DOI: 10.1515/agms-2023-0103 ISSN: 2299-3274
Pseudometric spaces: From minimality to maximality in the groups of combinatorial self-similarities
Viktoriia Bilet, Oleksiy Dovgoshey 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.