Letter to the editor (Q5896518)

The author gives a new proof, mainly based on the use of König's lemma, for the following theorem [\textit{H. A. Maurer}, \textit{A. Salomaa} and \textit{D. Wood}, Math. Syst. Theory 15, 251-265 (1982; Zbl 0508.68049)]: For languages \(L_ 1\) and \(L_ 2\), the equation \(L_ F(L_ 1)=L_ F(L_ 2)\) implies the equation \(L_{\infty}(L_ 1)=L_{\infty}(L_ 2)\), where \(L_{\infty}(L)=\{L'| L'<L\), L' is a language over a denumerable alphabet\(\}\) and \(L_ F(L)=\{L'| L'\) is finite and \(L'<L\}\). Here \(L'<L\) means that there is a letter-to-letter homomorphism h such that h(L')\(\subseteq L\).