  In this paper we investigate iteration of maps on lattices and the
corresponding polynomial-like iterative equation. Since a lattice need not have
a metric space structure, neither the Schauder fixed point theorem nor the
Banach fixed point theorem is available. Using Tarski's fixed point theorem, we
prove the existence of order-preserving solutions on convex complete
sublattices of Riesz spaces. Further, in $\mathbb{R}^n$ and $\mathbb{R}$,
special cases of Riesz space, we discuss upper semi-continuous solutions and
integrable solutions respectively. Finally, we indicate more special cases of
Riesz space for discussion on the iterative equation.
