On the use of binary operations for the construction of a multiply transitive class of block transformations

We continue to study the set of block transformations $\{\Sigma^F : F\in\mathcal B^*(\Omega)\}$ implemented by a binary network $\Sigma$ endowed with a binary operation $F$ invertible in the second variable. For an arbitrary $k\geqslant2$ we obtain necessary and sufficient conditions for $k$-transitivity of the set of transformations $\{\Sigma^F \colon F\in\mathcal B^*(\Omega)\}$, and propose an efficient method for checking whether these conditions hold. We also introduce two methods for construction of networks $\Sigma$ such that the sets of transformations $\{\Sigma^F\colon F\in\mathcal B^*(\Omega)\}$ are $k$-transitive.