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Communicated by S. Margolis Recently it has been noticed that many interesting combinatorial objects belong to a class of semigroups called left regular bands, and that random walks on these semigroups encode several well-known random walks. For example, the set of faces of a hyperplane arrangement is endowed with a left regular band structure. This paper studies the module structure of the semigroup algebra of an arbitrary left regular band, extending results for the semigroup algebra of thedoi:10.1142/s0218196707004219 fatcat:3qa7iqna5jflnndg3f4xha2pvu