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Work number - M 2 AWARDED

Presented Presented by Oles Honchar Dnipro National University

Doctor of Physics and Mathematics Pypka О.О., PhD (Mathematics) Yashchuk V.S.

The aim of the work is the investigation of the structure and properties of algebraic structures (of some types of groups, rings and algebras) under certain restrictions of different nature and character.

The authors, based on theoretical studies, obtained a series of results devoted to the study of the structure of finite and some classes of infinite groups, in which every subgroup (or some part of subgroups) can be either only of one type or of two opposite (in some sense) to each other. A consideration of a similar problem for some infinite-dimensional Leibniz algebras is proposed and implemented. In particular, a description of some Leibniz T-algebras is obtained.

Some natural relationships between the factors of (generalized) central series of groups, Lie rings, Lie algebras and Leibniz algebras are investigated. In particular, a series of analogues (automorphic, rank, etc.) and generalizations of theorems of Schur, Baer and Hall have been proved. Relations between the factors of (generalized) central series of these algebraic structures with different types of residuals, which defined in these structures, are established.

Two new algebraic structures are proposed, namely lattice groups and lattice rings. Their properties and relationships with L-fuzzy groups and L-fuzzy rings are investigated.

The description of 3-dimensional Leibniz algebras over finite fields was first investigated and obtained.

The results of this work are analogues or generalizations of the classical results of world-famous algebraists (Ph. Hall, R. Baer, S.N. Chernikov, B. Neumann, B.I. Plotkin, J. Wiegold, D. Robinson, I. Stewart, etc.).

The results, as well as the research methods, can be used in the further development of relevant directions of general algebra.

Number of publications: 84, including 1 monograph, 43 papers (18 – in foreign journals with impact factor), 26 conference abstracts. The total number of references to author's publications / h-index according to databases is respectively: Web of Science 27/3; Scopus –26/3; Google Scholar – 65/4.