# Mikhail Moiseevich Lesokhin

### Born: 13 August 1933 in Leningrad, USSR (now St Petersburg, Russia)

Died: 1 June 1998 in St Petersburg, Russia

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**Mikhail Moiseevich Lesokhin**'s name is sometimes written as Lesohin. His father was a shoemaker in Leningrad and Mikhail Moiseevich had a very hard childhood as a result of World War II. In 1941 the Germans invaded Russia and by September of that year German troops were on the outskirts of Leningrad cutting the city off from the rest of Russia. There was appalling suffering from shortages of supplies. About 660,000 people died in Leningrad during the siege from scurvy and starvation. The exceptionally bitter winter of 1941-42, when temperatures fell to -40° C, was one of extreme hardship. The difficulties continued after the end of the war and Mikhail Moiseevich had to suffer not only the general hardships of the time but also the loss of his father who died from starvation in 1944. It was through a piece of good fortune that Mikhail Moiseevich survived these difficult times. His mother requested permission to keep her young son with her as she worked in a military hospital and, somewhat unusually, permission was granted. The siege was broken in January 1944 when a Soviet offensive drove off the besieging Germans from the southern outskirts of the city. The German army fell back to the Panther Line in Estonia and the Russian army began the Baltic offensive. The hospital where Mikhail Moiseevich's mother worked was moved to this Baltic front and Mikhail Moiseevich, together with his mother, went too.

By this time the young boy should have been attending secondary school but the doctors in the hospital made sure that his education did not suffer by teaching him in their spare time. Even at this early stage the doctors realised that the young Mikhail Moiseevich had special mathematical talents but still more adventures were to come before he could return to a normal upbringing. After the Russian army triumphed (losing 260,000 men in the process) the hospital was moved again, this time to the Far East where hostilities were still taking place. Mikhail Moiseevich and his mother, therefore, had to spend time in the Far East before the hospital was closed and they were able to return to Leningrad. The fact that he was a brilliant pupil, together with the educational help he had received from the doctors at the hospital, meant that Mikhail Moiseevich was not disadvantaged educationally when he attended high school in Leningrad and when he graduated in 1951 he received the medal for the top pupil in the whole school.

In 1951 Lesokhin entered the Leningrad Herzen Pedagogical Institute where he studies mathematics and physics. He was awarded a Master's Degree with distinction in 1956 and became a doctoral student of Evgeniy Sergeyevich Lyapin. Lyapin was one of the founders of the algebraic theory of semigroups and one of the leading researchers on the topic. It was therefore natural that Lesokhin should undertake research on this topic and he submitted his dissertation

*Systems with external multiplication*in 1961. This deals with an abstract generalization of some features of the theory of characters of semigroups. Lesokhin published a number of papers on this topic beginning in 1958 and 1961, followed by

*Regularity of systems with exterior multiplication with regular first component*(Russian) (1963) and

*On the completeness of systems with exterior multiplication*(Russian) (1963). All of these papers and his dissertation consider triples of semigroups (

*A*,

*B*,

*C*) with an external multiplication which is a bilinear map from

*A*×

*B*to

*C*. In fact most of the time Lesokhin was undertaking research for his doctorate he was working at the Pedagogical Institute in Khabarovsk in the east of the USSR about 30 km from the Chinese border. He worked there for three years, returning to take up a post as an assistant in the Department of Higher Algebra in the Leningrad Herzen Pedagogical Institute in 1962. He worked in this department for the next 36 years being a full professor in the last part of this period.

Lesokhin's research is mostly in semigroup theory. In addition to the theory of characters of semigroups which he began to work on from the time he was a research student, he also made major contributions to the theory of approximations of semigroups. His first paper in this area was in 1963 when he published

*Homomorphisms into regular semigroups*(Russian). A semigroup

*A*is "approximately" in a class of semigroups

*P*if

*A*is imbeddable in the product of the members of

*P*. In this paper Lesokhin looks only at commutative semigroups where he finds necessary and sufficient conditions that a semigroup be approximately in the class of regular semigroups, the class of groups, the class of idempotent semigroups, and the class of finite groups.

Let us look briefly at two books that Lesokhin jointly authored. In 1967 he published

*Exercises in group theory*jointly with E S Lyapin and A Ya Aizenshtat. Eugene Schenkman, himself the author of a famous group theory textbook, writes:-

The other text we will mention isThis book is meant to give the reader practice with the techniques and methods of algebra. There are eight chapters, each with several sections. Each section has an introductory paragraph or page introducing the concepts and notions. This introduction is then followed by about30exercises, some giving insight into a specific set or group such as the quaternions,2×2matrix groups, and symmetric groups. Other exercises give a general theorem such as the theorem that a characteristic subgroup is normal or that a finite p-group has a centre.

*An introduction to mathematical linguistics*(Russian) (1982) written with K F Luk'yanenkov and R G Piotrovskii. The authors write:-

S I Kublanovsky, the auther if [1], was a student of Lesokhin. He writes:-Elements of language and speech system analysis as well as application peculiarities of fundamentals of modern mathematics to the setting of linguistic models are considered, major attention being focused on the arrangement of thesaurus semantic nets. The book is intended for linguists, specialists in the fields of cybernetics and computing machinery and postgraduates and students training in structural, applied and mathematical linguistics.

In [1] Lesokhin is described as:-Mikhail Moiseevich followed the mathematical growth of his students with great care and was sincerely pleased by their success. His encouragement and support were a great stimulus for us. At his seminars we learned to talk "like real mathematicians", not to be afraid to ask questions, to defend our answers using reasoning and mathematical arguments. Sometimes, when it seemed that everyone understood everything, there remained only one "dumb" participant, Mikhail Moiseevich himself. His students would explain to him with great feeling what he "didn't understand". If even Professor Lesokhin did not understand something, then we, his students, didn't have to feel ashamed that we too didn't understand something. We would ask questions and "confess". Mikhail Moiseevich was very skilful in destroying our inferiority complexes. We became surer and surer of our strength, of our ability to do something in mathematics.

and Kublanovsky also writes:-... a brilliant teacher, a significant mathematicians, a very kind person ...

Mikhail Moiseevich was a very cheerful man, bubbling with life. He treated people ... with great sympathy and respect, and they replied in kind.

**Article by:** *J J O'Connor* and *E F Robertson*

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