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Constructions of algebraic structures on compactifications of topological algebras


Author: Ciobanu Ina
Degree:doctor of physics and mathematics
Speciality: 01.01.04 - Geometry and topology
Year:2011
Scientific adviser: Mitrofan Cioban
doctor habilitat, professor
Institution: Tiraspol State University
Scientific council: DH 01-01.01.04-27.03.08
Institute of Mathematics and Computer Science

Status

The thesis was presented on the 2 March, 2011
Approved by NCAA on the 31 March, 2011

Abstract

Adobe PDF document – 0.18 Mb / in romanian

Keywords

universal algebra, topologic universal algebra, compactness, algebraic extension, free object, a-pseudocompact algebra

Summary

Thesis structure: The thesis is written in Romanian and consists of an introduction, 3 chapters, conclusions, 80 bibliography titles, 106 pages of main text and 8 figures. The obtained results are published in 9 scientific papers.

The aim of research: to study the conditions of the existence of the free objects in distinct classes of topological universal algebras; to investigate the algebraical extensions of topological universal algebras; to study the classes of topological universal algebras with conditions of compactness; to elaborate the conception of the algebraical extension of topological universal algebras; to determinate the necessary and sufficient conditions are given under which the Stone-ˇCech compactification of the topological universal algebra is algebraical extension.

Scientific noveltly and originality: As a result of the realization of the thesis objectives there was solved the Comfort-van Mill problem of the existence of the (U, V)-free groups for distinct classes U, V of the topological groups; it was introduced and investigated the conception of the a-pseudocompact algebra; it was demonstrated that the Mal’cev algebra is a-pseudocompact iff it is pseudocompact; it was established various conditions under which the Stone-ˇCech compactification of the topological universal algebra is algebraical extension; it was solved the Arhangel’skii problem about the structure of compactifications of the topological algebras.

The theoretical significance and applicative value of the thesis: The results of the thesis present a considerable interest for understanding the deepness of the concepts of compactness and free object in actual mathematics.

The ability to vary the classes of objects in which the free object is determinated is important for obtaining the conditions of their existence. These results can be applied in theoretical and applied research related to the theory of universal topological algebras, the theory of abstract automata and other.

The implementation of the scientific results: The results of the thesis can be used as content of some special courses for students and masters from mathematic specialities and can served as support for some master thesis.