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 @? A^*, xy^kz @? E if and only if xy^k^+^1z @? 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.