Some Algebraic Aspects of Morse Code Sequences

1 Morse code polynomials Morse code sequences are finite sequences of dots (•) and dashes (−). If a dot has length 1 and a dash has length 2 then the number of all such sequences of total length n−1 is the Fibonacci number Fn, which is defined as the sequence of numbers satisfying the recursion Fn = Fn−1 + Fn−2 with initial conditions F0 = 0 and F1 = 1. If a dash is assumed to have length j, j≥ 1, we get a simple generalization of Fibonacci numbers. Let MC be the set of all Morse code sequences. We interpret MC as a monoid with respect to concatenation whose unit element is the empty sequence ε. If we write a for a dot and b for a dash then MC consists of all words in a and b. Let P be the corresponding monoid algebra over C, i. e. the algebra of all finite sums ∑ v∈MC λvv with complex coefficients. The elements of P will be called Morse code polynomials.