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

Abstract

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.

Keywords

• combinatorial similarity
• discrete pseudometric
• equivalence relation
• strongly rigid pseudometric
• symmetric group
• primary 54e35
• secondary 20m05