Basic Concepts and Notations

In this preliminary chapter, we give a short account of some basic concepts and notations which will be used throughout the book. Section 1.1 contains the background knowledge in the general semigroup theory. Section 1.2 covers a short introduction to the theory of languages. In Sect. 1.3, we discuss dense subsets (languages) in semigroups. Dense subsets (languages) will play an important role throughout the book. In Sect. 1.4, we deal with a special but important class of words called primitive words which will be often used in the book. Some elementary properties of infinite words are involved in the last section.