  Suppose $G$ is a locally solid lattice group. It is known that there are
non-equivalent classes of bounded homomorphisms on $G$ which have topological
structures. In this paper, our attempt is to assign lattice structures on them.
More precisely, we use of a version of the remarkable Riesz-Kantorovich
formulae and Fatou property for bounded order bounded homomorphisms to allocate
the desired structures. Moreover, we show that unbounded convergence on a
locally solid lattice group is topological and we investigate some applications
of it. Also, some necessary and sufficient conditions for completeness of
different types of bounded group homomorphisms between topological rings have
been obtained, as well.
