The ordinal relation x ̃ρy ̃, x ̃=(x₁,…,x_n), y ̃=(y₁…,y_n)ϵ Rⁿ is completely determined by the relations (more, less, equal) of components x_i and y_i, i= 1,…,n. In terms of the representing functions g_ρ, a complete description of the ordinal relations ρ for the main classes used in multi-criteria choice problems is found. These are classes of L linear orders, W weak orders, S semi-orders, I interval orders, P partial orders, and A acyclic relations, L⊂W⊂S⊂I⊂P⊂A. We also studied the problem of design of (R₁ⁿ→R₂)-operators of group choice. It consists in describing the operators that aggregate any n-tuple of individual relations from the class R₁ into a group relation of the class R₂. The article solves this problem for all pairs of classes (R₁,R₂), R₁,R₂∈ {L,W,S,I,P,A} satisfying a naturally occurring condition R₁⊆R₂. Recognition problems of whether operator in question is a (R₁ⁿ→R₂)-operator of a given type, are investigated for NP-hardness and polynomiality.

multi-criteria choice, group choice, ordinal relation, description of ordinal relations, group choice operator, design of operators

Mathematical problems and theory of numerical methods