Euclidean division by d in base b

Abstract

Let $b⩾2$ be an integer. For each positive integer d, let $E_{d,b}$ be the Euclidean division by d in base b, that is, the function which associates to a word u in ${\{0,\dots ,b-1\}}^{⁎}$, representing an integer n in base b, the unique word of the same length as u representing the quotient of the division of n by d. We describe the pure sequential transducer realizing this function and analyze the algebraic structure of its syntactic monoid. We compute its size, describe its Green's relations and its minimum ideal. As a consequence, we show that it is a group if and only if d and b are coprime numbers, it is a p-group if and only if d is a power of p and b is congruent to 1 modulo p and it is an aperiodic monoid if and only if d divides some power of b. The uniform continuity of $E_{d,b}$ for the pro-group metric was studied by Reutenauer and Schützenberger in 1995. We launch a similar study for the uniform continuity of $E_{d,b}$ with respect to the pro-p metric, where p is a prime number.