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.

翻译：本文引入单项式可除图（MDD），这是一种支持新生成元插入与快速成员检测的单项式理想数据结构。MDD源于通过最大化共享相等子树得到的规范树表示，从而形成有向无环图。我们建立了成员检测与插入操作的基本复杂度界限，并通过实证研究分析了MDD的规模特征。作为应用实例，我们将MDD集成至Julia软件包AlgebraicSolving.jl的签名Gröbner基实现中。单项式理想的成员检测可用于发现部分零约化现象，而MDD的应用带来了显著的加速效果。