Lutsenko A. Quasigroups with inverse properties

The dissertation is devoted to the study of quasigroups with the properties of reversibility. With the development of cryptography and the search for new methods of information protection, it became possible to use such non-associative structures as quasi-groups. This led to the need to create and construct a structured theory of quasigroups. Quasigroups have various applications in differential geometry, automata theory, physics, theory of planning experiments, coding theory, cryptography, and other sciences related to mathematics. The object of research is quasigroups with the properties of reversibility and the manifolds they define. The subject of research is sets of translation quasigroups, corresponding varieties of quasigroups with reversibility properties, their parastrophic orbits and group isotopes. The purpose of the research is to obtain the connection of quasigroups with the properties of reversibility with sets of translations, the corresponding manifolds, and the classification of group isotopes. Scientific novelty of the obtained results. Basic scientific the results obtained by the author independently are new and consist of the following: — classes of quasigroups with properties of reversibility in directions have been found broadcasts; — the distribution of the corresponding classes of quasigroups on parastrophes is described orbits (beams) according to parastrophic symmetry; — it is proved that these classes of quasigroups with invertibility properties exist by many species and the corresponding identities were found; — invertibility functions are found for each variety of quasigroups z turnover properties; — a classification of group isotopes with reversibility properties was found and constructed bundles of manifolds with properties of reversibility; — matrix quasigroups and quasigroups are described.