Some properties of recognizable -subsets.

We study some properties of recognizable Z-subsets of A∗ and its subfamilies: the simple Z-subsets, the limited Z-subsets, the recognizable M-subsets, the simple M-subsets and the M-subsets which are nondeterministic complexities of finite automata. At first, we study some necessary conditions for membership in each one of these families and after we investigate the closure properties of these families under several operations. We also study the relations existing among some subfamilies of recognizable M-subsets and the families Hp (p⩾0) obtained by Simon.