Iteration and iterative equation on lattices

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.