We study both the Submonoid Membership problem and the Rational Subset Membership problem in finitely generated nilpotent groups. We give two reductions with important applications. First, Submonoid Membership in any nilpotent group can be reduced to Rational Subset Membership in smaller groups. As a corollary, we prove the existence of a group with decidable Submonoid Membership and undecidable Rational Subset Membership, confirming a conjecture of Lohrey and Steinberg. Second, the Rational Subset Membership problem in $H_3(\mathbb Z)$ can be reduced to the Knapsack problem in the same group, and is therefore decidable. Combining both results, we deduce that the filiform $3$-step nilpotent group has decidable Submonoid Membership.

翻译：我们研究了有限生成幂零群中的子幺半群成员判定问题与有理子集成员判定问题。我们给出了两个具有重要应用的归约。首先，任意幂零群中的子幺半群成员判定问题可以归约为更小群中的有理子集成员判定问题。作为推论，我们证明了存在一个群，其子幺半群成员判定可判定，而有理子集成员判定不可判定，从而证实了Lohrey和Steinberg的猜想。其次，$H_3(\mathbb Z)$中的有理子集成员判定问题可以归约为同一群中的背包问题，因而是可判定的。结合这两个结果，我们推导出丝状三步幂零群具有可判定的子幺半群成员判定问题。