On $\psi$-lattices in modular $(\phi ,\mathrm{\Gamma })$-modules

Let $F\mathrm{/}{\mathbb{Q}}_p$ be a finite field extension, let $k$ be a finite field extension of the residue field of $F$. Generalizing the $\psi$-lattices which Colmez constructed in étale $(\phi ,\mathrm{\Gamma })$-modules over $k[[t]][t^{-1}]$, we define, study and exemplify $\psi$-lattices in étale $(\phi ,\mathrm{\Gamma })$-modules over $k[[t_1,\dots ,t_d]][{\prod }_i{t}_i^{-1}]$ for arbitrary $d\in \mathbb{N}$.