Using binary operations to constructa transitive set of block transformations

I. V. Cherednik

Diskretnaya Matematika, Tome 31 (2019) no. 3, pp. 93-113.

Résumé

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 $\{\Sigma^F\colon F\in\mathcal B^*(\Omega)\}$.

Détail

Keywords: network, block transformation, transitive class of block transformations.

Bibliographie

[1] Belousov V. D., Osnovy teorii kvazigrupp i lup, Nauka, M., 1967

[2] Cherednik I. V., “Odin podkhod k postroeniyu tranzitivnogo mnozhestva blochnykh preobrazovanii”, Prikladnaya diskretnaya matematika, 38 (2017), 5–34