Prof David Rees

(29th May 1918 to 16th August 2013)

Although he is best
known in the wider mathematical world for his work in
commutative algebra, within semigroup theory David Rees is
viewed as one of the pioneers of the field. His research in
semigroups, although carried out in the 1940's, is still
influential today. Prof Rees was also one of the team of young
men and women who worked at Bletchley Park during the Second
World War where he made significant contributions to their
work.

Rees wrote only four papers in semigroups on his own and one further paper with J. A. Green. They are as follows

Rees wrote only four papers in semigroups on his own and one further paper with J. A. Green. They are as follows

- On semi-groups,
*Proc. Camb**ridge Philos. Soc.***36**(1940), 387--400. - Note on semi-groups,
*Camb**ridge Philos. Soc.***37**(1941), 434--435. - On the group of a set of partial
transformations,
*J. London**Math. Soc.***22**(1947), 281--284. - On the ideal structure of a
semi-group satisfying a cancellation law,
*Quart. J. Math. Oxford Ser.***19**(1948), 101--108. - On semigroups in which x^r = x,
(with J. A. Green),
*Proc. Cambr**idge Phil. Soc.*(1952)**48**, 35--40.

Let me say a little more about the
papers Rees wrote on his own.

The first two papers build on the
work of Suschkewitsch and describe completely 0-simple
semigroups in terms of what are now called Rees matrix
semigroups. This theorem is the first that any beginning
student of semigroup theory meets and is one of the most
influential results in the field. In recent years, the Rees
Theorem, as it is called, has finally been placed in the
context of a Morita theory of semigroups.

The third paper is really a
contribution to inverse semigroup theory before the class of
inverse semigroups had been formalized. Rees first defines
what we would now call an inverse semigroup of partial
bijections on a set. He is specifically interested in the
case where the inverse semigroup that arises does not
contain a zero. He then defines what we would now call the
minimum group congruence on the inverse semigroup thereby
obtaining a group. These constructions are then applied to a
special case. He begins with a cancellative monoid in which
any two principal left ideals intersect. He then constructs
the inverse monoid generated by all the injective functions
defined by right multiplication by an element of the monoid
and their partial bijection inverses. He then proves that
the original cancellative monoid is embeddable from the
group constructed from this inverse monoid. This gives a new
proof of the Ore embedding theorem. Not only is this a neat
application of inverse semigroup theory, the inverse monoids
that arise are in fact what would later be known as *E*-unitary.
The theory of such semigroups and their generalizations has
been an important theme in semigroup theory.

In his fourth paper, Rees deals with
left cancellative monoids *S*. His starting point is
the partially ordered set of principal right ideals of *S
*which I shall

denote by* P = P(S) *where the order is subset
inclusion. Let *aS* be any principal right ideal and
consider the order ideal of* P *generated by* aS;
*that is, the poset of all principal right ideals
contained in

isomorphisms from

To understand the significance of
this paper, it is worth bearing in mind that the most
interesting question relating to cancellative monoids before
then was whether they could be embedded in groups or not.
This paper is treating cancellative monoids in their own
right rather than as things to be embedded in groups.

The key concept of the paper is that
of uniformity and is a very early use of the notion of
self-similarity in algebra; it may well be the first. Today,
self-similarity is everywhere thanks to fractals but then it
must have been much less well-known. The paper doesn't
reveal its motivations but I would suspect that Rees might
first have noticed this property by studying free monoids
since these are all cancellative monoids. The case of the
free monoid on the two generators *a* and *b* is
illustrative. Draw the tree whose vertices are the strings
over the alphabet {*a,b*} with the empty string at the
top, and where from a vertex labelled by the string *w*
we attach two further vertices: namely *wa *and* wb*.
We obtain in this way the complete binary tree. The
self-similarity property that is the basis of Rees's
definition arises from the observation that the tree below
any vertex is isomorphic to the whole tree. The key step
would have been to observe that what is true of the free
monoid is in fact true of any left cancellative monoid.Some
evidence for this is provided by the last theorem in the
paper. He shows there how to construct all left cancellative
monoids where the poset of principal right ideals is order
isomorphic with the poset of natural numbers with their dual
ordering; all one needs is a group and an endomorphism of
that group. The natural numbers are, of course, the free
monoids on one generator.

The posterity of this paper was
significant. It played a decisive role in the development of
the theory of bisimple inverse monoids and their
generalizations. Less well-known is the connection between
this paper and the theory of self-similar group actions. In
the 1970's, J-F Perrot in his thesis set about generalizing
the final theorem in Rees's paper by replacing the free
monoid on one generator by an arbitrary free monoid. In the
process of analysing the structure of the resulting monoids,
some very odd group actions arose. It was only several
decades later that it became apparent that Perrot had
stumbled across the definition of what are now known as
self-similar group actions. These are the group actions on
free monoids that intertwine in a precise way with the
self-similar structure of the free monoids or, to use Rees's
terminology, with their uniform structure.

I don't think it too much of a
stretch to view Rees's fourth paper as being far more
significant than it first appears and certainly as it would
have appeared at the time. By using the notion of
self-similarity in algebra Rees had set the scene for the
theory of self-similar group actions.