Factor-powers of finite symmetric groups

To a transformation semigroup $(S,M)$ we assign a new semigroup $FP(S)$ called the factor-power of the semigroup $(S,M)$. Then we apply this construction to the symmetric group $S_n$. Some combinatorial properties of the semigroup $FP(S_n)$ are studied; in particular, we investigate its relationship with the semigroup of 2-stochastic matrices of order $n$ and the structure of its idempotents. The idempotents are used in characterizing $FP(S_n)$ as an extremal subsemigroup of the semigroup $B_n$ of all binary relations of an $n$-element set and also in the proof of the fact that $FP(S_n)$ contains almost all elements of $B_n$.