(In)dependence of the axioms of Λ-trees

A Λ \Lambda -tree is a Λ \Lambda -metric space satisfying three axioms (1), (2), and (3). We give a characterization of those ordered abelian groups Λ \Lambda for which axioms (1) and (2) imply axiom (3). As a special case, it follows that for the important class of ordered abelian groups Λ \Lambda that satisfy Λ = 2 Λ \Lambda =2\Lambda , (3) follows from (1) and (2). For some ordered abelian groups Λ \Lambda , we show that axiom (2) is independent of axioms (1) and (3) and ask whether this holds for all ordered abelian groups. Part of this work has been formalized in the proof assistant Lean {\mathsf{Lean}} .