  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.
