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Morphisms and properties of non-associative algebraic systems with Moufang type conditions


Author: Diduric Natalia
Degree:doctor of
Speciality: 01.01.06 - Mathematical logics, algebra and theory of numbers
Year:2022
Scientific adviser: Victor Şcerbacov
doctor habilitat, associate professor (docent), Institute of Mathematics and Computer Science
Institution: Moldova State University

Status

The thesis was presented on the 29 March, 2022
Approved by NCAA on the 1 July, 2022

Abstract

Adobe PDF document – 1.03 Mb / in romanian
Adobe PDF document – 1.03 Mb / in english

Thesis

CZU 512.548

Adobe PDF document 2.04 Mb / in romanian
104 pages

Keywords

quasigroup, loop, isotope, pseudo-automorphism, left Bol (right) quasigroup, Moufang quasigrup, WA-quasigroup, i-quasigroup, G-properties

Summary

Thesis structure: the thesis is written in Romanian and contains an introduction, four chapters, general conclusions and recommendations, 109 bibliographic titles, 104 pages (including 91 pages of basic text). The obtained results are published in 16 scientific papers.

Thesis field of study: algebra, in spesial, the theory of quasigroups with identities including Bol-Moufang-type identities, properties of non-associative algebraic systems.

The purpose and objectives of the paper. The aim of the paper is to investigate the properties of non-associative algebraic systems with Bol-Moufang type identities. To achieve this goal, the following objectives have been defined: research on the relations of WA-, CI-quasi-groups, transitive on the left and Neuman with the quasigroups Moufang, Bol on the left, on the right, etc.; research of the existence of unilateral unity in quasigroups with Bol-Moufang type identities, enumerated in the work of F. Fenyves “Extra loops II. On loops with identities of Bol-Moufang type”, (1969); research of morphisms, properties, relationships with other classes of quasigroups of newly defined quasigroups (i-quasigroups and WIP-generalized quasigroups); research on the G-properties of left transitive quasigroups and Neumann.

Scientific novelty and originality. All the results presented in the thesis are new and original. Diverse classes of quasigroups known earlier (WA-, CI-quasigroups, transitive left quasigroups, Neumann, etc.) were researched. Two new classes of quasigroups were introduced and researched (i-quasigroups, WIP-generalized quasigroups). Isotope group quasigroup classes were investigated. The properties of some classes of invertible quasigroups were described. Connections between the studied quasigroup classes and the classical quasigroups Moufang, Bol, etc. were investigated. The general forms of the automorphisms, pseudoautomorphisms and quasiautomorphisms of these quasigroups were determined.

The important scientific problem solved consists in the research of different morphisms (autotopies, pseudoautomorphisms, G-properties) and notions (distributant, nucleus) in non-asso-ciative algebraic systems with Bol-Moufang conditions that lead to the description of important new relationships between the classes studied by quasigroups (including newly introduced clas-ses).

The theoretical importance and applicative value of the thesis are determined by obtain-ing new results in the research of non-associative systems of the Bol-Moufang type. The paper is of theoretical character. The methods developed in the paper allowed solving the problems.

Implementation of scientific results. The results of the paper can be used in teaching spe-cialized courses for students, masters and doctoral students in mathematics.