# Monoidal networks

## TL;DR

This paper introduces monoidal networks, a new algebraic structure involving rings and groups indexed by ideals, motivated by the union-closed sets conjecture, and explores their properties and implications.

## Contribution

It defines monoidal networks, linking ring and group theory with the union-closed sets conjecture, and investigates their structural properties.

## Key findings

- Established the formal definition of monoidal networks.
- Connected monoidal networks to the union-closed sets conjecture.
- Explored structural properties and potential applications.

## Abstract

In this paper we define and study the notion of a monoidal network, which consists of a commutative ring $R$ and a collection of groups $\Gamma_I$, indexed by the ideals of $R$, with $\Gamma_I$ acting on the quotient $R/I$ and satisfying a certain lifting condition. The examination of these objects is largely motivated by, and initially arose from, the study of the union-closed sets conjecture. This connection is made precise and other aspects of these structures are investigated.