Gérard Lallement (1935–2006)
D. Perrin Institut Gaspard Monge, Universit Paris-Est
, 77454 Marne-la-Valle CEDEX 2,
J. Almeida ( ) Dep. Matemtica Pura/CMUP, Fac. Cincias, Univ. Porto, Rua do Campo Alegre 687, 4169-007 Porto,
Grard Lallement died suddenly on January 22, 2006. He was born in 1935 in
Metz, a city of Lorraine, a province disputed a long time between France and
Germany. His own name reflected this duality since it means the German in French.
He graduated from the University of Paris in 1961 and obtained the same year
the prestigious Agrgation in mathematics. He defended his doctoral thesis prepared
under the supervision of Paul Dubreil in 1966 and received the same year the Albert
Chtelet medal for research from the Acadmie des Sciences de Paris. He was during
four years a researcher at the Centre National de la Recherche Scientifique (1962
1966) and Matre de Confrences at the University of Reims two years (19671969).
He joined the Pennsylvania State University in 1969 and remained there until 2006.
He visited the University of Paris on several occasions, especially on sabbatical leave
in 1975 and 1995.
His contributions in mathematics are outstanding. He has worked on both classical
semigroup theory and the algebraic theory of automata. He may be considered as
a major actor of the interplay between semigroup theory and theoretical computer
science. In this direction, his book Semigroups and Combinatorial Applications
 played an important role.
Semigroup theory Among his many results in classical semigroup theory, we single
out the following ones.
His contribution to the property of residual finiteness. Recall that a semigroup is
called residually finite if any two distinct elements can be separated by a congruence
of finite index. In , he gave a constructive proof of the theorem of Malcev
asserting that commutative semigroups are residually finite. In , he considered the more
general case of nilpotent semigroups. Recall that a semigroup is said to be c-nilpotent
if the law
qc(x, y, z1, . . . , zc1) = qc(y, x, z1, . . . , zc1)
qi+1(x, y, z1, . . . , zi ) = qi (x, y, z1, . . . , zi1)zi qi (y, x, z1, . . . , zi1).
This notion was introduced by Neumann and Taylor in 1963 and they showed that
it coincides with the usual definition of nilpotency for groups. Grard Lallement has
studied this class of semigroups and shown in particular that finitely generated regular
nilpotent semigroups are residually finite. It is still not known if this statement holds
without the hypothesis that the semigroup is regular.
He has worked repeatedly on finitely presented semigroups and, in particular, on
one-relator semigroups and especially on the conjecture that all one-relator
semigroups have solvable word problem. He told once that, after a lecture of Adjan on
this subject, somebody in the audience asked whether a Soviet journal would publish
the solution in English if somebody from the West would solve the problem. Adjan
answered that before the problem was solved, everybody would publish in English in
the Soviet Union. He seems to be right? Grard Lallement published in  a
number of results on this subject, including a characterization of one-relator semigroups
containing an element of finite order, generalizing the result of Magnus, Karrass and
Solitar on one-relator groups.
Algebraic automata theory Grard Lallement has worked on the theory of
decomposition of semigroups initiated by the Krohn-Rhodes theorem proved in 1965. The
result describes the irreducible semigroups with respect to the operation of semidirect
product. It can be stated either for abstract semigroups (or monoids) or for
transformation semigroups. It can also be stated as a theorem on sequential transducers with
the composition of transducers replacing the semidirect product of semigroups. The
proof itself admits possible variants leading to different decomposition algorithms.
Grard Lallement has contributed to clarify this complicated domain. In , he gives
an algebraic proof of the result using the notion of wreath product of monoids instead
of semidirect product. The presentation given by Eilenberg in volume B of his book
Automata, Languages and Machines in 1976 gives account of his work. The
building blocks of these decompositions are either finite groups or elementary semigroups,
which have been characterized by Grard Lallement in .
Prefix codes and semigroups A number of his papers are devoted to a problem on
which the second author has worked himself, including jointly with Grard . The
general idea is to study the relation between a rational language and its syntactic
semigroup. General theorems of correspondence between combinatorial properties
of the language and the algebraic structure of the semigroup are known. One of the
most famous is Schtzenbergers theorem characterizing star-free languages as
corresponding to semigroups with trivial subgroups. The general framework was set by
Eilenberg as a theory of varieties of semigroups in correspondence with varieties of
formal languages. In a more specific framework, Grard Lallement has worked on
the correspondence with an emphasis on the languages of the form X where X is
a finite prefix code. He has in particular studied the case where the corresponding
syntactic semigroup is a union of groups , an inverse semigroup  or a regular
Grard Lallement, as said above, played a key role in the interaction between
mathematics and theoretical computer science by his commitment into the two-way
fertilization between semigroup theory and automata theory.
He joined the editorial board of this journal in 1974 and became a managing
editor in 1979 and a council member when the system was implemented in 1991. He
remained an active member of the editorial board until the year 2000 when he became
a honorary editor.
Grard was also an outstanding professor and lecturer as outlined in the obituary
published by Penn State University: His greatest professional joy was as a teacher,
as evidenced by the generation of students who remember his class fondly. He
celebrated 50 years of teaching in 2005. His courses were demanding and difficult, but
ultimately rewarding for all students who worked hard and persevered. He served as
director of graduate studies in 1978 and had himself a number of research students
including Michael Keenan, Elaine Milito, Janusz Konieczny, and the first author. He
was chairman of his department from 1982 to 1985.
Let us conclude this brief account of his scientific work by a more personal touch.
He was also a key person in the communication between Europe and North America.
Many people, including both authors, found a splendid hospitality in his family when
traveling to the United States and he was for many of us a Pygmalion welcoming
young researchers to an American campus, including memorable journeys around the
countryside of Pennsylvania. The scientific community is thus missing since 2006 a
prominent member as well as a cheerful, independent and deeply original personality.