  The notion of graph acyclicity has been extended to several different notions
of hypergraph acyclicity, in increasing order of generality: gamma acyclicity,
beta acyclicity, and alpha acyclicity, that have met a great interest in many
fields. We prove the equivalence between the numerous characterizations of each
notion 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 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.
