On orthosymmetric bilinear maps

We use a theorem by Buskes–Kusraev on the automatic symmetry of an order bounded orthosymmetric bilinear map to give a complete description of order bounded derivations on a pseudo f-algebra. This generalizes a well known theorem by Colville, Davis, and Keimel. Then, we investigate the commutator [, ]_Φ of an orthosymmetric bilinear map using an order theoretical approach. This leads to a generalization of the Buskes–Kusraev result to a larger class of orthosymmetric bilinear maps.