Abstract: Given a complete lattice $(L,\le)$ equipped with an order-re\-ver\-sing involution $'$ (also called a complete De Morgan algebra), we find conditions for the existence of a residuated binary operation $\ast$ on $L$ such that the given order-reversing involution is determined by the residuation associated to $\ast$. As a consequence, in the case of completely distributive lattices with an order-reversing involution, we find a necessary and sufficient condition for the desired residuated binary operation $\ast$ to exist.

AMS Subject Classification: 06D72, 06D10, 06D15, 06D20.