Affiliation:
1. Department of Theory of Functions Institute of Applied Mathematics and Mechanics of NASU, Dobrovolskogo str. 1 , Slovyansk, 84100 , Ukraine
2. Department of Mathematics and Statistics University of Turku, Fin-20014 , Turku , Finland
Abstract
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.
Reference19 articles.
1. V. Bilet and O. Dovgoshey, Completeness, closedness and metric reflections of pseudometric spaces, Topology Appl. 327 (2023), Article ID 108440, 14 p.
2. V. Bilet and O. Dovgoshey, When all permutations are combinatorial similarities, Bull. Korean Math. Soc. 60 (2023), no. 3, 733–746.
3. V. Bilet, O. Dovgoshey, M. Küçükaslan, and E. Petrov, Minimal universal metric spaces, Ann. Acad. Sci. Fenn. Math. 42 (2017), no. 2, 1019–1064.
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5. O. Dovgoshey, Semigroups generated by partitions, Int. Electron. J. Algebra 26 (2019), 145–190.