Weak Similarities of Finite Ultrametric and Semimetric Spaces

Weak similarities form a special class of mappings between semimetric spaces. Two semimetric spaces X and Y are weakly similar if there exists a weak similarity Φ: X → Y. We find a structural characteristic of finite ultrametric spaces for which the isomorphism of its representing trees implies a weak similarity of the spaces. We also find conditions under which the Hasse diagrams of balleans of finite semimetric spaces are isomorphic.