We introduce monomial divisibility diagrams (MDDs), a data structure for monomial ideals that supports insertion of new generators and fast membership tests. MDDs stem from a canonical tree representation by maximally sharing equal subtrees, yielding a directed acyclic graph. We establish basic complexity bounds for membership and insertion, and study empirically the size of MDDs. As an application, we integrate MDDs into the signature Gröbner basis implementation of the Julia package AlgebraicSolving.jl. Membership tests in monomial ideals are used to detect some reductions to zero, and the use of MDDs leads to substantial speed-ups compared to the existing representation by lists of generators with divmasks.

翻译：我们提出了单项式整除图（MDDs），一种用于单项式理想的数据结构，支持新生成元的插入与快速成员测试。MDDs源于一种典范树形表示，通过最大化共享相等子树来生成有向无环图。我们建立了成员测试与插入操作的基本复杂度界限，并对MDDs的规模进行了实证研究。作为应用，我们将MDDs集成到Julia包AlgebraicSolving.jl的签名Gröbner基实现中。单项式理想中的成员测试用于检测某些归零约化，与现有基于带divmasks生成元列表的表示相比，使用MDDs可实现显著的速度提升。