On the Endomorphism Monoid of Certain Ultrametric Speces

Abstract

In this work we investigate the endomorphism monoid of certain ultrametric spaces. According to our main result, if X =<X,e> is an ultrametric space such that the range of e is finite, then the set of locally finite endomorphisms is dense in the endomorphism monoid of X and the endomorphism monoid of X has a dense, locally finite submonoid. This can be regarded as a homomorphism oriented counterpart of some ecently obtained results about the existence of dense, locally finite subgroups of the automorphism group of certain homogeneous structures. Further, as a byproduct, we obtain Hrushovski style extension theorems for the ages of certain ultrametric spaces, but here, instead of partial isomorphisms we extend partial homomorphisms.