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

Abstract Let $$\mathbb {L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> be a Dedekind complete unital f -algebra. We prove the Riesz-Kantorovich formulas for order bounded $$\mathbb {L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> -module homomorphisms from a directed partially ordered $$\mathbb {L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> -module with the Riesz Decomposition Property into a Dedekind complete $$\mathbb {L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> -vector lattice satisfying an additional mild condition.