One approach to constructing a transitive class of block transformations

Let $\Omega$ be an arbitrary finite set, and $\mathcal Q(\Omega)$ be the collection of all the binary quasigroups defined on the set $\Omega$. Denote by $\Sigma^F$ the map $\Omega^n\to\Omega^n$, $n\in\mathbb N$, that is defined by the network $\Sigma$ with one binary operation $F$ on the set $\Omega$. In this paper, we present a criterion for the bijectivity of all mappings from the class $\{\Sigma^F\colon F\in\mathcal Q(\Omega)\}$ and define conditions for the transitivity of this class.