The notion of graph acyclicity has been extended to several notions of hypergraph acyclicity. In increasing order of generality: gamma acyclicity, beta acyclicity, and alpha acyclicity have met a great interest in many fields. For each notion, we prove the equivalence between the numerous characterizations with a new, simpler proof, in a self-contained manner. For that purpose, we introduce new notions of alpha, beta, and gamma leaf that allow one to define new "rule-based" characterizations of each notion. The combined presentation of the notions is completed with a study of their respective closure properties. New closure results are established, and alpha, beta, and gamma acyclicity are proved optimal w.r.t. their closure properties.
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