Using binary operations to constructa transitive set of block transformations

We study the set of transformations $\{\Sigma^F\colon F\in\mathcal B^*(\Omega)\}$ implemented by a network $\Sigma$ with a single binary operation $F$, where $\mathcal B^*(\Omega)$ is the set of all binary operations on $\Omega$ that are invertible as function of the second variable. We state a criterion of bijectivity of all transformations from the family $\{\Sigma^F\colon F\in\mathcal B^*(\Omega)\}$ in terms of the structure of the network $\Sigma$, identify necessary and sufficient conditions of transitivity of the set of transformations $\{\Sigma^F\colon F\in\mathcal B^*(\Omega)\}$, and propose an efficient way of verifying these conditions. We also describe an algorithm for construction of networks $\Sigma$ with transitive sets of transformations $\{\Sigma^F\colon F\in\mathcal B^*(\Omega)\}$.