Watson-Crick strong bi-catenation on words

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.