Authors: Tara Brough ^1,2; Laura Ciobanu ^3,4; Murray Elder ^5,6; Georg Zetzsche ⁷

• 1 Universidade de Lisboa = University of Lisbon
• 2 Universidade de Lisboa = University of Lisbon = Université de Lisbonne
• 3 Université de Neuchâtel
• 4 Université de Neuchâtel = University of Neuchatel
• 5 University of Newcastle [Australia]
• 6 University of Newcastle [Callaghan, Australia]
• 7 Laboratoire Spécification et Vérification [Cachan]

For a language $L$, we consider its cyclic closure, and more generally the language $C^{k}(L)$, which consists of all words obtained by partitioning words from $L$ into $k$ factors and permuting them. We prove that the classes of ET0L and EDT0L languages are closed under the operators $C^k$. This both sharpens and generalises Brandstädt's result that if $L$ is context-free then $C^{k}(L)$ is context-sensitive and not context-free in general for $k \geq 3$. We also show that the cyclic closure of an indexed language is indexed.