On networks over finite rings

Abstract

The (control) networks over finite rings are proposed and their properties are investigated. Based on semi-tensor product (STP) of matrices, a set of algebraic equations are provided to verify whether a finite set with two binary operators is a ring. As an application, all rings with 4 elements are obtained. It is then shown that the STP-based technique developed for logical (control) networks are applicable to (control) networks over finite rings. A (control) sub-network over a proper ideal of the bearing ring is constructed. Certain properties are revealed, showing that a network over the ideal behaves like a subsystem over an invariant subspace. Product rings are then introduced, which provides a tool for both constructing product networks and decomposing complex networks. Finally, the representation of networks over finite rings is considered, which investigates how many finite networks can be expressed as networks over finite rings, showing that the technique developed in this paper is widely applicable.