# Department of Algebra and Mathematical Logic

The algebra studies common properties of operations and transformations with sets satisfying to some laws. These laws, i.e. axioms, which define these operations, can be completely different, depending on a problem. In this case we get such algebraic objects, as groups, rings, fields, alternative algebras, Lie algebras etc.

As a rule these algebraic objects appear consensually, but by consideration of deep problems of mathematics, physics and other sections of science. For example, the concept of group appeared in algebra by consideration of the solutions of algebraic equations in radicals (term "group" was defined by Evarist Galois who has applied designed by him a group theory to the solution of a concrete problem of algebraic equations).

The heyday of algebraic researches on algebra in the Kazan university began in 30-50-th years of the 20th century. In that times an outstanding mathematician - N.G.Chebotarev began to work, who was member-correspondent of Academy of Science in USSR, managing department of algebra till 1947. The Kazan school these years conducted radical researches on algebra received a world admission. It has elaborated the theory of fields and theory of Galois, theory of Lie groups and algebras (N.G.Chebotarev, professor V.V.Morozov, who managed the department of algebra since 1947 till 1970, and also their schoolboys).

Now these traditions are prolonged by V.V.Morozov schoolboys U.B. Ermolaev, I.I. Sakhaev and also other employees of chair, who works in different areas of algebra. On chair of algebra there are fruitful researches and on other scientific direction - logic theory. At pre-revolutionary times in Kazan university the activities in mathematical logic were also executed by P.S. Proreckii, N.A. Vasiliev, N.N. Parfentiev. The first course of lectures on mathematical logic in Russia was also read at the Kazan university by professor Proreckii. However there were no stable traditions in this field. It is possible to say about indirect eligibility at formation of school of logic theory in Kazan State University in 70th. The source of this eligibility was served first of all by professor Morozov (1910-1975), miscellaneous scientist who had a high level of mathematical culture.

The researches on logic theory at the Kazan university have deployed with full force in second half of 70th, when from the isolated area it has turn into a speciality of many young mathematicians. Both an increasing concern to problems of logic theory and popularity of its ideas and methods could be explained, on the one hand, by intensive development of computer science and programming, and with other by infiltration of a postulational method into natural mathematical sciences and cybernetics.

Main role in quantitative increase and dilating of subjects on mathematical logic researches at the Kazan university has played the logical seminar under a management of M.M. Arslanov, schoolboy of V.V. Morozov. The seminar helped to grow mathematicians, working in different areas of logic theory in miscellaneous cities of Russia and the CIS countries.

Many members of the department are also working in the department of algebra and mathematical logic of NIIMM organized by our department in 1994 (Head of Department, M.M. Arslanov, the section is under the scientific and methodological direction of the department of algebra).