Shuffle on trajectories was introduced by Mateescu et al. as a method of generalizing several studied operations on words, such as the shuffle, concatenation and insertion operations. This natural construction has received significant and varied attention in the literature. In this thesis, we consider several unexamined areas related to shuffle on trajectories.
We first examine the state complexity of shuffle on trajectories. We find that the density of the set of trajectories is an appropriate measure of the complexity of the associated operation, since low density sets of trajectories yield less complex operations.
We introduce the operation of deletion along trajectories, which serves as an inverse to shuffle on trajectories. The operation is also of independent interest, and we examine its closure properties. The study of deletion along trajectories also leads to the study of language equations and systems of language equations with shuffle on trajectories.
The notion of shuffle on trajectories also has applications to the theory of codes. Each shuffle on trajectories operation defines a class of languages. Several of these language classes are important in the theory of codes, including the prefix-, suffix-, biprefix-codes and the hypercodes. We investigate these classes of languages, decidability questions, and related binary relations.
We conclude with results relating to iteration of shuffle and deletion on trajectories. We characterize the smallest language closed under shuffle on trajectories or deletion along trajectories, as well as generalize the notion of primitive words and primitive roots. Further examination of language equations are also possible with the iterated counterparts of shuffle and deletion along trajectories.