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Research of topological groupoids with multiple identities


Author: Josu Natalia
Degree:doctor of
Speciality: 01.01.04 - Geometry and topology
Year:2015
Scientific adviser: Liubomir Chiriac
doctor habilitat, professor, Tiraspol State University
Scientific consultant: Mitrofan Cioban
doctor habilitat, professor
Institution: Tiraspol State University

Status

The thesis was presented on the 24 November, 2015
Approved by NCAA on the 22 December, 2015

Abstract

Adobe PDF document – 0.50 Mb / in romanian

Thesis

CZU 515.1(043.3)

Adobe PDF document 0.50 Mb / in romanian
128 pages

Keywords

topological groupoid, multiple identities, topological quasigroup, medial groupoid, paramedial groupoid, homogenous isotope, right topological loop, special Cartesian product, n-topological groupoid with continuous division

Summary

Thesis structure: the thesis is written in Romanian and contains introduction, 3 chapter, conclusions, glossary, 177 references, 110 pages of basic text. The main result of the thesis was published in 35 scientific works.

Field of study of the thesis: the influence of the algebraic structures on the topological properties of the topological groupoids with multiple identities and continuous division.

Thesis aim and objectives: mastering the studying methods of the topological groupoids with some algebraic properties; elaborating research methods regarding topological groupoids with multiple identities; establishing the conditions for the continuous homomorphisms of the n-topological groupoids with continuous division to be open; solving the problem for which one open compact subset from right paramedial topological loop contain one open compact paramedial right subloop; establishing the conditions for which (n, m)-homogenous isotope of topological groupoids with algebraic property P has the same property; determining the conditions for which on the set can be defining one binary operation such that the new algebraic structure becomes non associative quasigroup with special algebraic properties.

Scientific innovation and originality: there have been determined the conditions for the continuous homomorphisms of the n-topological groupoids with continous division to be open; there have been analyzed the algebraic properties of (n, m)-homogenous isotope of topological groupoids; there have been established some properties of primitive subgroupoid with divisions of topological primitive groupoid with division; there have been determined the conditions for which one open compact subset from right paramedial topological loop contain one open compact paramedial right subloop; there have been established for which on the set can be defining one binary operation such that the new algebraic structure becomes non associative quasigroup with special algebraic properties.

The important scientific problem solved:that was solved consists in the development of several research methods for topological groupoids with divisions, which led to the determination of correlations between the properties of algebraic and topological groupoids with multiple identities and continuos division.

The theoretical significance and applicative value of the thesis: there have been elaborated the new concepts, methods and new constructions which contributed to achieving goals and objectives of the research. The basic research of the work are new.

The methodology applied, the concepts and methods developed in work allowed to find the solution of concrete problems or some aspects of the problem formulated by M.M.Choban and L.L.Chiriac. Mathematical tools developed and applied led to solving problems in topological algebras.

The implementation of the scientific results: the results from this work can be used in the theory of topological groupoids and quasigroups, theory of automata and in elaborating optional courses.