Classification of Noncounting Events

An event E is a subset of the free monoid A* generated by the finite alphabet A. E is noncounting if and only if there exists an integer k ~> 0, called the order of E, such that for any x, y, z E A*, xykz ~ E if and only if xyk+lz E E. From semigroup theory it follows that the number of noncounting events of order < 1 is finite. Each such event is regular and the finite automata accepting such events over a fixed alphabet are homomorphic images of a universal automaton. Star-free regular expressions for such events are easily obtainable. It is next shown that the number of distinct noncounting events of order >~2 over any alphabet with two or more letters is infinite. Furthermore, there exist noncounting events which are of any "arbitrary degree of complexity," e.g. not recursively enumerable.