
Algebra, theory of numbers, and mathematical logic
This direction of researching was initiated
at the Institute of Mathematics in 1960 by one of the most prominent mathematicians
of the 20th century — Anatolij
Ivanovich Mal'tsev (19091967). A. I. Mal'tsev is the author
of profound works on application of mathematical logic to algebra and
mathematical cybernatics. According to american mathematicians A. Tarskij,
L. Henkin and D. Monk, A. I. Mal'tsev pointed out
the way from logic to algebra. "Mal'tsev conditions", "Mal'tsev
theories", "Mal'tsev theorems", "Mal'tsev algebras"
are widely used in modern mathematical language. The theory of axiomatizable
classes of algebraic systems, developed by A. I. Mal'tsev and
A. Tarskij, led to overall synthesis of ideas of algebra and mathematical
logic, and with new strength (now — in 20th) demonstrated the
unity and integrity of mathematical science. At present, in the frames
of the theory of axiomatizable classes, the theory of varieties, which
became the central part of general algebra, received wide development.
The theory of quasivarieties and Horn logic are also successfully developed.
Profound investigations are carried on in the theory of models, which
became an independent part of mathematical logic. This theory studies
the connections between formal languages and their interpretations or
models.
At present, investigations in mathematical logic and algorithm theory
are headed by Academician Yuri
Leonidovich Ershov. Due to intensive development of computers, the
notion of algorithm became one of the most important notions in modern
mathematics. Along with studying the theory of numberings, which dates
back to A. I. Mal'tsev, the problems of elementary classification
of algebraic objects and problems of algorithmic decidability or undecidability
of elementary theories of various classes of algebraic systems are studied.
Problems of generalized theory of computability are investigated with
the aim of further development of classical recursion theory.
Investigations in algorithm theory, constructive models and algebras,
applied logic are headed by Corresponding Member of RAS Sergej
Savost'janovich Goncharov. He and his followers develop the theory
of algorithmic dimensions, solve the problems of existing for decidable
and constructive models. On this way the problems, founded by A. I. Mal'tsev,
M. Morly, K. Ash, Yu. L. Ershov and other famous mathematicians,
were solved. Profound investigations are carried on in the fields of nonclassical
logics and semantic programming.
Investigations in the general theory of rings were originated by a prominent
algebraist Corresponding Member of the USSR Acad. Sci. Anatolij Illarionovich
Shirshov (19211981). He created a large scientific school on the theory
of rings and Lie algebras. In this school both classical leads and new
ones, united in the frames of the theory of rings close to assotiative
and the algebraic theory of projective planes, are now successfully developing.
Some particular researchers are connected with quantum mechanics and geometry.
Corresponding Member of the USSR Acad. Sci. Mikhail Ivanovich Kargapolov
(19281976) was the direct organizer of investigations in the theory of
groups in Mal'tsev's logicalgebraic school. Under his guidance, a large
group of professors formed a group of scientists, which successfully works
in Siberia just now. At present, investigations in the theory of groups
continue in the field of finite groups, decidable and generally decidable
groups, matrix groups, latticeordered groups and groups of authomorphisms,
profinite groups. The publication "Unsolved Problems of the Group
Theory. Kourovka Notebook" received worldwide fame.
Since 1962, the Siberian LogicAlgebraic School publishes a specialized
journal "Algebra
and Logic", which is translated into English.

