Lattices of uniformly continuous functions

An explicit representation of the order isomorphisms between lattices of uniformly continuous functions on complete metric spaces is given. It is shown that every lattice isomorphism T:U(Y)→U(X) is given by the formula (Tf)(x)=t(x,f(τ(x))), where τ:X→Y is a uniform homeomorphism and t:X×R→R is defined by t(x,c)=(Tc)(x). This provides a correct proof for a statement made by Shirota sixty years ago.