A $K$-affine structure on $B^{sm}$ compatible with the given ${\bf Z}$-affine structure is a $GL(n,{{\bf Z}})\ltimes(K^{\times})^{n}$-torsor on $B^{sm}$ such that the application of $val^{\times n}$ to $(K^{\times})^{n}$ gives the initial $GL(n,{{\bf Z}})\ltimes{{\bf R}}^{n}$-torsor.