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CNAA / Theses / 2017 / July /

Self-ortogonal quasigroups: connections with paratopies of some orthogonal systems


Author: Ceban Dina
Degree:doctor of
Speciality: 01.01.06 - Mathematical logics, algebra and theory of numbers
Year:2018
Scientific adviser: Parascovia Sârbu
doctor, associate professor (docent), Moldova State University
Institution: Moldova State University

Status

The thesis was presented on the 7 July, 2017
Approved by NCAA on the 11 May, 2018

Abstract

Adobe PDF document – 0.65 Mb / in romanian

Thesis

CZU 512.548

Adobe PDF document 1.79 Mb / in romanian
137 pages

Keywords

quasigroup, orthogonal system, self-orthogonal quasigroup, paratopy, minimal identity, universal property.

Summary

The language of the Thesis is Romanian. It comprises 117 base pages and has the following structure: Introduction, 3 Chapters, General Conclusions and Recommendations, Bibliography with 142 References and an annex. Research outcomes were reflected in 20 scientific publications.

Field of study: theory of binary and -ary quasigroups.

The purpose and objectives. The purpose of the Thesis is to describe the orthogonal systems consisting of three ternary quasigroups and ternary selectors, admitting at least one nontrivial paratopy. To achieve this purpose the following objectives are established: the founding of all such systems, the characterization of paratopies of these systems, the study of the identities implied by paratopies and the parastrophic-orthogonal (self-orthogonal) quasigroups of different arity, satisfying such identities.

Novelty and scientific originality. In the present Thesis, for the first time, there are determined all orthogonal systems consisting of three ternary quasigroups and ternary selectors, admitting at least one nontrivial paratopy and all paratopies of these systems are described; all identities implied by the paratopies are found and classified. The description of orthogonal systems consisting of three ternary quasigroups and ternary selectors, admitting at least one nontrivial paratopy, is a generalization of the V. Belousov result about the paratopies of orthogonal systems consisting of two binary quasigroups and binary selectors. For this purpose a general method was used, which can be applied for any finite arity. Estimations of the spectra of self-orthogonal -ary quasigroups are obtained, binary and ternary quasigroups with identities which imply the orthogonality of their parastrophes are studied.

The main solved scientific problem consists in describtion of orthogonal systems of three ternary quasigroups and ternary selectors, admitting at least one nontrivial paratopy.

The significance of theoretical and practical values of the work. The results concerning the description of the paratopies of orthogonal systems of quasigroups represent an important step in the study of the transformations of orthogonal systems of -ary operations and of the identities implying the orthogonality of parastrophes of an -ary quasigroup.

Implementation of the scientific results. Orthogonal systems of n-quasigroups, n≥2, are used in the theory of MDS-codes, in criptography, planning experiments, in combinatorics, in the theory of algebraic k-nets etc. The results may be applied as a support for teaching courses in higher education.