On decomposition of choice functions into a system of simpler functions

The paper is concerned with decomposition of the choice function $C(X)$ into a system of simpler functions $\{C_i(X)\}$. For the case of the function $C(X)$ being a Pareto choice function and the functions $C_i(X)$, scalar criteria (univalent or of general form) necessary and sufficient conditions are given for representability of $C(X)$ in the system $\{C_i(X)\}$. Estimates of the minimal number of functions $\{C_i(X)\}$ are obtained.