On morphic generation of regular languages (Q1083217)

Let Reg be the family of regular languages and H (resp. \(H^{-1})\) be the family of all morphisms (resp. inverse morphisms). It is known that Reg\(\neq H H^{-1}(L_ 0)\) for all \(L_ 0\in Reg\) [\textit{A. Salomaa}, Theory of automata (1969; Zbl 0193.329)], but \(Reg=H H^{-1} H(a^*b)=H^{-1} H H^{-1}(b)\) [see \textit{M. Latteaux} and \textit{J. Leguy}, Lect. Notes Comput. Sci. 154, 420-432 (1983; Zbl 0523.68067), and \textit{P. Turakainen}, Ann. Univ. Turku., Ser. A I 186, 118-128 (1984; Zbl 0541.68045)]. The present paper settles the problem whether \(Reg=H^{-1} H(L_ 0)\) for some \(L_ 0\in Reg\), proving that for no regular language \(L_ 0\) such an equality holds (there are even finite languages outside \(H^{-1} H(L_ 0))\).