  We generalize results concerning $\mathrm{C}_0$-semigroups on Banach lattices
to a setting of ordered Banach spaces. We prove that the generator of a
disjointness preserving $\mathrm{C}_0$-semigroup is local. Some basic
properties of local operators are also given. We investigate cases where local
operators generate local $\mathrm{C}_0$-semigroups, by using Taylor series or
Yosida approximations. As norms we consider regular norms and show that bands
are closed with respect to such norms. Our proofs rely on the theory of
embedding pre-Riesz spaces in vector lattices and on corresponding extensions
of regular norms.
