Dual decompositions and continuity of partial orders in algorithm choice theory

The usefulness of stating is shown and a solution is obtained of the problem of decompositing, in some form, of a continuous partial order in a family of scalarized partial orders which are immersions of the initial partial order. The proposed decomposition is performed by using scalar functions which can be either 0 or 1. Relations are investigated between structural and topological properties of the initial order and of orders which make the decomposition.