and making ${\mathbb{A}}$ a $\Lambda$-metric space in the sense of Definition \[Def\_metric\].

\[Def\_LambdaBuilding\] Let $X$ be a set and ${\mathcal{A}}$ a collection of injective charts $f:{\mathbb{A}}\hookrightarrow X$. We call the images $f({\mathbb{A}})$ of the charts $f$ in ${\mathcal{A}}$ *apartments* of $X$ and we define *Weyl simplices, hyperplanes, half-apartments, ... of $X$* to be images of such in ${\mathbb{A}}$ under a chart in ${\mathcal{A}}$. The set $X$ is a *(generalized) affine building* with *atlas* ${\mathcal{A}}$ if the following conditions are satisfied

1.  The atlas is invariant under pre-composition with elements of $ {{{W}}_T}$.

2.  Given two charts $f,g\in{\mathcal{A}}$ with $f({\mathbb{A}})\cap g({\mathbb{A}})\neq\emptyset$. Then $f^{-1}(g({\mathbb{A}}))$ is a closed convex subset of ${\mathbb{A}}$ and there exists $w\in {{{W}}_T}$ with $f\vert_{f^{-1}(g({\mathbb{A}}))} = (g\circ w )\vert_{f^{-1}(g({\mathbb{A}}))}$.

3.  For any pair of points in $X$ there is an apartment containing both.

Given a $\Lambda$-metric on the model space, axioms $(A1)-(A3)$ imply the existence of a $\Lambda$-valued distance on $X$, that is a function $d:X\times X\mapsto \Lambda$ satisfying all conditions of the definition in \[Def\_metric\] but the triangle inequality. The distance of points $x,y$ in $X$ is the distance of their preimages under a chart $f$ of an apartment containing both.

1.  Given two Weyl chambers in $X$ 