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Prof David Rees
FRS

(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
1. On semi-groups, Proc. Cambridge Philos. Soc. 36 (1940), 387--400.
2. Note on semi-groups, Cambridge Philos. Soc. 37 (1941), 434--435.
3. On the group of a set of partial transformations, J. London Math. Soc. 22 (1947), 281--284.
4. On the ideal structure of a semi-group satisfying a cancellation law, Quart. J. Math. Oxford Ser. 19 (1948), 101--108.
5. On semigroups in which x^r = x, (with J. A. Green), Proc. Cambridge 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 aS. We denote this poset by P_{a}. The question Rees asks is what the relation is between the posets P and P_{a}. Observe that  xS \subseteq yS if and only if axS \subseteq ayS. It follows that P and P_{a} are order-isomorphic. He calls any poset satisfying this condition uniform. Today, we would be more inclined to say that these posets are self-similar. Having introduced this notion he proves his main theorem: every uniform poset arises in this way. The proof is very easy: with each uniform poset, we associate the monoid S(P) consisting all order
isomorphisms from P to the principal ideals of P. This turns out to be a left cancellative monoid such that P(S(P)) is isomorphic with P. As a follow up, Rees shows that the monoids S(P) enjoy a certain universal property. To achieve this, he develops a theory of a special class of homomorphisms which are determined by generalizations of normal subgroups (what he call right normal divisors).

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.