Introduction

Formal language theory is overlapped by a close relative among the family of mathematical disciplines. This is the specialty known as Combinatorics on Words. We must use a few of the most basic concepts and propositions of this field. A nonnull word, q, is said to be primitive if it cannot be expressed in the form xk with x a word and k > 1. Thus, for any alphabet containing the symbols a and b, each of the words a, b, ab, bab, and abababa is primitive. The words aa and ababab are not primitive and neither is any word in (aba)+ other than aba itself. One of the foundational facts of word combinatorics, which is demonstrated here in Section 6.2, is that each nonnull word, ω, consisting of symbols from an alphabet Σ, can be expressed in a unique way in the form ω = qn where q is a primitive word and n is a positive integer. The uniqueness of the representation, ω = qn, allows a useful display of the free semigroup Σ+, consisting of the nonnull words formed from symbols in Σ, in the form of a Cartesian product, Q × N, where Q is the set of all primitive words in Σ+ and N is the set of positive integers. Each word ω = qn is identified with the ordered pair (q, n). This chapter provides the groundwork for investigations of concepts that arise naturally in visualizing languages as subsets of Q × N. In the suggested visualizations, the order structure of N is respected. We regard N as labeling a vertical axis (y-axis) that extends upward only.