi}w)$. ∎

3.1. Extension to Languages

In this section we extend the $\leftrightarrows_{\phi}$ operation to languages. We use the notation $L_{\phi}$ to denote the set $L\cup\phi(L)$. Given $L_{1},L_{2}\subseteq\Sigma^{*}$ define,

[TABLE]

and $L_{1}\leftrightarrows_{\phi}\emptyset=\emptyset\leftrightarrows_{\phi}L_{2}=\emptyset$ and $L_{1}\leftrightarrows_{\phi}^{0}L_{2}=L_{1}\cup\phi(L_{1})\cup L_{2}\cup\phi(L_{2})$. The iterated strong -bi-$\phi$-catenation operation $\leftrightarrows_{\phi}^{i}$ for $i\geq 1$ and languages $L_{1}$ and $L_{2}$ is defined as $L_{1}\leftrightarrows_{\phi}^{i}L_{2}=(L_{1}\leftrightarrows_{\phi}^{i-1}L_{2})\leftrightarrows_{\phi}L_{2}$. The $i$-th $\leftrightarrows_{\phi}$-power of a non-empty language $L$ is defined as

[TABLE]

The $+$-closure of a non-empty language $L$ with respect to a bw-operation $\leftrightarrows_{\phi}$, denoted by $L^{\leftrightarrows_{\phi}(+)}$ is defined as

[TABLE]

We say that $L$ is $\leftrightarrows_{\phi}$-closed if for any $u$ and $v$ in $L$, $u\leftrightarrows_{\phi}v$ is a subset of $L$. We say that a binary operation $\leftrightarrows_{\phi}$ is $plus$-$closed$ if for any non-empty language $L\subset\Sigma^{*}$, $L^{\leftrightarrows_{\phi}(+)}$ is also $\leftrightarrows_{\phi}$-closed.

We first observe that, $u\leftrightarrows_{\phi}u=u\otimes u$ and hence, $u^{\leftrightarrows_{\phi}(n)}=u^{\otimes(n)}$ for all $n\geq 0$. Thus, for $u=ATC$ and $\theta=\theta_{DNA}$ we have,

[TABLE]

We observe the following.

Lemma 3.4**.**

For a language $U,V\subset\Sigma^{*}$,

[TABLE]

Proof.

For $U,V\subseteq\Sigma^{*}$, we have,

[TABLE]

∎

We now have the following observation which characterizes the form of words in $L^{\leftrightarrows_{\phi}(n)}$ when the strong-$\phi$-bi-catenation operation is applied iteratively.

Proposition 1**.**

For a language $L\subset\Sigma^{*}$, $L^{\leftrightarrows_{\phi}(n)}$ is the collection of all words of the form $u_{1}u_{2}...u_{n}$ where $u_{i}\in L_{\phi}$ and $n\geq 2$.

Proof.

We use induction on $n$. For $n=2$,

[TABLE]

Now assume that $L^{\leftrightarrows_{\phi}(n)}=\{u_{1}u_{2}...u_{n}~:~u_{i}\in L_{\phi}\}$. For $n+1$,

[TABLE]

Hence the result. ∎

Proposition 4**.**

Let $L\subset\Sigma^{*}$. For any morphic or antimorphic involution,

[TABLE]

Proof.

Using the above result (Proposition 1), we have

[TABLE]

Repeating the $\leftrightarrows_{\phi}$ operation $m$ times and using above result (Proposition 1) we have,

[TABLE]

∎

Corollary 3.4.1**.**

The operation $\leftrightarrows_{\phi}$ is plus-closed; i.e., for any $u,v\in L^{\leftrightarrows_{\phi}(+)}$, we have $u\leftrightarrows_{\phi}v\in L^{\leftrightarrows_{\phi}(+)}$.

Proof.

Let $u,v\in L^{\leftrightarrows_{\phi}(+)}$. Then, there exist $n$ and $m$ such that $u\in L^{\leftrightarrows_{\phi}(n)}$ and $v\in L^{\leftrightarrows_{\phi}(m)}$. By Proposition 4, we have $u\leftrightarrows_{\phi}v\in L^{\leftrightarrows_{\phi}(n+m)}$. Thus, $u\leftrightarrows_{\phi}v\in L^{\leftrightarrows_{\phi}(+)}$. ∎

One can also easily observe that for a regular (context-free, context-sensitive) language $L$, $\phi(L)$ is also regular (context-free, context-sensitive respectively). Thus, from Lemma 3.4, we conclude the following.

Theorem 3.5**.**

The families of regular, context-free and context-sensitive languages are closed under the operation of strong bi-$\phi$-catenation.

3.2. $\leftrightarrows_{\theta}$-closed Languages

A language $L$ is closed under the mapping $\phi$ if $x\in L$ implies $\phi(x)\in L$ i.e., $L=L_{\phi}$ and is closed under catenation if $u,v\in L$, imply $uv\in L$. A language $L$ is $\leftrightarrows_{\phi}$-closed if $u,v\in L$ imply $u\leftrightarrows_{\phi}v\subseteq L$. It was shown in Corollary 3.4.1 that the operation $\leftrightarrows_{\phi}$ is plus-closed.

Lemma 3.6**.**

If $L$ is closed under $\phi$ and catenation then $L$ is closed under $\leftrightarrows_{\phi}$.

Proof.

If $L$ is closed under $\phi$ then $L=L_{\phi}$ and if $L$ is closed under catenation then, $L^{2}=L$. From Lemma 3.4 we observe that, $L\leftrightarrows_{\phi}L=L_{\phi}L_{\phi}=L^{2}=L$. Hence, $L$ is closed under $\leftrightarrows_{\phi}$. ∎

The converse of Lemma 3.6 is not true in general. For example, consider the alphabet $\{a,b\}$ and an antimorphic involution $\theta$ such that $\theta(a)=a$ and $\theta(b)=b$. Let $L=\{ab\}\cup\{x~:~x\in\{a,b\}^{+},~|x|\geq 3\}$. Note that, $L$ is closed under catenation and $L$ is closed under $\leftrightarrows_{\theta}$ but $L$ is not closed under $\theta$ as $\theta(ab)=ba\notin L$.

We now give an example of a language $L$ such that $L$ is closed under $\leftrightarrows_{\phi}$.

Example 2**.**

Consider the alphabet $\{a,b\}$ and $\phi$ be an (anti)morphic involution that maps $a$ to $b$ and vice-versa. Let $L=\{w~:~|w|_{a}=|w|_{b}\}\subseteq\Sigma^{+}$. Note that for any $x\in L$, $\phi(x)\in L$ and for $x,y\in L$, $xy\in L$. Hence by Lemma 3.6, $L$ is closed under $\leftrightarrows_{\phi}$.

Lemma 3.7**.**

Let $L$ be such that $L$ is $\leftrightarrows_{\phi}$ closed. Then, $L_{1}L_{2}L_{3}\cdots L_{n}\subseteq L$ for $L_{i}\in L_{\phi}$ for $1\leq i\leq n$ and $n\geq 2$.

Proof.

We first observe from Lemma 3.4 that,

[TABLE]

Since, $L$ is closed under $\leftrightarrows_{\phi}$, we have $L\leftrightarrows_{\phi}L\subseteq L$ which implies that $L_{1}L_{2}\subseteq L$ for $L_{1},L_{2}\in L_{\phi}$. One can easily prove by induction that, $L^{n}\leftrightarrows_{\phi}L^{n}=L_{1}L_{2}L_{3}\cdots L_{n}$ for $L_{i}\in L_{\phi}$ for $1\leq i\leq n$ and $n\geq 2$. Since $L_{1}L_{2}\subseteq L$ for $L_{1},L_{2}\in L_{\phi}$ we have that $L_{1}L_{2}L_{3}\cdots L_{n}\subseteq L$ for $L_{i}\in L_{\phi}$ for $1\leq i\leq n$ and $n\geq 2$ and hence the result. ∎

Lemma 3.8**.**

Let $L$ be such that $L$ is $\leftrightarrows_{\phi}$ closed. Then, the following are true.

1. (1)

$L$* is closed under catenation.* 2. (2)

$L^{R}$* is closed under $\leftrightarrows_{\phi}$.* 3. (3)

$\phi(L)$* is closed under $\leftrightarrows_{\phi}$.* 4. (4)

For all $A,B\in L_{\phi}^{n}$, $A\leftrightarrows_{\phi}B\subseteq L$.

Proof.

Given that $L$ is closed under $\leftrightarrows_{\phi}$ (i.e.) for all $u,v\in L$, we have $u\leftrightarrows_{\phi}v\subseteq L$. Then let,

[TABLE]

1. (1)

Note that, $u\leftrightarrows_{\phi}v\subseteq L$ implies that $uv\in L$ for all $u,v\in L$. Hence, $L$ is closed under catenation. 2. (2)

For $u,v\in L$ we have $u^{R},v^{R}\in L^{R}$. Then, $A=u\leftrightarrows_{\phi}v\subseteq L$ and $A^{R}=\{u^{R}v^{R},\phi(u^{R})v^{R},u^{R}\phi(v^{R}),\phi(u^{R})\phi(v^{R}),v^{R}u^{R},\phi(v^{R})u^{R},\\ v^{R}\phi(u^{R}),\phi(v^{R})\phi(u^{R})\}=u^{R}\leftrightarrows_{\phi}v^{R}\subseteq L^{R}$. Hence, $L^{R}$ is closed under $\leftrightarrows_{\phi}$. 3. (3)

It is easy to observe that, $u\leftrightarrows_{\phi}v=\phi(u)\leftrightarrows_{\phi}\phi(v)$ and $A=u\leftrightarrows_{\phi}v\subseteq L$ implies $\phi(A)\subseteq\phi(L)$. But, $A=u\leftrightarrows_{\phi}v=\phi(u)\leftrightarrows_{\phi}\phi(v)=\phi(A)\subseteq\phi(L)$. Thus, $\phi(L)$ is closed under $\leftrightarrows_{\phi}$. 4. (4)

Since $L$ is closed under $\leftrightarrows_{\phi}$, we have by Lemma 3.7, $A\leftrightarrows_{\phi}B=L^{n}\leftrightarrows_{\phi}L^{n}=L_{1}L_{2}L_{3}\cdots L_{n}\subseteq L$ for all $n\geq 2$ and $A,B\in L_{\phi}^{n}$, $L_{i}\in\{L,\phi(L)\}$. Hence, the result.

∎

We now have the following example.

Example 3**.**

Consider the alphabet $\{a,b,c\}$ and $\phi$ an (anti)morphic involution that maps $a$ to $b$ and vice-versa and $\phi(c)=c$. Let $L_{1}=\{w~:~|w|_{a}+|w|_{b}=|w|_{c}\}$ and $L_{2}=\{w~:~|w|_{a}=|w|_{b}=|w|_{c}\}$. Note that for any $x\in L_{1}$, $\phi(x)\in L_{1}$ and for $x,y\in L_{1}$, $xy\in L_{1}$. Hence, $L_{1}$ is closed under $\leftrightarrows_{\phi}$. Similarly one can verify that $L_{2}$ is closed under $\leftrightarrows_{\phi}$.

It is clear from the above example that in general for a given $\leftrightarrows_{\phi}$-closed language $L_{1}$, $L_{1}^{c}$ is not $\leftrightarrows_{\phi}$-closed.

Lemma 3.9**.**

Let $L_{1},L_{2}\subseteq\Sigma^{+}$ be such that $L_{1}$ and $L_{2}$ are closed under $\leftrightarrows_{\phi}$. Then the following are true.

1. (1)

$L_{1}^{c}$* is not closed under $\leftrightarrows_{\phi}$.* 2. (2)

$L_{1}\cap L_{2}$* is closed under $\leftrightarrows_{\phi}$.* 3. (3)

$L_{1}\cup L_{2}$* is not closed under $\leftrightarrows_{\phi}$.*

Proof.

1. (1)

Consider the language $L_{1}=\{w~:~|w|_{a}=|w|_{b}\}$ discussed in Example 2. Then, $L_{1}^{c}=\{w~:~|w|_{a}\neq|w|_{b}\}$ and for $u=aba,v=bab\in L_{1}^{c}$, we have $uv=ababab\in u\leftrightarrows_{\phi}v$ but $uv\notin L_{1}^{c}$. Thus, for a given $L$ which is closed under $\leftrightarrows_{\phi}$, $L_{1}^{c}$ is not necessarily closed under $\leftrightarrows_{\phi}$. 2. (2)

Given that $L_{1}$ and $L_{2}$ are closed under $\leftrightarrows_{\phi}$. Let $u,v\in L_{1}\cap L_{2}$. Then, $u\leftrightarrows_{\phi}v\in L_{1}\cap L_{2}$. Thus, $L_{1}\cap L_{2}$ is closed under $\leftrightarrows_{\phi}$. 3. (3)

Consider $L_{1}$ and $L_{2}$ from Example 3. Note that, $L_{1}=\phi(L_{1})$, $L_{2}=\phi(L_{2})$ and $abc,bcca\in L_{1}\cup L_{2}$. But, $abcbcca\in abc\leftrightarrows_{\phi}bcca\nsubseteq L_{1}\cup L_{2}$. Hence, $L_{1}\cup L_{2}$ is not $\leftrightarrows_{\phi}$-closed.

∎

We now define the $\leftrightarrows_{\phi}$-Iterative closure of a language $L$ denoted by $cl_{\leftrightarrows_{\phi}}(L)$

Definition 4**.**

For a given language $L\subseteq\Sigma^{+}$, we define the $\leftrightarrows_{\phi}$-Iterative closure of a language $L$ denoted by $cl_{\leftrightarrows_{\phi}}(L)=\bigcup_{i\geq 0}L_{i}$ where $L_{0}=L_{\phi}$,

[TABLE]

We have the following observation which is clear from Definition 4.

Lemma 3.10**.**

For $L\subseteq\Sigma^{*}$,

[TABLE]

Note that for each $i\geq 0$, $L_{i}$ defined above is $\phi$-closed. Also, observe that the iterative closure of a language $L$, denoted by $cl_{\leftrightarrows_{\phi}}(L)$ is $\leftrightarrows_{\phi}$-closed.

Example 4**.**

Consider the alphabet $\{a,b\}$ and $\theta$ an antimorphic involution such that $\theta(a)=b$ and vice-versa. Let $L=\{ab\}$. Note that, $L_{\theta}=L=L_{0}$. Then, $L_{1}=\{abab\}$, $L_{2}=\{(ab)^{2},(ab)^{3},(ab)^{4}\}$ and $L_{n}=\{(ab)^{i}~:~2\leq i\leq 2n\}$. Hence, $cl_{\leftrightarrows_{\theta}}(L)=\{(ab)^{i}~:~i\geq 1\}$

Theorem 3.11**.**

The families of regular, context-free and context sensitive languages are closed under the iterative $\leftrightarrows_{\phi}$-closure operation.

4. Conjugacy of words with respect to $\leftrightarrows_{\phi}$

The conjugate of a word is one of the basic concept in combinatorics of words. A word $u$ is called a conjugate of $v$ if both $u$ and $v$ satisfy the word equation $u\boldsymbol{\cdot}w=w\boldsymbol{\cdot}v$ for some word $w\in\Sigma^{*}$ where $\boldsymbol{\cdot}$ represents the basic catenation operation. This catenation operation can be replaced by any binary operation $\circ$ to define a $\circ$- conjugate of a given word (i.e.) $u$ is a $\circ$-conjugate of $v$, if there exists a $w\in\Sigma^{*}$ such that $u\circ w=w\circ v$. Depending on the operation $\circ$, $u\circ w$ may be a singleton or a set. The authors in [22], studied properties of $u$ and $v$ when $u$ is a $\otimes$-conjugate of $v$.

In this section, we discuss conditions on words $u,w\in\Sigma^{+}$, such that $u$ is a $\leftrightarrows_{\phi}$-conjugate of $w$, i.e., $u\leftrightarrows_{\phi}v=v\leftrightarrows_{\phi}w$ for some $v\in\Sigma^{+}$. The special case when $u=w$ always holds true by definition, as the operation $\leftrightarrows_{\phi}$ is commutative. Thus, we can say that $u$ $\leftrightarrows_{\phi}$-commutes with $v$ for all $u,v\in\Sigma^{*}$. We prove a necessary and sufficient condition for $\leftrightarrows_{\phi}$-conjugacy (Theorem 4.2). Since the Watson-Crick complementarity function $\theta_{DNA}$ is an antimorphic involution, in the remainder of this paper we only investigate antimorphic involution mappings $\phi=\theta$.

Proposition 5**.**

Let $u,v,w\in\Sigma^{+}$ be such that $uv=vw$ and $u\leftrightarrows_{\theta}v=v\leftrightarrows_{\theta}w$. Then, one of the following hold true.

1. (1)

$u=s^{m}=w$* and $v=s^{n}$ for some $s\in\Sigma^{+}$.* 2. (2)

$u=p^{m}=w$* and $v=p^{n}$, for $p\in P_{\theta}$.* 3. (3)

$u=xy$, $v=(xy)^{i}x$ and $w=yx$ for $x,y\in P_{\theta}$ and $i\geq 0$.

Proof.

By definition, for $u,v,w\in\Sigma^{+}$,

[TABLE]

and similarly,

[TABLE]

Given that $uv=vw$ and $u\leftrightarrows_{\theta}v=v\leftrightarrows_{\theta}w$. Then, by Lemma 2.1, we have $u=xy$, $v=(xy)^{i}x$ and $w=yx$. We now have the following cases.

1. (1)

If $u\theta(v)=v\theta(w)$ then, $u\theta(v)=(xy)(\theta(x)\theta(y))^{i}\theta(x)=(xy)^{i}x\theta(x)\theta(y)$. If $i\neq 0$ then, $x,y\in P_{\theta}$ and $xy=yx$ and hence, $u,v$ and $w$ are powers of a common $\theta$-palindrome. If $i=0$ then, $xy\theta(x)=x\theta(x)\theta(y)$ and by Proposition 1, $y=st$ and $\theta(x)=(st)^{j}s$ where $s,t\in P_{\theta}$ and hence, $x\in P_{\theta}$. Thus,

[TABLE]

and

[TABLE]

Since, $u\leftrightarrows_{\theta}v=v\leftrightarrows_{\theta}w$, we have either $\theta(y)xx=xx\theta(y)$ or $\theta(y)xx=yxx$. If $\theta(y)xx=xx\theta(y)$ then, by Lemma 2.3, $x=p^{m_{1}}$, $y=p^{m_{2}}$ for $p\in P_{\theta}$. Thus, $u=p^{m}=w$, $v=p^{n}$ for $p\in P_{\theta}$. If $\theta(y)xx=yxx$ then, $y\in P_{\theta}$ which implies that $u=xy$, $v=x$ and $w=yx$ for $x,y\in P_{\theta}$. 2. (2)

The case when $u\theta(v)=\theta(v)w$ is similar to case (1) and we omit it. 3. (3)

If $u\theta(v)=\theta(v)\theta(w)$ then, $u\theta(v)=xy(\theta(x)\theta(y))^{i}\theta(x)=$ $(\theta(x)\theta(y))^{i}\theta(x)\theta(x)\theta(y)$ $=\theta(v)\theta(w)$. If $i=0$ then, $x\in P_{\theta}$ and the case is similar to the previous one. If $i\neq 0$ then, $x,y\in P_{\theta}$ and $yx=xy$ and hence, $y=p^{j_{1}}$, $x=p^{j_{2}}$. Thus, $u=w=p^{m}$ and $v=p^{n}$ for $p\in P_{\theta}$. 4. (4)

If $u\theta(v)=wv$ then, $v=\theta(v)$ and $u=w$ which implies that $u=xy=w=yx$ which implies that $x$ and $y$ are powers of a common word. If $i=0$ then, $v=x\in P_{\theta}$ and

[TABLE]

[TABLE]

and the case is similar to the previous one. If $i\neq 0$ then, $v=\theta(v)$ implies that $(xy)^{i}x=(\theta(x)\theta(y)^{i}\theta(x)$ which implies that $x,y\in P_{\theta}$. Thus, in both cases we get, $x$ and $y$ to be powers of a common $\theta$-palindromic word. 5. (5)

If $u\theta(v)=w\theta(v)$ then, $u=w$ which implies that $u=xy=w=yx$. Hence, $x$ and $y$ are powers of a common word. Thus, $u=s^{m}=w$ and $v=s^{n}$ for some $s\in\Sigma^{+}$. Then,

[TABLE] 6. (6)

If $u\theta(v)=\theta(w)v$ then, $u=\theta(w)=xy=\theta(x)\theta(y)$ and $v=\theta(v)$. Thus, $u=xy$, $v=(xy)^{i}x$ and $w=yx$ where $x,y\in P_{\theta}.$ 7. (7)

The case when $u\theta(v)=\theta(w)\theta(v)$ is similar to the previous case and we omit it.

∎

A similar proof works for the next result and hence, we omit it.

Proposition 6**.**

Let $u,v,w\in\Sigma^{+}$ be such that $uv=v\theta(w)$ and $u\leftrightarrows_{\theta}v=v\leftrightarrows_{\theta}w$. Then, one of the following hold true.

1. (1)

$u=\theta(w)=(pq)^{j+1}p$, $v=(pq)^{j}p$ for some $p,q\in\Sigma^{*}$ and $j\geq 0$. 2. (2)

$u=w=(pq)^{j+1}p$, $v=(pq)^{j}p$ for some $p,q\in P_{\theta}$ and $j\geq 0$. 3. (3)

$u=\alpha^{m}=w$* and $v=\alpha^{n}$, for $\alpha\in P_{\theta}$.* 4. (4)

$u=w=xy$* and $v=(xy)^{i}x$, for $x,y\in P_{\theta}$ and $i\geq 0$.*

We now have the following result which is used in Proposition 7.

Lemma 4.1**.**

Let $x,y\in\Sigma^{+}$ be such that $xx\theta(y)=\theta(y)\theta(x)x$ for an antimorphic involution $\theta$. Then, $x$ and $y$ are powers of a common $\theta$-palindromic word.

Proof.

Given that $xx\theta(y)=\theta(y)\theta(x)x$. Then by Lemma 2.1, we have $xx=pq$, $\theta(y)=(pq)^{i}p$ and $\theta(x)x=qp$ for some $p,q\in\Sigma^{+}$. If $|x|\leq|p|$ then, $x=p_{1}=p_{2}q$ where $p=p_{1}p_{2}$. Then, $\theta(x)x=\theta(p_{2}q)p_{1}=qp$ which implies that $q\in P_{\theta}$ and $p_{1}p_{2}=\theta(p_{2})p_{1}$. By Lemma 2.1, we have, $p_{1}=\alpha(\beta\alpha)^{j}$ and $p_{2}=\beta\alpha$ for some $\alpha,\beta\in P_{\theta}$, $j\geq 0$. Thus, $x=\alpha(\beta\alpha)^{j}=p_{2}q=\beta\alpha q$ which implies that $j\neq 0$ and $\alpha\beta=\beta\alpha$. Hence, by Lemma 2.2, $\alpha$ and $\beta$ are powers of a common word. Therefore, $x$ and $y$ are powers of a common $\theta$-palindromic word. The case when $|x|\geq|p|$ is similar and we omit it. ∎

Proposition 7**.**

Let $u,v,w\in\Sigma^{+}$ be such that $uv=\theta(v)w$ and $u\leftrightarrows_{\theta}v=v\leftrightarrows_{\theta}w$. Then, one of the following hold true.

1. (1)

$u=\theta(w)$* and $v=\gamma w$ for some $\gamma\in P_{\theta}$.* 2. (2)

$u=(xy)^{j+1}p=w$* and $v=yx$ for $x,y\in P_{\theta}$ and $j\geq 0$.* 3. (3)

$u=xy=\theta(w)$, $v=\theta(x)$ for $y\in P_{\theta}$. 4. (4)

$u=xy=\theta(w)$, $v=x$ for $x,y\in P_{\theta}$. 5. (5)

$u=\alpha^{m}=w$* and $v=\alpha^{n}$, for $\alpha\in P_{\theta}$.* 6. (6)

$u=\theta(t)s^{k}=\theta(w)$* and $v=s^{n}t$ where $s=t\theta(t)$.*

Proof.

By definition, for $u,v,w\in\Sigma^{+}$,

[TABLE]

and similarly,

[TABLE]

Given that $uv=\theta(v)w$ and $u\leftrightarrows_{\theta}v=v\leftrightarrows_{\theta}w$. Then by Proposition 1, we have either $u=\theta(w)$ and $v=\gamma w$ for some $\gamma\in P_{\theta}$ or $u=xy$, $v=\theta(x)$ , $w=y\theta(x)$ for some $x,y\in\Sigma^{*}$. If $u=xy$, $v=\theta(x)$ , $w=y\theta(x)$ for some $x,y\in\Sigma^{*}$ then,

[TABLE]

and similarly,

[TABLE]

We have the following subcases.

1. (1)

If $\theta(x)xy=\theta(x)y\theta(x)$ then, $xy=y\theta(x)$ and by Lemma 2.1, $x=pq$, $y=(pq)^{j}p$ for some $p,q\in P_{\theta}$ which implies that, $u=(pq)^{j+1}p=w$ and $v=qp$. 2. (2)

If $\theta(x)xy=\theta(x)x\theta(y)$ then, $y\in P_{\theta}$. Thus, $u=xy$, $v=\theta(x)$, $w=y\theta(x)$ for $y\in P_{\theta}$. 3. (3)

If $\theta(x)xy=xx\theta(y)$ then, $u=xy$, $v=x$ and $w=yx$ for $x,y\in P_{\theta}$. 4. (4)

If $\theta(x)xy=y\theta(x)\theta(x)$ then by Lemma 4.1, $x$ and $y$ are powers of a common $\theta$-palindromic word and hence, $u,v$ and $w$ are powers of a common $\theta$-palindromic word. 5. (5)

If $\theta(x)xy=y\theta(x)x$ then by Lemma 2.2, $y$ and $\theta(x)x$ are powers of a common word say $s$. Then, $y=s^{m}$ and $\theta(x)x=s^{n}$. If $\theta(x)=s^{n_{1}}$ then, $s\in P_{\theta}$ and $u,v$ and $w$ are powers of a $\theta$-palindromic word $s$. If $\theta(x)=s^{n_{1}}s_{1}$ then, $x=s_{2}s^{n_{1}}$ for $s=s_{1}s_{2}$ and $2n_{1}+1=n$ which implies that $s_{1}=\theta(s_{2})=t$ and $s=t\theta(t)$. Thus, $u=\theta(t)s^{n_{1}+m}$, $v=s^{n_{1}}t$ and $w=s^{m+n_{1}}t$. 6. (6)

If $\theta(x)xy=x\theta(y)\theta(x)$ then, $x\in P_{\theta}$ and $xy=\theta(y)x$ and by Lemma 2.1, we have, $\theta(y)=pq$, $x=(pq)^{j}p$ where $p,q\in P_{\theta}$. Thus, $u=(pq)^{j+1}p$, $v=(pq)^{j}p$ and $w=qp(pq)^{j}p$. Therefore, $u\leftrightarrows_{\theta}v=v\leftrightarrows_{\phi}w$ implies that $pq=qp$ and by Lemma 2.2, $p$, $q$ and hence, $u,v$ and $w$ are powers of a common $\theta$-palindromic word. 7. (7)

The case when $\theta(x)xy=x\theta(y)x$ is similar to the previous and we obtain $u,v$ and $w$ to be powers of a common $\theta$-palindromic word.

∎

The proof of the following is similar to that of Proposition 7.

Proposition 8**.**

Let $u,v,w\in\Sigma^{+}$ be such that $uv=\theta(v)\theta(w)$ and $u\leftrightarrows_{\theta}v=v\leftrightarrows_{\theta}w$. Then, one of the following hold true.

1. (1)

$u=w$* and $v=\gamma\theta(w)$ for some $\gamma\in P_{\theta}$.* 2. (2)

$u=(xy)^{j+1}p=w$* and $v=yx$ for $x,y\in P_{\theta}$ and $j\geq 0$.* 3. (3)

$u=xy$, $v=\theta(x)$, $w=y\theta(x)$ for $y\in P_{\theta}$. 4. (4)

$u=xy$, $v=x$, $w=yx$ for $x,y\in P_{\theta}$. 5. (5)

$u=\alpha^{m}=w$* and $v=\alpha^{n}$, for $\alpha\in P_{\theta}$.* 6. (6)

$u=\theta(t)s^{k}$, $v=s^{n}t$ and $w=s^{k}t$ where $s=t\theta(t)$.

Based on the above results (Propositions 5, 6, 7 and 8), we give a necessary and sufficient condition on words $u$, $v$ and $w$ such that $u\leftrightarrows_{\theta}v=v\leftrightarrows_{\theta}w$.

Theorem 4.2**.**

Let $u,v,w\in\Sigma^{+}$. Then, $u\leftrightarrows_{\theta}v=v\leftrightarrows_{\theta}w$ iff one of the following holds:

1. (1)

$u=w$, $v\in\Sigma^{+}$ or $v\in P_{\theta}$. 2. (2)

$u=\theta(w)$* and either $v\in\Sigma^{+}$ or $v\in P_{\theta}$ or $v=\gamma w$ for some $\gamma\in P_{\theta}$.* 3. (3)

$u=s^{m}=w$* and $v=s^{n}$, for $m,n\geq 1$ and either $s\in P_{\theta}$ or $s\in\Sigma^{+}$.* 4. (4)

$u=\theta(w)=(pq)^{j+1}p$, $v=(pq)^{j}p$ for some $p,q\in\Sigma^{*}$ and $j\geq 0$. 5. (5)

$u=w=(pq)^{j+1}p$, $v=(pq)^{j}p$ for some $p,q\in P_{\theta}$ and $j\geq 0$. 6. (6)

$u=w=xy$* and $v=(xy)^{i}x$, for $x,y\in P_{\theta}$ and $i\geq 0$.* 7. (7)

$u=(xy)^{j+1}x=w$* and $v=yx$ for $x,y\in P_{\theta}$ and $j\geq 0$.* 8. (8)

$u=xy=\theta(w)$, $v=\theta(x)$ for $y\in P_{\theta}$. 9. (9)

$u=xy=\theta(w)$, $v=x$ for $x,y\in P_{\theta}$. 10. (10)

$u=\theta(t)s^{k}=\theta(w)$* and $v=s^{n}t$ where $s=t\theta(t)$.*

5. Solutions to $u\leftrightarrows_{\theta}L=L\leftrightarrows_{\theta}v$

In this section we discuss solutions to the equation $u\leftrightarrows_{\theta}L=L\leftrightarrows_{\theta}v$ where $u,v\in\Sigma^{+}$ and $L\subseteq\Sigma^{+}$ which is a generalization of the equation $u\leftrightarrows_{\theta}w=w\leftrightarrows_{\theta}v$ where now $w$ is replaced with a set. Section 4 gave a complete characterization of words $u$ and $v$ when $L$ is a singleton. In this section we give solutions to the eqution $u\leftrightarrows_{\theta}L=L\leftrightarrows_{\theta}v$ under some special cases.

We first recall the following from [2] which characterizes languages such that $uL=Lv$ for non empty words $u$ and $v$.

Proposition 9**.**

[2]** Let $u,v\in\Sigma^{+}$ and $L\subseteq\Sigma^{+}$. Then $uL=Lv$ iff there exists $x,y\in\Sigma^{*}$ with $|xy|\geq 1$ such that $u=(xy)^{i}$ and $v=(yx)^{i}$ for some $i\geq 1$ and $L=\{x(yx)^{j}~:~j\geq 0\}$.

The following result gives solution to some simultaneous involution conjugate equations ([17]).

Proposition 10**.**

[17]** Let $x,y\in\Sigma^{+}$ and $\theta$ be an antimorphic involution with $xy=\theta(y)\theta(x)$ and $x\theta(y)=y\theta(x)$. Then, $x=(\alpha\beta)^{m}$, $y=\alpha(\beta\alpha)^{n}$ with both $\alpha,\beta\in P_{\theta}$ for some $m\geq 1$ and $n\geq 0$.

We also recall the following results from [4] which deals with some language equations incorporating the involution function.

Proposition 11**.**

[4]** Let $\theta$ be an antimorphic involution, $u,v\in\Sigma^{+}$ and $L\subseteq\Sigma^{+}$. If $\theta(L)u=vL$, then for $x\in\Sigma^{+}$, $y,z\in\Sigma^{*}$ with $xy\in\mathcal{Q}$, $v=(xy)^{i}z$, $u=z(\theta(y)\theta(x))^{i}$ for some $i\geq 1$ and

[TABLE]

We use the following lemma.

Lemma 5.1**.**

For an antimorphic involution $\theta$, if either $xxy=yx\theta(x)$ or $xxy=y\theta(x)x$ then, $x$ and $y$ are powers of a common $\theta$-palindromic word.

Proof.

We only prove the case when $xxy=yx\theta(x)$ as the proof for $xxy=y\theta(x)x$ is similar and we omit it. Let $xxy=yx\theta(x)$. Then by Lemma 2.1, we have $xx=pq$, $y=(pq)^{j}p$ for some $j\geq 0$ and $x\theta(x)=qp$ for some $p,q\in\Sigma^{+}$. If $|x|\leq|p|$ then, $x=p_{1}=p_{2}q$ for $p=p_{1}p_{2}$. Then, $x\theta(x)=p_{1}\theta(q)\theta(p_{2})=qp$ which implies that $p_{2}\in P_{\theta}$ and $p_{1}\theta(q)=qp_{1}$ and by Lemma 2.1, there exists $\alpha,\beta\in P_{\theta}$ such that $q=\alpha\beta$ and $p_{1}=(\alpha\beta)^{k}\alpha$. Thus, we have $x=p_{1}=(\alpha\beta)^{k}\alpha=p_{2}q=p_{2}\alpha\beta$ which implies that $\alpha\beta=\beta\alpha$ and by Lemma 2.3, $\alpha$ and $\beta$ are powers of a common word. Hence, $x$ and $y$ are powers of a common $\theta$-palindromic word. The proof for the case when $|x|\geq|p|$ is similar.

∎

Corollary 5.1.1**.**

For an antimorphic involution $\theta$, if either $x(xy)^{i}=(yx)^{i}\theta(x)$ or $x(xy)^{i}=(y\theta(x))^{i}x$ for $i\geq 1$ then $x,y$ are powers of a common $\theta$-palindromic word.

Proof.

We only prove for one of the given equation as the proof of the other one is similar. Given that $x(xy)^{i}=(yx)^{i}\theta(x)$. The case when $i=1$ is proved in Lemma 5.1. Let $i\geq 2$. If $|x|\leq|y|$ then $y=xp=q\theta(x)$ and by Proposition 1 either $y=xs\theta(x)$ where $s\in\Sigma^{*}$ or $y=us\theta(u)$ where $x=us$ and $s\in P_{\theta}$. In both cases, $x(xy)^{i}=(yx)^{i}\theta(x)$ implies that, $xxs=s\theta(x)x$ and by Lemma 5.1, both $s$, $x$ and hence, $y$ are powers of a common $\theta$-palindromic word. The case when $|x|\geq|y$ is similar and we omit it. ∎

Theorem 5.2**.**

Let $u\in\Sigma^{+}$ and $L\subseteq\Sigma^{+}$ such that $u\leftrightarrows_{\theta}L=L\leftrightarrows_{\theta}v$ and $uL=Lv$. Then, one of the following hold true.

1. (1)

$u=s^{m}$, $v=s^{n}$ and $L=\{s^{k}~:~k\geq 0\}$ for some $s\in\Sigma^{+}$. 2. (2)

$u=s^{m}$, $v=s^{n}$ and $L=\{s^{k}~:~k\geq 0\}$ for some $s\in P_{\theta}$. 3. (3)

$u=(xy)^{i}$, $v=(yx)^{i}$ for some $i\geq 1$ and $L=\{x(yx)^{j}~:~j\geq 0\}$ where $x,y\in P_{\theta}$.

Proof.

Given that $uL=Lv$ and by Proposition 9, there exists $x,y\in\Sigma^{*}$ with $|xy|\geq 1$ such that $u=(xy)^{i}$ and $v=(yx)^{i}$ for some $i\geq 1$ and $L=\{x(yx)^{j}~:~j\geq 0\}$. Since $u\leftrightarrows_{\theta}L=L\leftrightarrows_{\theta}u$, we have,

[TABLE]

[TABLE]

We now have the following cases.

1. (1)

Let $(xy)^{j}x(xy)^{i}=(yx)^{m}x(yx)^{n}$ where $i+j=m+n$, $i,m\geq 1$. If $j\neq 0$ then, $xy=yx$ which implies by Lemma 2.2 that both $x$ and $y$ are powers of a common word. If $j=0$ then, $xxy=yxx$ and by Lemma 2.3, $x$ and $y$ are powers of a common word. Hence in both cases, $u=s^{m}$, $v=s^{n}$ and $L=\{s^{k}~:~k\geq 0\}$ for some $s\in\Sigma^{+}$. 2. (2)

Let $(xy)^{j}x(xy)^{i}=(yx)^{m}\theta(x)(\theta(y)\theta(x))^{n}$ where $i+j=m+n$, $i,m\geq 1$. We now have the following subcases.

• •

If $j\neq 0$ then, $xy=yx$ which implies by Lemma 2.2 that both $x$ and $y$ are powers of a common word $s\in\Sigma^{+}$. If in addition $n\neq 0$ then, $xy\in P_{\theta}$ which implies that $s\in P_{\theta}$. If $n=0$ then, $x\in P_{\theta}$ and hence, both $x$ and $y$ are powers of a common word $s\in\Sigma^{+}$.

• •

If both $j=0$ and $n=0$ then, $i=m$ and $x(xy)^{i}=(yx)^{i}\theta(x)$ by Corollary 5.1.1, $x$ and $y$ are powers of a common $\theta$-palindromic word.

• •

If $j=0$ and $n\neq 0$ then, both $x(xy)^{i}=(yx)^{m}\theta(x)(\theta(y)\theta(x))^{n}$ which implies that $xy\in P_{\theta}$ and $x(xy)^{i}=(yx)^{i}\theta(x)$. Then, by Corollary 5.1.1, $x$ and $y$ are powers of a common $\theta$-palindromic word. 3. (3)

Let $(xy)^{j}x(xy)^{i}=\theta(x)(\theta(y)\theta(x))^{n}(yx)^{m}$ where $i+j=m+n$, $i,m\geq 1$. Then, $xy=yx$ and $x\in P_{\theta}$ which implies by Lemma 2.2 that both $x$ and $y$ are powers of a common $\theta$-palindromic word. 4. (4)

Let $(xy)^{j}x(xy)^{i}=(\theta(x)\theta(y))^{m}x(yx)^{n},$ where $i+j=m+n$, $i,m\geq 1$. We now have the following subcases.

• •

If $j\neq 0$ then, $x,y\in P_{\theta}$. If in addition $n=0$ then $x(xy)^{i}=(\theta(x)\theta(y))^{i}x=(xy)^{i}x$ and if $n\neq 0$, we also get $xy=yx$. Hence, by Lemma 2.2 both $x$ and $y$ are powers of a common $\theta$-palindromic word.

• •

If $j=0$ then, $x\in P_{\theta}$ and $xy=yx$. Hence, by Lemma 2.2 both $x$ and $y$ are powers of a common $\theta$-palindromic word. 5. (5)

Let $(xy)^{j}x(xy)^{i}=x(yx)^{n}(\theta(x)\theta(y))^{m},$ where $i+j=m+n$, $i,m\geq 1$ which implies that both $x,y\in P_{\theta}$. If $i\neq m$ then, $xy=yx$ and by Lemma 2.2 both $x$ and $y$ are powers of a common $\theta$-palindromic word. 6. (6)

Let $(xy)^{j}x(xy)^{i}=\theta(x)(\theta(y)\theta(x))^{m+n}$, where $i+j=m+n$, $i,m\geq 1$. Then, $xy\in P_{\theta}$ and we have the following subcases.

• •

If $j=0$ then, $x(xy)^{i}=\theta(x)(\theta(y)\theta(x))^{m+n}$ which implies that $x\in P_{\theta}$ and $xy=\theta(y)x$. Hence, by Lemma 2.1, $y=qp$, $x=(pq)^{t}p$ for $p,q\in P_{\theta}$. Then,

[TABLE]

Also, observe that $[(qp)(pq)^{t}p]^{i}[(pq)^{t+1}p]^{j}(pq)^{t}p\in L\leftrightarrows_{\theta}v$. Since, $u\leftrightarrows_{\theta}L=L\leftrightarrows_{\theta}v$, we have either $qp=pq$ or $qpp=ppq$, which implies that both $p$ and $q$ are powers of a common word.

• •

If $j\neq 0$ and $n\neq 0$ then, $x,y,xy\in P_{\theta}$ and by Lemma 2.2, both $x$ and $y$ are powers of a common $\theta$-palindromic word. 7. (7)

Let $(xy)^{j}x(xy)^{i}=\theta(x)(\theta(y)\theta(x))^{n}(\theta(x)\theta(y))^{m}$, where $i+j=m+n$, $i,m\geq 1$. Then, $x,y\in P_{\theta}$. If $i\neq m$ then, $xy=yx$ and by Lemma 2.2, both $x$ and $y$ are powers of a common $\theta$-palindromic word.

Hence, the result. ∎

We now have the following which follows directly from Proposition 9 and Theorem 5.2.

Theorem 5.3**.**

Let $u\in\Sigma^{+}$ and $L\subseteq\Sigma^{+}$ such that $u\leftrightarrows_{\theta}L=L\leftrightarrows_{\theta}v$ and $uL=L\theta(v)$. Then, one of the following hold true.

1. (1)

$u=s^{m}$, $v=\theta(s)^{n}$ and $L=\{s^{k}~:~k\geq 0\}$ for some $s\in\Sigma^{+}$. 2. (2)

$u=s^{m}$, $v=s^{n}$ and $L=\{s^{k}~:~k\geq 0\}$ for some $s\in P_{\theta}$. 3. (3)

$u=(xy)^{i}=v$* for some $i\geq 1$ and $L=\{x(yx)^{j}~:~j\geq 0\}$ where $x,y\in P_{\theta}$.*

One can easily observe from Remark 1, the following.

[TABLE]

for $u_{1}\in u_{\theta}$ and $L_{1}\in L_{\theta}$. Hence by Proposition 11 we conclude the following.

Theorem 5.4**.**

Let $u\in\Sigma^{+}$ and $L\subseteq\Sigma^{+}$ such that $u\leftrightarrows_{\theta}L=L\leftrightarrows_{\theta}v$ and $uL=\theta(L)v$. Then, for $x\in\Sigma^{+}$, $y,z\in\Sigma^{*}$ with $xy\in\mathcal{Q}$, $u=(xy)^{i}z$, $v=z(\theta(y)\theta(x))^{i}$ for some $i\geq 1$ and

[TABLE]

Proof.

Observe that,

[TABLE]

[TABLE]

Given that $uL=\theta(L)v$, which implies that $\theta(v)L=\theta(L)\theta(u)$ and by Proposition 11, $u=\theta(v)$. Hence,

[TABLE]

[TABLE]

Thus, by Proposition 11, for $x\in\Sigma^{+}$, $y,z\in\Sigma^{*}$ with $xy\in\mathcal{Q}$, $u=(xy)^{i}z$, $v=z(\theta(y)\theta(x))^{i}$ for some $i\geq 1$ and

[TABLE]

∎

We now have the following which follows directly from Proposition 11 and Theorem 5.4.

Theorem 5.5**.**

Let $u\in\Sigma^{+}$ and $L\subseteq\Sigma^{+}$ such that $u\leftrightarrows_{\theta}L=L\leftrightarrows_{\theta}v$ and $uL=\theta(L)\theta(v)$. Then, for $x\in\Sigma^{+}$, $y,z\in\Sigma^{*}$ with $xy\in\mathcal{Q}$, $u=(xy)^{i}z=v$ for some $i\geq 1$ and

[TABLE]

Lemma 5.6**.**

Let $u,v\in\Sigma^{+}$ and $L\subseteq\Sigma^{+}$. The following are true.

1. (1)

If $uL=vL$ then $u=v$. 2. (2)

If $uL=\theta(v)L$ then $u=\theta(v)$. 3. (3)

If $uL=v\theta(L)$. then $u=v$ and $L=\theta(L)$. 4. (4)

If $uL=\theta(v)\theta(L)$ then, $u=\theta(v)$ and $L=\theta(L)$.

Proof.

We only prove the first implication as the others are similar. Let $w\in L$ be such that $|w|\leq|x|$ for all $x\in L$. Since $uL=vL$, $uw=vx$ for some $x\in L$ and $|x|\geq|w|$ which implies that $|u|\geq|v|$. Also, there exists a $y\in L$ such that $|w|\leq|y|$ and $uy=vw$. If $|x|>|w|$ then, $|u|>|v|$ and hence, $|wv|<|uy|$ a contradiction. Similarly, we can show that $|x|\nless|w|$. Hence, $|w|=|x|$ which implies that $|u|=|v|$. Since, $uL=vL$ we conclude that $u=v$. ∎

By Lemma 5.6, we conclude the following.

Theorem 5.7**.**

Let $u\in\Sigma^{+}$ and $L\subseteq\Sigma^{+}$ such that $u\leftrightarrows_{\theta}L=L\leftrightarrows_{\theta}v$. The following are true.

1. (1)

If $uL=vL$ then $u=v$. 2. (2)

If $uL=\theta(v)L$ then $u=\theta(v)$. 3. (3)

If $uL=v\theta(L)$. then $u=v$ and $L=\theta(L)$. 4. (4)

If $uL=\theta(v)\theta(L)$ then, $u=\theta(v)$ and $L=\theta(L)$.

6. Conclusions

This paper defines and investigates the binary word operation strong-$\phi$-bi-catenation which, when iteratively applied to words $u$ and $v$ generates words in the set $\{u,\phi(u),v,\phi(v)\}^{+}$. The operation was naturally extended to languages (Section 3.1) and we investigated some of its properties. Future topics of research include extending the $\leftrightarrows_{\phi}$-conjugacy on words (Section 4 ) to $\leftrightarrows_{\phi}$-conjugacy on languages as well as exploring an associative version of strong-$\theta$-bi-catenation.