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Purpose – From the perspective of set theory, this paper establishes a rigorous mathematical definition of L systems, and proves the common characteristics of simple L systems by providing a theoretical framework for a systematic research of L systems. Additionally, explanations for the characteristics of systems’ emergence based on L systems are provided, and some design methods of L systems are developed and interesting cases of design are constructed. Design/methodology/approach –Set theory is employed in this research as the fundamental logic of reasoning and thread of thinking. Findings –Based on a rigorous mathematical definition of L systems, we derive the common characteristics of simple L systems and provide explanations for the characteristics of systems’ emergence based on L systems. Specifically, to produce complicated fractal figures using simple L systems, one should avoid using any such L mapping whose restriction on S is a bijection from S to S. Practical implications – This work lies down the theoretical basis for giving a rigorous mathematical framework of L system which is widely applied. Originality/value – This work is the first of its kind that investigates the L system using a rigorous mathematical definition based on set theory.