The category of ultrametric spaces is isomorphic to the category of complete, atomic, tree-like, and real graduated lattices LAT*

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).