Reduction theorems in the social choice theory

In the paper, combinatorial theorems relating to the theory of social choice are obtained. These theorems describe general conditions under which the problem on the preserving the preference set $\mathcal{D}$ by an arbitrary aggregation rule $f$ and the problem on the compatibility of the preference set $\mathcal{D}$ with a pair $(f,\mathcal{C})$ can be reduced to similar problems for two specific aggregation rules: the majority rule maj and the “counting rhyme” rule cog. Results are obtained within the framework of clone approach in the theory of social choice proposed by S. Shelah and developed by the authors.