Relative crystalline representations and weakly admissible modules

Let $k$ be a perfect field of characteristic $p>2$, and let $K$ be a finite totally ramified extension over $W(k)[\frac{1}{p}]$. Let $R_0$ be an unramified relative base ring over $W(k)⟨X_1^{\pm 1},\dots ,X_d^{\pm 1}⟩$, and let $R=R_0{\otimes }_{W(k)}{\mathcal{O}}_K$. We define relative $B$-pairs and study their relations to weakly admissible $R_0[\frac{1}{p}]$-modules and ${\mathbf{Q}}_p$-representations. As an application, when $R = $\mathcal{O}$_K[![Y]!]$with$k = \overline{k}$,weshowthateveryrank$2$horizontalcrystallinerepresentationwithHodge-Tateweightsin$[0, 1]$ whose associated isocrystal over $W(k)[\frac{1}{p}]$ is reducible arises from a $p$-divisible group over $R$. Furthermore, we give an example of a $B$-pair which arises from a weakly admissible $R_0[\frac{1}{p}]$-module but does not arise from a ${\mathbf{Q}}_p$-representation.