  We prove that every lattice homomorphism acting on a Banach space
$\mathcal{X}$ with the lattice structure given by an unconditional basis has a
non-trivial closed invariant subspace. In fact, it has a non-trivial closed
invariant ideal, which is no longer true for every positive operator on such a
space. Motivated by these later examples, we characterize tridiagonal positive
operators without non-trivial closed invariant ideals on $\mathcal{X}$
extending to this context a result of Grivaux on the existence of non-trivial
closed invariant subspaces for tridiagonal operators.
