Positive systems play an important role in systems and control theory and have found many applications in multiagent systems, neural networks, systems biology, and more. Positive systems map the nonnegative orthant to itself (and also the nonpositive orthant to itself). In other words, they map the set of vectors with zero sign variations to itself. In this article, discrete-time linear systems that map the set of vectors with up to k - 1 sign variations to itself are introduced. For the special case k = 1 these reduce to discrete-time positive linear systems. Properties of these systems are analyzed using tools from the theory of sign-regular matrices. In particular, it is shown that almost every solution of such systems converges to the set of vectors with up to k - 1 sign variations. It is also shown that these systems induce a positive dynamics of k-dimensional parallelotopes.