Infinite hierarchy of expressions containing shuffle closure operator (Q1107331)

The author [ibid. 25, 363-367 (1987; Zbl 0633.68070)] has shown that languages generated by shuffle expressions with no nested \(\odot\) operator form a proper subclass of all the shuffle languages. Now we are going to show that the \(\odot\)-depth hierarchy is infinite, that is, for each n there exists a language generated by an expression which contains exactly n nested \(\odot\) operators. The example of such a language is \(L_ n=(x_ nL_{n-1}y_ nz_ n)^{\odot}\), where \(L_ 1=(x_ 1y_ 1z_ 1)^{\odot}\).