Uniform Residuated Lattices and their Cauchy Completions

Distance function defined by Chang is an important tool for describing closeness and constructing topologies and uniformities on MV-algebras. Unfortunately, this function on residuated lattices is not good enough as on MV-algebras since it is not compatible with operations on residuated lattices. Based on this fact, the axioms of similarity operators and semi-norms are introduced on residuated lattices. By using the above two tools, uniformities and topologies are induced, respectively. Residuated lattices equipped with these uniformities (topologies) are proved to be uniform (topological) residuated lattices. Finally, two kinds of sequential completions for these uniformities are given and they are isomorphic.