On decompositions of regular events

This paper studies decompositions of regular events into star events, i.e. events of the form W = V*. Mathematically, the structure of a star event is that of a monoid. First it is shown that every regular event contains a finite number of maximal star events, which are shown to be regular and can be effectively computed. Necessary and sufficient conditions for a regular event to be the union of its maximal star events are found. Next, star events are factored out from arbitrary events, yielding the form W=V*T. For each W there exists a unique largest V* and a unique smallest T; an algorithm for finding suitable regular expressions for V and T is developed. Finally, an open problem of Paz and Peleg is answered: Every regular event is decomposable as a finite product of star events and prime events.