The Riesz-Kantorovich formulas for $\mathbb{L}$-vector lattices

Let $\mathbb{L}$ be a Dedekind complete unital $f$-algebra. We prove the Riesz-Kantorovich formulas for order bounded $\mathbb{L}$-module homomorphisms from a directed partially ordered $\mathbb{L}$-module with the Riesz Decomposition Property into a Dedekind complete $\mathbb{L}$-vector lattice satisfying an additional mild condition.