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

Abstract

Abstract We continue to study the set of block transformations $ \{{\it\Sigma}^F : F\in\mathcal B^*({\it\Omega})\} $ implemented by a binary network Σ endowed with a binary operation F invertible in the second variable. For an arbitrary k⩾2 we obtain necessary and sufficient conditions for k-transitivity of the set of transformations $ \{{\it{\it\Sigma}}^F \colon F\in\mathcal B^*({\it\Omega})\} $ , and propose an efficient method for checking whether these conditions hold. We also introduce two methods for construction of networks Σ such that the sets of transformations $ \{{\it\Sigma}^F\colon F\in\mathcal B^*(\Omega)\} $ are k-transitive.