Semilinear metric semilattices on $\mathbb{R}$-trees

Abstract

We introduce the notion of metric semilattice on the metric space and prove the criterion of $\mathbb{R}$-tree as connected geodesic metric space $X$ admitting the partial order, such that $X$ is semilinear metric semilattice. Also we state the homeomorphism between topological space of orders defining upper semilinear metric $\vee$-semilattices on locally compact complete $\mathbb{R}$-tree $X$ and its metric compactification ${\bar{X}}_m$. As an application we construct the example of locally complete non-homogeneous similarity-homogeneous space showing essentiality of the condition of locally compactness in V.N. Berestovski\vı's conjecture on the structure of such spaces. Constructed metric space is $\mathbb{R}$-tree, where every point is a branching point. It is the metric fibration but is not topological product with factor $\mathbb{R}$ and does not satisfy the Berestovski\vı's conjecture.

14 pages