When all Permutations are Combinatorial Similarities

TL;DR

This paper characterizes all semimetrics on a set where every permutation acts as a combinatorial self similarity, revealing the structure of such metrics in abstract semimetric spaces.

Contribution

It provides a complete description of semimetrics on any set where all permutations are combinatorial self similarities.

Findings

Identifies conditions under which all permutations are self similarities.

Characterizes the structure of semimetrics with this property.

Extends understanding of symmetry in semimetric spaces.

Abstract

Let $(X, d)$ be a semimetric space. A permutation $\Phi$ of the set $X$ is a combinatorial self similarity of $(X, d)$ if there is a bijective function $f \colon d(X^2) \to d(X^2)$ such that $$ d(x, y) = f(d(\Phi(x), \Phi(y))) $$ for all $x$, $y \in X$. We describe the set of all semimetrics $\rho$ on an arbitrary nonempty set $Y$ for which every permutation of $Y$ is a combinatorial self similarity of $(Y, \rho)$.