Watson-Crick conjugates of words and languages

In this work, we explore the concept of Watson-Crick conjugates, also known as $θ$-conjugates (where $θ$ is an antimorphic involution), of words and languages. This concept extends the classical idea of conjugates by incorporating the Watson-Crick complementarity of DNA sequences. Our investigation initially focuses on the properties of $θ$-conjugates of words. We then define $θ$-conjugates of a language and study closure properties of certain families of languages under the $θ$-conjugate operation. Furthermore, we analyze the iterated $θ$-conjugate of both words and languages. Finally, we discuss the idea of $θ$-conjugate-free languages and examine some decidability problems related to it.