Algebra is the language of modern mathematics. As a branch of mathematics it is much broader than elementary algebra , studying what happens, beyond arithmetics of normal numbers when different rules of operations and relations are used. It leads to constructions and concepts arising from them, including terms, polynomials, equations. When the rules of addition and multiplication are generalized, their precise definitions lead to the notions of algebraic structures such as groups, rings, and fields which are actually studied in the area of mathematics called abstract algebra. Algebra is one of the main branches of mathematics, together with geometry, analysis, topology, combinatorics and number theory.
The department of Mathematics at CIIT, Abbottabad has a strong tradition of research in algebra. Algebraic research at CIIT covers a wide spectrum of interests, from pure group theory, semigroup theory, ring theory and fuzzy set theory to applications of algebraic techniques in computer science, Biology (genetics), coding & cryptography and quantum physics. The department is a fertile breeding ground for research, with regular organized seminars and study groups complemented by plenty of opportunities for informal discussion. Algebra group has diversified and acquired strengths in several areas, notably in group generalizations, semigroup generalizations, commutative and non-commutative algebra, fuzzy ordered semigroups and multidirectional applications of these algebraic structures.
The algebra group is active in a variety of research area including;
Non-associative algebras with genetic realizations
Left almost semigroups (LA-semigroups)
Left almost rings (LA-rings), Left almost modules
Fuzzy ordered semigroups
Soft sets theory and its application to non-associative structures
The department is also always on the lookout for strong M.S/M.Phil students in various areas of algebra and several graduate students have completed M.S/M.Phil degrees in recent years.
Ring Theory, Group Theory and generalizations, Fuzzy Algebra, Soft Sets, Genetic algebra