GROUPOIDS

At the end of the previous chapter we had a brief indication that it might be useful to consider objects similar to groups, but where the product of two elements is not always defined. In this section we consider these objects in detail.

A partial multiplication on a set G is a function from some subset X of G × G to G. If (x, y)∈X we denote the value of the function on (x, y) by xy or by x.y. We say that xy is defined to mean that (x, y)∈X.

The element e is an identity for a partial multiplication if ex - x whenever ex is defined and also ye - y whenever ye is defined. There may be many identities, but it is clear from the definition that if e and f are identities with ef defined then e - f.

Definition A groupoid is a set G with a partial multiplication such that:

1. (associative law) if one of (ab)c and a(bc) is defined then so is the other and they are equal; also, if both ab and bc are defined then (ab)c is defined,

2. (existence of identities) for any a, there are identities e and f with ea and af defined,

3. (existence of inverses) for any a, and e and f as in (2), there is an element a−1 such that aa−1 - eand a−1a - f.

We nearly always want our groupoids to be non-empty. I leave it to the reader to decide which properties stated should have the empty groupoid given as an exception.

DECISION PROBLEMS IN GENERAL

Can we tell whether or not an element of a free group is in the commutator subgroup? Whether or not it is a commutator? Given a finite presentation, can we tell whether or not an element of the free group is 1 in the group presented? Can we tell whether or not two finite presentations present isomorphic groups?

These questions, and other similar ones, are obviously of interest. We shall see later that the first two questions have the answer “Yes”, while the other two have the answer “No”. The techniques used are those of combinatorial group theory, but have no specific connection with the topological approach. I include this chapter because I find the material particularly interesting.

In order to make these questions precise, we need to be clearer about what is meant by “We can tell … ”. This should mean that we can tell by some purely mechanical process, not requiring thought. In other words, we want to be able to feed the data to a computer and have the computer arrive at the answer as to whether or not the required property holds for the given data.

Readers might think that a computer's ability to arrive at the answer for given data would depend very much on the computer. This turns out not to be so, though the speed and efficiency of the computer's answer will depend on the computer.

More precisely, there is a class of functions (called partial recursive functions) with the following properties.

I have recently been working on a book about generating functions. It will be called ‘Generatingfunctionology,’ and it is intended to be an upper-level undergraduate, or graduate text in the subject. The object is to try to impress students with the beauty of this subject too, so they won't think that only bijections can be lovely.

In one section of the book I will discuss combinatorial identities, and the approach will be this. First I'll give the ‘Snake Oil’ method, in the spirit of a unified approach that works on many relatively simple identities. It involves generating functions. Second, I will write about a much more powerful method that works on ‘nearly all’ identities, including all classical hypergeometric identities and many, many binomial coefficient identities.

These two approaches will be mirrored here, in that this article will mostly be about the Snake Oil method, whereas the talk that I will give at the conference will be about the much more powerful method of WZ pairs [WZ], which at this writing is still under development. A brief summary of the WZ results appears in section (II) below.

Aside from these developments, there have been other unifying forces at work in the field of identities. The expository paper of Roy [Ro], shows how even without a computer one can recognize many binomial identities as cases of just a very few identities in the theory of hypergeometric series. The work of Knuth [Kn] shows how a few rules about binomial coefficients and their handling can, in skilled hands, prove many difficult identities.

INTRODUCTION

Several years ago I was asked a seemingly innocuous question: What are the minimal-weight vectors of the code of an affine plane? I thought the answer would be that they were, just as in the projective case, simply the scalar multiples of the lines; indeed, that may be true and the question is still open. I managed to prove this (for arbitrary affine planes) only for those of prime order.

The question is deeper than it at first seems. If, for example, one could prove that the minimal-weight vectors of the code of an arbitrary affine plane were simply the scalar multiples of the lines, one would have a proof of the fact [15] that a projective plane of order ten has no ovals; indeed, one would prove that no projective plane of order congruent to two modulo four, except the Fano plane, could have an oval. (The minimal-weight vectors of the code of a desarguesian affine plane are the scalar multiples of the lines of the plane but the only known proof relies heavily on algebraic coding theory.)

These considerations led J. D. Key (who asked the original question) and me to what seems to be a fruitful approach to affine and projective planes and to what we hope will be a fruitful approach to the theory of designs. The purpose of this paper is to explain these matters. Much of the work we have done has already appeared and thus the present paper will rely heavily on four joint papers: Arcs and ovals in hermitian and Ree unitals, Affine and projective planes, Baer subplanes, ovals and unitals, and Translation planes and derivation sets.

Since its beginning in 1969 the British Combinatorial Conference has grown into an established international meeting. This year the twelfth conference is being held in Norwich under the auspices of the School of Mathematics at the University of East Anglia. Participants come from a great number of countries worldwide and represent a multitude of interests in combinatorial theory.

This volume contains the contributions of the principal speakers. They were invited to prepare a survey paper for this book and to deliver a lecture in an area of their expertise. In this way it is hoped to make available a valuable source of reference to the current state of art in combinatorics. The speakers have produced their papers well in advance so that they are now all available in time for the conference.

This book has been produced to a tight schedule. I am grateful to the authors for their cooperation and to the referees for their assistance and comments about the papers. The British Combinatorial Conference is largely selffinancing but on behalf of the committee I would like to thank the London Mathematical Society, Norwich Union and Peat Marwick McLintock for their financial support.

Structured sets

Every statistical design consists of two sets and a function between them. One set, T, consists of the treatments: they are under the experimenter's control, and the purpose of the experiment is to find out about them. The elements of the second set, Ω, are called plots for historical reasons. A plot is the smallest experimental unit to which an individual treatment is applied: it may or may not be a plot of land. In general the experimenter has less direct control over the attributes of the plots, and is not interested in finding out about the plots per se. Unless otherwise stated, Ω and T are always finite. Finally there is the design map ϕ from Ω to T which allocates treatments to plots; if treatment t is allocated to plot ω then ωϕ = t.

Typically one or both of the sets T and Ω is structured. I cannot give a formal definition of this concept, but examples of structures on a set include: a set of partitions; a transitive permutation group; a set of subsets; a graph; an association scheme. Many, but not all, of these structures can be specified as a set of (binary) relations with certain properties. A set will be said to be unstructured if the relations in its structure are just the equality relation E and the uncaring relation U (the direct square of the whole set) or U \ E.