Weak similarities of finite ultrametric and semimetric spaces

TL;DR

This paper investigates the structural properties of finite ultrametric and semimetric spaces, establishing conditions under which their representing trees and Hasse diagrams imply weak similarities or isomorphisms.

Contribution

It provides new criteria linking the isomorphism of representing trees and Hasse diagrams to weak similarities in finite ultrametric and semimetric spaces.

Findings

Structural characteristic of finite ultrametric spaces identified.

Conditions for isomorphic Hasse diagrams of semimetric spaces established.

Link between tree isomorphism and weak similarity demonstrated.

Abstract

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 $\Phi\colon X\to Y$. 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.