Lattice structure on bounded homomorphisms between topological lattice rings

Suppose $X$ is a topological ring. It is known that there are three classes of bounded group homomorphisms on X whose topological structures make them again topological rings. First, we show that if $X$ is a Hausdorff topological ring, then so are these classes of bounded group homomorphisms on $X$. Now, assume that $X$ is a locally solid lattice ring. In this paper, our aim is to consider lattice structure on these classes of bounded group homomorphisms; more precisely, we show that, under some mild assumptions, they are locally solid lattice rings. In fact, we consider bounded order bounded homomorphisms on $X$. Then we show that under the assumed topology, they form locally solid lattice rings. For this reason, we need a version of the remarkable Riesz–Kantorovich formulae for order bounded operators in Riesz spaces in terms of order bounded homomorphisms on topological lattice groups.