This article concerns Ehresmann structures in the partition monoid ${\mathcal{P}}_X$. Since ${\mathcal{P}}_X$ contains the symmetric and dual symmetric inverse monoids on the same base set X, it naturally contains the semilattices of idempotents of both submonoids. We show that one of these semilattices leads to an Ehresmann structure on ${\mathcal{P}}_X$ while the other does not. We explore some consequences of this (structural/combinatorial and representation theoretic), and in particular characterise the largest left-, right- and two-sided restriction submonoids. The new results are contrasted with known results concerning relation monoids, and a number of interesting dualities arise, stemming from the traditional philosophies of inverse semigroups as models of partial symmetries (Vagner and Preston) or block symmetries (FitzGerald and Leech): “surjections between subsets” for relations become “injections between quotients” for partitions. We also consider some related diagram monoids, including rook partition monoids, and state several open problems.

本文涉及分区半体中的Ehresmann结构 ${\mathcal{P}}_X$。自从${\mathcal{P}}_X$在同一个基集X上包含对称和对偶逆反半对半定理，它自然包含两个亚类群的幂等式的半格。我们表明，这些半晶格之一导致Ehresmann结构在${\mathcal{P}}_X$而另一个则没有。我们探讨了这种情况的一些后果（结构/组合和表示理论），尤其是表征最大的左侧，右侧和两侧限制亚类动物。新的结果与有关关系半定式的已知结果形成对比，并且产生了许多有趣的对偶性，源于反半群的传统哲学，即部分对称（Vagner和Preston）或块对称（FitzGerald和Leech）的模型：关系的“子集”变成分区的“商之间的注入”。我们还考虑了一些相关的图单边形，包括rook分区单边形，并陈述了几个未解决的问题。