Semigroup Presentations and Number Sequences

Abstract

In this paper, we look at some intriguing connections between Fibonacci-like sequences of numbers and semigroup presentations; in this section, we explain some of the basic terms we will be using, and give a general background to the subject. Suppose that we have a semigroup S generated by a set X, so that, if we let X ⁺ denote the set of all non-empty words in X, then every element of S is represented by at least one element of X ⁺. In general, a congruence on X ⁺ is an equivalence relation ≈ on X⁺ such that α ≈ α′, β ≈ β′ ⇒ αβ ≈ α′β′ ∀ α, α′, β. β′ ∈ X ⁺.