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

Abstract

The group of combinatorial self-similarities of a pseudometric space

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.