Loop agreement is a family of wait-free tasks that includes set agteement and approximate agreement tasks. This paper presents a complete classification of loop agreement tasks. Each loop agreement task can be assigned an algebraic signature consisting of a finitely-presented group G and a distinguished element g in G. This signature completely characterizes the task's computational power. If G and H are loop agreement tasks with respective signatures and then G implements H if and only if there exists a group homomorphism O: G ->H carrying g to h.
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