Remarks on the Riesz-Kantorovich formula

The Riesz-Kantorovich formula expresses (under certain assumptions) the supremum of two operators $S, T : X \to Y$ where $X$ and $Y$ are ordered linear spaces as $$S \vee T (x) = \sup_{0 \leqslant y \leqslant x} [S (y) + T (x - y)].$$ We explore some conditions that are sufficient for its validity, which enables us to get extensions of known results characterizing lattice and decomposition properties of certain spaces of order bounded linear operators between $X$ and $Y$.