  We are interested in the problem of translating between two representations
of closure systems, namely implicational bases and meet-irreducible elements.
Albeit its importance, the problem is open. Motivated by this problem, we
introduce splits of an implicational base. It is a partitioning operation of
the implications which we apply recursively to obtain a binary tree
representing a decomposition of the implicational base. We show that this
decomposition can be conducted in polynomial time and space in the size of the
input implicational base. In order to use our decomposition for the translation
task, we focus on the case of acyclic splits. In this case, we obtain a
recursive characterization of the meet-irreducible elements of the associated
closure system. We use this characterization and hypergraph dualization to
derive new results for the translation problem in acyclic convex geometries.
