Treelike lattices and ultrametric spaces
by
Alex J. Lemin
Moscow State University of Civil Engineering

A metric space is called ultrametric (non-Archimedean - in German literature, isosceles - in Russian) if its metric satisfies the strong triangle Axiom d(x, z) <= max[d(x, y), d(y, z)].

These spaces play an important role in various areas of mathematics such as number theory (rings Z_p and field Q_p of p-adic numbers), algebra (non-Archimedean valued rings and fields), real analysis (the Baire space B_\aleph0), general topology (generalized Baire spaces), complex analysis (rings M(U) of mero-morphic functions over an open region U), computer linguistics (a set of words of computer language, equipped with the metric inherited from B_\aleph0), p-adic analysis (the ground field \Omega_p), the theory of topological groups, p-adic functional analysis, the theory of p-adic analytic manifolds, and so on. Ultrametric spaces were described up to homeomorphism in [1], up to uniform equivalence in [2, 4] and up to isometry in [6]. A description of complete ultrametric spaces up to uniform homeomorphisms can be given in purely algebraic manner (there is a one-to-one functor to a category of Boolean algebras, [4]).

Theorem. There exists an isomorphism functor L: ULTRAMETR - > LAT* between the category of ultrametric spaces and non-expanding maps and the category LAT* of complete, atomic, treelike, and real graduated lattices and isotonic, semi-continuous, non-extending maps.

Recall that a map a map f: (X, d) - > (Y, d') is non-expanding if it enlarges no distance, i.e., d(x, y) >= d'(fx, fy) for any x and y in X). Isomorphisms in ULTRAMETR are non other than isometries.

Corollary. Lattices L(X) and L(Y) are isomorphic iff the spaces X and Y are isometric.

We describe properties of functor L its action on particular types of spaces (finite, bounded, totally bounded, complete, spherically complete, compact, etc), its relation to categorical operations and actions of other functors. We discuss its relations to Euclidean geometry, Lebesgue measure and integral theory, category theory and p-adic analysis [3, 5]. We give a complete solution for the Gelfand Problem (to describe all finite ultrametric spaces up to isometry) and state a few open problems. Exempli gratia

Problem. What complete, atomic, treelike lattices can be made real graduated, i.e., can be endowed with a monotonic, semi-continuous real-valued function that equals zero at atoms?

References

1. J. de Groot. Non-Archimedean metrics in topology, - Proc. A.M.S., 7:6 (1956), 948-956.
2. A. J. Lemin. Proximity on isosceles spaces, - Russian Math Surveys, 39:1 (1984), 143-144.
3. -. Isometric imbedding of isosceles (non-Archimedean) spaces in Euclidean spaces, - Soviet Math. Dokl. 32:3 (1985), 740-744.
4. -. Boolean algebras describe ultrametric spaces up to uniform equivalence, - AAA60, Dresden, p.17
5. -. Isometric embedding of ultrametric (non-Archimedean) spaces in Hilbert space and Lebesgue space, - in "p-adic Functional Analysis" (Lecture Notes in Pure and Applied Math), Marcel Dekker, 2001
6. -, V. Lemin. On a universal ultrametric space, - Topology and its Applications, 103;6 (2000), 339-345

Date received: January 4, 2001