The category of ultrametric spaces is isomorphic to the category of complete, atomic, tree-like, and real graduated lattices LAT*
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Abstract.It is proved that for any ultrametric space (X, d),
the set L(X) of its closed
balls is a lattice
$$ (\mathbf{L}(X), \bigcap, \mathrm{sup}, r(B)) $$
.
It is complete, atomic, tree-like, and real graduated.
For any such lattice
$$ (L, \bigwedge, \bigvee, r) $$
, the set A(L)
of its atoms can be naturally equipped with
an ultrametric
$$ \Delta (x,y) $$
.
These assignments are inverse of one another:
$$ (\mathbf{A}(\mathbf{L}(X)), \Delta) = (X,d)\quad
\mathrm{and}\quad (L, \bigwedge, \bigvee, r) =
(\mathbf{L}(\mathbf{A}(L)), \bigcap, \mathrm{sup}, r(B)) $$
where the first equality means an isometry while the second one is a lattice isomorphism.
A similar correspondence established for morphisms, shows that there is an isomorphism of
categories. The category ULTRAMETR of ultrametric spaces
and non-expanding maps is isomorphic to the category LAT*
of complete, atomic, tree-like, real graduated lattices and
isotonic, semi-continuous, non-extensive maps. We describe properties of the isomorphism
functor and its relations to the categorical operations and action of other functors. Basic
properties of a space (such as completeness, spherical completeness, total boundedness,
compactness, etc.) are translated into algebraic properties of the corresponding lattice
L(X).
[1] Kiiti Morita. Normal families and dimension theory for metric spaces , 1954 .
[2] Vladimir A. Lemin,et al. Finite Ultrametric Spaces and Computer Science , 2001 .