  For finite-dimensional linear semigroups which leave a proper cone invariant
it is shown that irreducibility with respect to the cone implies the existence
of an extremal norm. In case the cone is simplicial a similar statement applies
to absolute norms. The semigroups under consideration may be generated by
discrete-time systems, continuous-time systems or continuous-time systems with
jumps. The existence of extremal norms is used to extend results on the
Lipschitz continuity of the joint spectral radius beyond the known case of
semigroups that are irreducible in the representation theory interpretation of
the word.
