We study linear, possibly over- or under-determined, differential-algebraic equations that have the same solution behavior as linear differential-algebraic equations with well-defined strangeness index. In particular, we give three different characterizations for differential-algebraic equations, namely by means of solution spaces, canonical forms, and derivative arrays. We distinguish two levels of generalization, where the more restrictive case contains an additional assumption on the structure of the set of consistent inhomogeneities.