Betweenness, Metrics and Entropies in Lattices

We investigate a class of metrics on lattices that are compatible with the partial order defined by the lattice using the ternary relation of betweenness that can be naturally defined on a metric space. The relationships between entropy-like functions and metrics defined on lattices are studied and we show the links that exists between various properties of entropies and properties of metrics. Applications to metrics defined on the lattice of partitions of finite sets are examined.