  We consider convex monotone $C_0$-semigroups on a Banach lattice, which is
assumed to be a Riesz subspace of a $\sigma$-Dedekind complete Banach lattice.
Typical examples include the space of all bounded uniformly continuous
functions and the space of all continuous functions vanishing at infinity. We
show that the domain of the classical generator of a convex semigroup is
typically not invariant. Therefore, we propose alternative versions for the
domain, such as the monotone domain and the Lipschitz set, for which we prove
invariance under the semigroup. As a main result, we obtain the uniqueness of
the semigroup in terms of an extended version of the generator. The results are
illustrated with several examples related to Hamilton-Jacobi-Bellman equations,
including nonlinear versions of the shift semigroup and the heat equation. In
particular, we determine their symmetric Lipschitz sets, which are invariant
and allow to understand the generators in a weak sense.
