Using binary operations to construct a transitive set of block transformations

Igor V. Cherednik

doi:10.1515/dma-2020-0035

Article

Using binary operations to construct a transitive set of block transformations

Igor V. Cherednik

Published/Copyright: December 11, 2020

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From the journal Discrete Mathematics and Applications Volume 30 Issue 6

Abstract

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

Keywords: network; block transformation; transitive class of block transformations

Originally published in Diskretnaya Matematika (2019) 31,№3, 93–113 (in Russian).

References

[1] Belousov V.D., Fundamentals of quasigroups and loops theory M.: Nauka, 1967 (in Russian).Search in Google Scholar

[2] Cherednik I. V., “One approach to transitive set construction of block transformations”, Prikladnaya Diskretnaya Matematika 38 (2017), 5–34 (in Russian).10.17223/20710410/38/1Search in Google Scholar

Received: 2018-12-24

Revised: 2019-08-15

Published Online: 2020-12-11

Published in Print: 2020-12-16

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Articles in the same Issue

https://doi.org/10.1515/dma-2020-0035

Keywords for this article

network; block transformation; transitive class of block transformations