Let be a partially ordered metric space, that is, a metric space (X, m) equipped with a partial order on X. We say that a T (0)-quasi-metric d on X is m-splitting provided that . Furthermore d is said to be -producing provided that d is m-splitting and the specialization partial preorder of d is equal to . It is known and easy to see that if is a partially ordered metric space that is produced by a T (0)-quasi-metric and is a total order, then there exists exactly one producing T (0)-quasi-metric on X. We first will give an example that shows that a partially ordered metric space can be uniquely produced by a T (0)-quasi-metric although is not total.