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RESEARCH PROJECT

144011G: “ALGEBRAIC STRUCTURES AND INFORMATION PROCESSING METHODS”

RESEARCH TOPICS

From the project application:

The subject of research are algebaric structures and computing and information processing methodologies based on algebraic structures and the concepts and methods resulting from them.

In what follows we give a short review of the main planned research topics:

(1) Basic research on algebraic structures

(1.1) Universal algebra: varieties, generalized varieties and pseudovarieties, unary algebras and connections to automata and semigroups, limits and subdirect products of algebras, congruences and weak congruences, applications in computer science;

(1.2) Ordered sets and lattices: ordered algebraic structures, quasi-orders, conceptual structures, weak congruence lattices.

(1.3) Semirings and rings: varieties of associative and Lie rings, varieties of semirings, bases of identities of semirings, applications of semirings in theory of formal languages and automata.

(1.4) Semigroups: decomposition and composition methods (subdirect and semilattice decompositions, ideal extensions, etc.), identities and varieties, regularity conditions, involution semigroups.

(1.5) Discrete structures: graphs (random and Henson graphs, etc.), monoids of endomorphisms of graphs, hypergraphs, tournaments and hypertournaments, apllications of graphs and hypergraphs in theory of formal languages and automata.

(1.6) Algebraic-topological structures: covering and other properties of algebraic-topological structures (groups, vector spaces, etc.), connections with game theory, Ramsey theory, descriptive set theory, theory of cardinal numbers, applications in theories of formal languages and fuzzy sets.

(2) Investigation of computing and information processing methodologies based on algebraic structures and related concepts

(2.1) Formal languages and automata: scattered subwords and related problems, two-dimensional languages and shifts, laws of algebras of two-dimensional languages, tree automata and connections with algebraic logic, weighted automata.

(2.2) Symbolic and algebraic computation and applications: symbolic and algebraic computational methods for application in matrix computations, mathematical programming and optimization, methods of representation and computation of generalized inverses of matrices and their symbolic implementation, methods of linear, nonlinear and dynamic programming and their symbolic implementation.

(2.3) Fuzzy sets and applications: intuitionistic fuzzy sets, poset and lattice valued fuzzy sets, fuzzy relations, fuzzy equivalence relations and fuzzy quasi-orders, fuzzy matrices and related concepts, fuzzy languages and automata, applications in medical sciences, economics, management, etc.

Importance of the planned research one reflects in constantly growing needs for algebraic methods and concepts that would be used in modelling numerous phenomena in mathematics, computer science and other scientific fields, especially those in information processing, where these methods and concepts are the most powerful.