A modular triple characterization of circuit signatures

We study the modular triples of circuits of a matroid and use them to characterize four types of circuit signatures, three of which are known (weak orientations, orientations, and ternary signatures) and one of which is new (lifting signatures). Lifting signatures allow us to specify a linear class of circuits in a matroid, and thereby the lift of the matroid, by labeling the elements from a group in the manner of Dowling and Kelly.