  Positive systems play an important role in systems and control theory and
have found many applications in multi-agent 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 note, 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. Also, the operation of such systems on
$k$-dimensional parallelotopes are studied.
