On extensions of partial isometries

In this paper we define a notion of S-extension for a metric space and study minimality and coherence of S-extensions. We show that every S-extension can be identified with an algebraic object. We use this algebraic representation to give a complete characterization of all finite minimal S-extensions of a given finite metric space and a complete characterization of all minimal coherent S-extensions. We also define a notion of ultraextensive metric spaces and show that every countable metric space can be extended to a countable ultraextensive metric space. %As an application, we show that every countable subset of the Urysohn metric space can be extended to a countable dense ultraextensive subset of the Urysohn space. We also show that the isometry group of an infinite ultraextensive metric space has a dense locally finite subgroup, generalizing several previously known results. We also study compact ultrametric spaces and show that every compact ultrametric space can be extended to a compact ultraextensive ultrametric space.