... Higman was on leave for part of the time and John was supervised instead by Gordon Preston, who had recently returned to live in Oxford after a two-year stint with A H Clifford in New Orleans. In view of this, it was not surprising that John found himself steered into semigroup theory and he wrote his D.Phil. thesis on amalgamation theory - a new topic then and one that has remained of central interest to him throughout his career.

I met Gordon Preston first in 1959, when I arrived in Oxford as a graduate student working under the supervision of Graham Higman. Gordon lived in Oxford - Shrivenham, his place of work, is just 30km away - and he was a regular attender at the Higman algebra seminar. For the final year of my study at Oxford, when Higman was on sabbatical leave in Chicago, Preston was my supervisor, and it is a pleasure now to be able to pay public tribute to the quality of the help and encouragement I received from him during that period.

... when I moved to the newly-created Stirling University in 1967, [John Howie] accompanied me as a Senior Lecturer. [For] three years, ... he made an enormous contribution to many aspects of the life of this rapidly expanding institution ...

John's research has been entirely in the algebraic theory of semigroups. It is not possible to give a detailed critique of this here but, as can be seen from his list of publications, two main areas stand out: amalgamations and transformation semigroups. In one of his early papers, he showed that certain regular semigroups of transformations could be generated by their idempotents; and the notion of idempotent-generation was then taken up eagerly by others. Much of his later work on transformations has had a combinatorial and indeed number-theoretic flavour. Many of his ideas were adapted and developed by his research students, with whom he often published joint work. Amongst his early papers, we might single out two interesting examples which do not fall under the main headings above: his work on idempotent-separating congruences on inverse semigroups and that on epimorphisms and dominions (with John Isbell). Other work outside the aforementioned categories concerns semigroup presentations.

The author's aim is to provide an updated introduction to purely algebraic semigroups, i.e., semigroups with no further structure. The book more than succeeds in this aim. First, it is a top-rate book for its main intended audience, students with some knowledge of algebra and mathematical sophistication, who are approaching semigroups for the first time. The material is made absolutely clear and the style is interesting and informative. Of particular interest and importance are the many discussions of how sections of the theory are related to one another. In places, an air of excitement is generated as the author takes the reader through the possible thought processes of the researcher trying to produce a result, a mere five years earlier. Second, a study of the entire book will provide for the potential researcher a strong background in material not previously available in book form.

The year 1976 saw the publication of his monograph: 'An introduction to semigroup theory'. It would be hard to over-estimate the impact that this made on the subject. For many years previously the main repository of knowledge in this field had been the pioneering English-language text by Clifford and Preston: 'The algebraic theory of semigroups' (vol. 1, 1961; vol. 2, 1967). It had attempted to provide a comprehensive account of the subject and, although immaculately written, might have been somewhat daunting to a newcomer! John's book was less ambitious in its coverage: moreover, it included recent topics and had a clearly-defined theme (regular semigroups). Coupled with the fact that it was written in his typically appealing style, it was a winner from the start. Certainly it must have attracted many students worldwide to the subject.

The original book described itself as pitched at the undergraduate/graduate level, while the present work is called a graduate introduction. This perhaps says more about the lower standards expected of undergraduates than about a true change in the nature of the book, but it has freed the author, allowing him to be a little more thorough and uncompromising. ... the book remains the most suitable on which to base a modern introductory course in algebraic semigroups.

The most striking feature of 'Real Analysis' is not so much its content (which is absolutely standard fare) but the author's Ferrar-like concern for the reader's understanding which shines through on every page of this carefully written and carefully paced text.

This is a view of semigroup theory, past, present and future, by eminent semigroup theorist J M Howie. His thoughts and comments are captured in a succinct and informative paper which looks at the history of semigroup theory and its development over the years as he notes the most influential results, papers and books. He also touches on subjects such as what is meant by good mathematics and, indeed, good semigroup theory, and quality in mathematics. He believes that semigroup theory has the potential to be an important and indispensable part of mathematics, but that it will not happen unless links with other parts of mathematics are actively pursued. He comments that, in the future, he would like to see semigroup theory develop in ways that establish links with other areas of learning, both within and beyond pure mathematics.

... for services to education.