Metric spaces in which the distances are not real numbers have been studied by several people (2, 3, 4, 7, 9). Any ring R together with a mapping, X —> ϕ(X), of R into a lattice A with 0 and 1 satisfying
is called a "lattice-valued ring," where the operations union, ∪, and intersection, ∩, are the usual lattice operations. The mapping ϕ is called a "valuation" and A is a 'Valuation lattice." If R is a lattice-valued ring and a mapping d is defined by