In this paper we give separation theorems for convex sets of a product space. Separation is carried out by linear operators. With these theorems we prove several assertions on conjugate operators and subdifferentials of operators (which map a vector space into an order complete vector lattice), where we use the definition of conjugate operators as done by Zowe. Simultaneously we generalize some of his results to such operators. Moreover we prove that an order complete vector lattice is a vector lattice with certain separation properties.