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Abstract
The paper introduces a new class of L systems, where it is possible to continue derivations from certain specified subwords of the words obtained. Such L systems (called L systems with fragmentation or just JL systems) are of interest both from biological and formal language theory point of view. The paper deals with JL systems without interactions, discusses the basic properties of the language families obtained, as well as their position in the L hierarchy. Finally, two infinite hierarchies of language families are obtained by limited fragmentation, the notions being analogous to those of ultralineanty and finiteness of index for context-free languages.