Sections of Submonoids of Nilpotent Groups

TL;DR

This paper demonstrates that products of finitely generated submonoids in a group can be represented as sections of finitely generated submonoids in a larger group, providing new insights into nilpotent groups and undecidable membership problems.

Contribution

It establishes a novel connection between submonoid products and sections in a larger group, offering a new proof related to undecidable membership problems in nilpotent groups.

Findings

Every product of finitely generated submonoids is a section of a finitely generated submonoid in a larger group.

Provides a converse to Bodart's reduction.

Shows existence of a submonoid with undecidable membership problem in a nilpotent group of class 2.

Abstract

We show that every product of f.g.\ submonoids of a group $G$ is a section of a f.g.\ submonoid of $G{\times}H_5(\mathbb{Z})$, where $H_5(\mathbb{Z})$ is a Heisenberg group. This gives us a converse of a reduction of Bodart, and a new simple proof of the existence of a submonoid of a nilpotent group of class 2 with undecidable membership problem.