StatusThe thesis was presented on the 19 April, 2011
Approved by NCAA on the 8 July, 2011
Abstract– 0.45 Mb / in romanian
The thesis is written in English, contains introduction, 6 chapters, conclusions, 210 references, 200 pages of the basic text. The main results of the thesis are published in 45 scientific works. Keywords: topological universal algebra, variety, topological quasigroup, multiple identities, omogen isotope, resoluble space, medial and paramedial grupoid, fuzzy al- gebra.
The thesis is dedicated to the study: The in uence of the algebraic structures on the topological properties of the topological universal algebras. In particular, the topological algebraic systems and its applications in diverse fields is investigated. The purpose of the work resides in: mastering the studying methods of the topolo- gies on free algebras generated by pseudocompact and countable compact spaces; de- scribing the compact subsets of the free topological algebras and that of k-algebras; elaborating research methods regarding the topological quasigroups with multiple iden- tities; establishing a general theory on the decomposition of the topological groupoids with invertibility properties; solving the homomorphism problem for fuzzy algebras. Methodology of the research: the constructions and the methods of proofs are based on the notions of topological algebra, free algebra, variety, quasigroup with multiple identities, solvable space, fuzzy algebra.
The scientific innovation is determined by the solving of the following problems: there have been elaborated studying methods of topologies on free algebras gener- ated by pseudocompact and countable compact spaces; there have been determined the conditions for the continuous homomorphisms of the topological groupoids with continuous division to be open; there have been described the compact subsets of the free k-algebras; there have been established some topological properties, which are pre- served under the MK-equivalence relation; there have been introduced and analyzed the quasigroups with multiple identities; there has been elaborated a method of con- struction of the Haar measure on medial quasigroups; there has been elaborated a method of decomposition of special topological groupoids; there has been constructed a universal covering on topological E-algebras with continuous signature; are given a general solution of the homomorphism problem for fuzzy algebras.
The theoretical value of the work: there have been development of the general
theories, elaborated the new concepts, methods and constructions which contributed to
achieving goals and objectives of the research. The basic results of the work are new.
The practical value: the methodology applied, the concepts and methods developed
in work allowed to find the solution of concrete problems or some aspects of the prob-
lems formulated by A.I Mal'cev, L.S. Pontrjagin, M.M Choban. Mathematical tools
developed and applied led to solving problems in various areas of modern mathematics.
Implementation: The results from this work can be used in the theory of topological
universal algebras, of topological quasigroups, of automata, of fuzzy algebras, and in
elaborating optional courses.