Monoidal networks

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 Γ I , indexed by the ideals of R, with Γ 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.