Bulletin of the
Korean Mathematical Society BKMS

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 \(f \colon d(X \times X) \to d(X \times X)\) 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 \((Y, \rho)\).

Keywords: Combinatorial similarity, discrete metric, semimetric, strongly rigid metric, symmetric group

MSC numbers: Primary 54E35; Secondary 20M05

Supported by: Viktoriia Bilet was partially supported by the Grant of the NAS of Ukraine for the project 0121U111851 ``Regularity for solutions of elliptic and parabolic equations and asymptotic properties of metric spaces at infinity'' and by the Grant EFDS-FL2-08 of the found The European Federation of Academies of Sciences and Humanities (ALLEA). Oleksiy Dovgoshey was supported by Volkswagen Stiftung Project ``From Modeling and Analysis to Approximation''.