The enumeration of meet-irreducible elements based on hierarchical decompositions of implicational bases

We study the well-known problem of translating between two representations of closure systems, namely implicational bases and meet-irreducible elements. Albeit its importance, the problem is open. In this paper, we introduce splits of an implicational base. It is a partitioning operation of the implications which we recursively apply 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. Focusing on the case of acyclic splits, 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.