From similarity to distance: axiom set,monotonic transformations and metric determinacy

How to normalise similarity metric to a metric space for a clusterization? A new system of axioms describes the known generalizations of distance metrics and similarity metrics, the Pearson correlation coefficient and the cosine metrics. Equivalent definitions of order-preserving transformations of metrics (both monotonic and pivot-monotonic) are given in various terms. The metric definiteness of convex metric subspaces $\mathbb{R}^n$ and $\mathbb{Z}$ among the pivot-monotonic transformations is proved. Faster formulas for the monotonic normalization of metrics are discussed.