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.

翻译：我们关注闭包系统的两种表示形式——蕴含基与不可约交元——之间的转换问题。尽管该问题至关重要，但目前尚未解决。受此问题启发，我们引入蕴含基的分裂操作。该操作对蕴含式进行划分，并通过递归应用构建一棵表示蕴含基分解的二叉树。我们证明，该分解可在输入蕴含基规模的多项式时间和空间内完成。为了将分解用于转换任务，我们聚焦于无环分裂情形。在此情形下，我们获得了关联闭包系统不可约交元的递归刻画。利用该刻画与超图对偶化方法，我们推导出无环凸几何中转换问题的新结果。