Pseudometric spaces: From minimality to maximality in the groups of combinatorial self-similarities

The group of combinatorial self-similarities of a pseudometric space ( X , d ) \left(X,d) is the maximal subgroup of the symmetric group Sym ( X ) {\rm{Sym}}\left(X) whose elements preserve the four-point equality d ( x , y ) = d ( u , v ) d\left(x,y)=d\left(u,v) . Let us denote by ℐP {\mathcal{ {\mathcal I} P}} the class of all pseudometric spaces ( X , d ) \left(X,d) for which every combinatorial self-similarity Φ : X → X \Phi :X\to X satisfies the equality d ( x , Φ ( x ) ) = 0 , d\left(x,\Phi \left(x))=0, but all permutations of metric reflection of ( X , d ) \left(X,d) are combinatorial self-similarities of this reflection. The structure of ℐP {\mathcal{ {\mathcal I} P}} -spaces is fully described.