Properties of recognizable $\mathcal{M}$-subsets of a free monoid

Abstract

We study some properties of recognizable $\mathcal{M}$-subsets of a free monoid ($\mathcal{M}$RecA ^*) and of two of its subfamilies: the simple $\mathcal{M}$-subsets ($\mathcal{M}$SRecA ^*) and the $\mathcal{M}$-subsets which are nondeterministic complexities of finite automata ($\mathcal{M}$CRecA ^*). We show that $\mathcal{M}$CRecA ^* $\subseteq /\mathcal{M}$SRecA ^* $\subseteq /\mathcal{M}$RecA ^*. The main result of this paper shows that every recognizable $\mathcal{M}$-subset of A⁺ is the sum of finitely many simple $\mathcal{M}$-subsets of A ⁺. We also study the closure properties of the above families under some operations.

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