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In this article we introduce the notion of a regular partition of a Coxeter group. We develop the theory of these partitions, and show that the class of regular partitions is essentially equivalent to the class of automata (not necessarily finite state) recognising the language of reduced words in the Coxeter group. As an application of this theory we prove that each cone type in a Coxeter group has a unique minimal length representative. This result can be seen as an analogue of Shi'sarXiv:2107.09962v3 fatcat:sjl6ceywyvdejil3fxvowfu63u