Roots of star events

A regular event W is a star event iff there exists another event V such that W = V*. In that case, V is called a root of W. It is shown that every regular event has a unique minimum root, which is contained in every other root of W. An algorithm for finding the minimum root is presented, and the root is shown to be regular. The results have applications to languages, codes, canonical forms for regular expressions, simplification of expressions, decomposition of sequential machines, and semigroup theory.