Two types of lattices that describe ultrametric spaces
by
Alex J. Lemin
Moscow State University of Civil Engineering

Theorem 1. There exists a contra-variant functor \lambda: Ultram --> Bool from the category of ultrametric spaces and uniformly continuous maps Ultram onto a subcategory Bool* of the category of Boolean algebras Bool such that algebras \lambda(X) and \lambda(Y) are isomorphic iff the completions of spaces X and Y are uniformly homeomorphic.

Corollary 1. Functor \lambda is one-to-one over the class of complete ultrametric spaces.

Functor \lambda can be defined as a composition \lambda = S \sigma F of a forgetful functor F (from Ultram to a category of proximity spaces Prox, [1]), the Smirnov compactification functor \sigma (from Prox to a category of compacta Comp, [4, 6]), and Marshall Stone's duality functor S (between zero-dimensional compacta 0-dim-Comp and Bool, [3, 7]). It can be also defined explicitly. We describe its properties, its relations to Euclidean and Hilbert geometry [2], and give an algebraic characterization of algebras \lambda(X) from Bool*.

Theorem 2 [5]. 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.

Corollary 2. Algebras L(X) and L(Y) are isomorphic iff the spaces X and Y are isometric.

We describe properties of L, its action on particular classes of spaces (such as complete, totally bounnded, compact spaces, etc) and its relations with actions of other functors in METR, [5].

References
1. J. Lemin. Proximity on isosceles spaces, - Russian Math. Surveys, 39:1 (1984), 143-144.
2. -. Isometric imbedding of isosceles (non-Archimedean) spaces in Euclidean spaces, - Soviet Math Doklady 32:3 (1985), 740-744.
3. -. Ultrametric spaces and Boolean algebras, - Fifth Int. Con "Topology and App". Zagreb, 1990
4. -. The Smirnov compactification functor is one-to-one over the class of complete first countable spaces, - "Topology and its applications", 38 (1991), 201-204.
5. -. The category of ultrametric spaces is isomorphic to the category of complete, atomic, tree-like, real graduated lattices LAT*, (to appear)
6. Yu. M. Smirnov, On proximity spaces, - Math. Sbornik, 31 (1952), 543-574 (in Russian), A.M.S. Trans. Ser. 2, 38, p.5-35.
7. M. Stone. Applications of the theory of Boolean rings to general topology, - Trans AMS 41 (1937), 375-481

Date received: May 18, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeu-11.