In this paper, we investigate convex semigroups on Banach lattices.
First, we consider the case, where the Banach lattice is $\sigma$-Dedekind complete and
satisfies a monotone convergence property, having L$^p$--spaces in mind as a typical
application. Second, we consider monotone convex semigroups on a Banach lattice,
which is a Riesz subspace of a $\sigma$-Dedekind complete Banach lattice, where we consider
the space of bounded uniformly continuous functions as a typical example. In
both cases, we prove the invariance of a suitable domain for the generator under
the semigroup. As a consequence, we obtain the uniqueness of the semigroup in
terms of the generator. The results are discussed in several examples such as semilinear
heat equations (g-expectation), nonlinear integro-differential equations (uncertain
compound Poisson processes), fully nonlinear partial differential equations (uncertain
shift semigroup and G-expectation).
