Ebook: Semigroups and Automata. SELECTA Uno Kaljulaid (1941–1999)

We have the pleasure to offer to the Mathematical Public the Selecta of the eminent, late Estonian algebraist Uno Kaljulaid. It contains mainly papers published in Kaljulaid's lifetime. Many of them were originally written in Russian, a few also in Estonian, and have now been translated into English, mainly, by one of us, J. Peetre

Latter on referred to as Senior Editor.

Heritage. In addition to this published material, Kaljulaid left a large number of manuscripts in various states of completion. They are currently in the custody of the Senior Editor. For instance, there is an almost complete paper on right order groups, surveying the subject in its historical development, starting with D. Hilbert; some material on Petri nets, etc., things that, apparently, occupied Kaljulaid in his last years. Hopefully, part of it can also be made public, at a later stage, perhaps in the form of Selecta II.

Let us now highlight some of the main items of the present Volume.

Contents. We offer here the English translation of Kaljulaid's 1979 Tartu/Minsk Candidate thesis [K79a], which originally was typewritten in Russian and manufactured in not so many copies. The thesis was devoted to representation theory in the spirit of his thesis advisor B. I. Plotkin: representations of semigroups and algebras, especially extension to this situation, and application of the notion of triangular product of representations for groups introduced by Plotkin. We include also two summaries of the thesis [K77a] and [K79b].

Through representation theory, Kaljulaid became also interested in automata theory, which at a later phase became his main area of interest.

Another field of research concerns combinatorics.

Besides being an outstanding and most dedicated mathematician Uno Kaljulaid was also very much interested in the history of mathematics. In particular, he took a vivid interest in the life and work of the great 19th century Dorpat-Tartu algebraist Th. Molien (see Chapter V). Perhaps he saw in Molien a kindred soul, as neither of the two got quite the recognition from their Alma Mater, which they for sure deserved; in Molien's case, he had to go into voluntary exile in Tomsk, Siberia.

Kaljulaid was also very interested in the teaching and exposition, or popularization of mathematics; he had several outstanding research students. Some of his more popular-scientific papers were published in an Estonian language journal Matemaatika ja Kaasaeg (Mathematics and Our Age). Amongst there is a whole series of papers about algebraic matters, culminating in a brilliant, elementary – although partly rather philosophical – essay devoted to Galois theory [K75a]. Another such series is his excellent essay of Diophantine Geometry [K68a,69b], in various installments, followed by his éloge [K68b] to another of his teachers Yu. I. Manin. We believe that the inclusion of these papers here will make the Volume more interesting for beginners, and perhaps even contribute to attracting young people to mathematics, in Estonia and elsewhere.

Presentation. The papers in the Volume are assembled in chapters according to the theme.

Important matters or notions have often, with some consequence, been set in italics, sometimes upon their first appearance, or else where they are defined.

Rather rare quotes in other languages than English are usually followed by a translation within parentheses.

References to items of Uno Kaljulaid come in the form [Kx], where x (a year) is taken modulo 1900, and refer to the bibliography. References to other mathematicians come in the form [y], where y runs through 1,2,3,…, independently in each separate paper.

In case of books translated into Russian, the Russian translation is often indicated, along with the original for the benefit of the Readers reading Russian or having access to the Russian book. In transliterating the Cyrillic into English we use, with some consequence, the system in Mathematical Reviews, as set forth on p. 1–2 of the book [1].

Some facts about Estonia and Estonian mathematics. It should perhaps also be recalled here that Estonia is the northern most of the three Baltic Republics, facing the Finnish Gulf in the north, bordering to Latvia in the south and to Russia in the East. Its population is about 1.3 million, most of them Estonians, many living in the capital Tallinn; there is also a large Russian speaking minority. The Estonians speak a language somewhat affined to Finnish and not at all related to the language of their southern neighbors the Latvians and the Lithuanians. Estonians were mentioned already by the Roman writer Tacitus (c. 55–117) who spoke of them as the Aestorum gentes. However, around the beginning of the 13th century the Estonians were still among those few people in Europe who had not accepted Christianity. In a devastating war (1208–1227), against German, Swedish and Danish Crusaders, the new religion was forced upon them. The last stronghold of the Estonians, the Castle of Valjala on the island of Saaremaa, was conquered by a Crusader's army, coming from Pärnu and marching over the frozen archipelago, in February, 1227. Then the Estonians became united, together with the Latvians, in a state ruled by the Order of the Brethren of the Sword, later known as the Teutonic Order, while the native population came to live, for centuries, in serfdom. The rule of the Order lasted until mid 16th century. At later times, Estonia was governed, alternatingly, by Swedes, Poles, and Russians. The situation of the indigenous deteriorated ever more and was particularly low towards the end of the 18th century, farmers were freely sold to the highest bidding landowner; one could even draw a parallel to the Belgian Congo at a much later epoch. However, in the mid of the 19th century a national awakening took place. After hard struggles, the Estonians managed to form an independent country of their own in 1918–20, in the aftermath of World War, when all empires collapsed, the Russian one included. In the advent of the Molotov-Ribbentrop treaty in August, 1939 it was annexed by the Soviet Union in June, 1940, and regained its independence in 1991, during the fall of the Soviet empire.

For more details about the above, and also some information about mathematics in Estonia until 1940, with a tradition going back to the Academia Gustaviana in Tartu, founded by the Swedish King Gustavus Adolphus in 1632, closed down in 1656, when the city was captured by the Russians, and then followed by the Academia Carolina (1690–1710)

Probably, few mathematicians are aware of that the first ever to teach about Newton's cosmology was the Swede Sven Dimberg in Tartu [3].

Acknowledgements. The appearance of the present compilation would not have been possible without the generous assistance of a large number of friends and colleagues, students, secretaries, librarians, family members, etc. – from Novosibirsk in the East to Iowa in the West. To all of them we express here our sincere thanks. The following list of names (in alphabetic order) comprises probably only a fraction of all:

Gert Almkvist, Marianne Blauert, Leonid Bokut, Kerstin Brandt, Michael Cwikel, Martina Eicheldinger, Miroslav Engliš, Jan Gustavsson, László Filep, Eila Ritva Jansson, Margreth Johnsson, Kalle Kaarli, Dan and Christer Kiselman, Andi Kivinukk, Richard Koch, Petr Krylov, Ruvim Lipyanskii, Indrek Martinson, Caroline Myrberg, Aleksandr Nikolskii, Inga-Britt Peetre, Jakob-Sebastian Peetre, Monika Perkmann, Ann-Christin Persson, Ulf Persson, Professor Pater Anders Piltz O.P., Boris Plotkin, Olga Sokratova, Sven Spanne, Gunnar Sparr, Michael David Spivak, Annika Tallinn, Hellis Tamm, Marje Tamm, Enn Tamme, Erki Tammiksaar, Gunnar Traustason, Michael Tsfasman, Victor Ufanrovski, Aleksandr Zubkov.

Amongst institutions, we mention in particular the following:

Eesti Loodusuurijate Selts (Estonian Naturalists' Society, Tartu, Estonia); Verlag Heyn (Klagenfurt, Austria).

We have had an invaluable aid from many libraries, amongst others:

Mathematical libraries of Lund, and the one of Uppsala (named the Beurling library); Lund University, Giesen, and Heidelberg; the library of the Mittag-Leffler Institute; the library of the Institute of Cybernetics at Tallinn University of Technology.

Finally, we express our great esteem for the generosity of our sponsors, the Royal Physiographic Society of Lund, taking over all costs of publication and the European Union's Fifth Framework Programme project IST-2001-37592 (eVikings II) that partially supported the editing of this book and the related visits of Jaan Penjam to Lund.

The Editors

References

[1] A., J. Lohwater. Russian-English Dictionary of the mathematical sciences. American Mathematical Society, Paris, 1961.

[2] Ü. Lumiste and J. Peetre. Edgar Krahn, A centenary volume 1894–1961. IOS Press, Providence, Rhode Island, 1994.

[3] Ü. Lumiste and H. Piirimäe. Newton's Principia in the curricula of the University of Tartu (Dorpat) in the early 1690's. In: R. Vihalemm (ed.), Estonian studies in the history and philosophy of science. Kluwer Academic Publishers, Dordrecht, Boston, New York and London, 2001, 1–18. Swedish translation, based on enlaged 1981 Estonian version: J. Peetre – S. Rodhe, Normat (to appear).