John Mackintosh Howie

1936 - 2011

John Mackintosh Howie (1936–2011): The Architect of Modern Semigroup Theory

John Mackintosh Howie was a preeminent Scottish mathematician whose work defined the modern landscape of algebraic semigroup theory. As the Regius Professor of Mathematics at the University of St Andrews for over three decades, Howie was not only a researcher of international standing but also a transformative figure in mathematics education and university administration.

1. Biography: From Chryston to the Regius Chair

John Mackintosh Howie was born on August 30, 1936, in Chryston, North Lanarkshire, Scotland. The son of a Church of Scotland minister, Howie’s upbringing emphasized intellectual rigor and public service.

Education:

He attended the University of Aberdeen, graduating in 1958 with a first-class honors degree in Mathematics. He then moved to Balliol College, Oxford, to pursue a DPhil. At Oxford, he studied under the legendary group theorist Graham Higman. His doctoral thesis, completed in 1962, focused on the theory of semigroups—a relatively young field at the time compared to the more established study of groups.

Academic Career:

1961–1967: Lecturer at the University of Glasgow.
1967–1970: Senior Lecturer at the University of Stirling.
1970–1997: Appointed Regius Professor of Mathematics at the University of St Andrews at the remarkably young age of 33. This chair, established by Charles II in 1668, is one of the most prestigious mathematical posts in the United Kingdom.
Post-1997: Following his retirement from the chair, he remained active as Professor Emeritus until his death in 2011.

2. Major Contributions: Semigroups and Transformations

Howie’s primary intellectual contribution was in Algebraic Semigroup Theory. While a "group" in mathematics requires every element to have an inverse (like addition with positive and negative integers), a "semigroup" only requires the operation to be associative (like the set of all positive integers under addition).

Idempotent-Generated Semigroups:

Howie’s most famous discovery came in 1966. He proved a fundamental theorem regarding the full transformation semigroup (T_n), which consists of all mappings from a finite set to itself. He demonstrated that any transformation that is not a permutation can be expressed as a product of idempotents (elements e such that e² = e). This was a groundbreaking result that linked the structure of transformations to the simpler building blocks of idempotents, sparking a vast subfield of research.

Extension Theory and Embeddings:

He made significant strides in understanding how semigroups could be embedded into other structures. He provided the necessary and sufficient conditions for a semigroup to be embedded into a "completely simple semigroup," a result that helped classify the hierarchy of algebraic structures.

3. Notable Publications

Howie was a gifted writer known for his "Scottish clarity"—a style that was precise, economical, and highly readable.

"The subsemigroup generated by the idempotents of a full transformation semigroup" (1966): His seminal paper published in the Journal of the London Mathematical Society.
An Introduction to Semigroup Theory (1976): This became the "bible" for researchers in the field. It was lauded for its pedagogical excellence and remained the standard text for twenty years.
Fundamentals of Semigroup Theory (1995): A comprehensive expansion of his 1976 work, incorporating decades of new research. It remains the definitive reference for graduate students today.
Real Analysis (2001) and Complex Analysis (2003): Written for the Springer Undergraduate Mathematics Series (SUMS), these textbooks are celebrated for making difficult concepts accessible to undergraduates worldwide.

4. Awards & Recognition

Howie’s contributions were recognized both for their mathematical depth and their impact on society.

The Keith Medal (1979–1981): Awarded by the Royal Society of Edinburgh for his outstanding contributions to mathematics.
CBE (Commander of the Order of the British Empire) (1993): Awarded for his services to education.
President of the London Mathematical Society (1988–1990): One of the few Scottish-based mathematicians to hold this prestigious UK-wide office.
Fellow of the Royal Society of Edinburgh (Elected 1971): He served as the Society's Vice-President from 1992 to 1995.

5. Impact & Legacy: The "Howie Report" and St Andrews

Howie’s legacy is twofold: he shaped the mathematical landscape and the Scottish educational landscape.

The Howie Report:

In the early 1990s, Howie chaired a committee tasked with reviewing the upper secondary education system in Scotland. The resulting "Howie Report" (1992) criticized the "Higher" system for being too narrow and rushed. While his most radical suggestions (like a two-tier baccalaureate) were not fully adopted, the report led to the "Higher Still" reforms, which significantly broadened the curriculum and introduced more flexible pathways for Scottish students.

The St Andrews School of Algebra:

Under his leadership, the University of St Andrews became a world-class center for algebra. He was instrumental in founding the St Andrews Groups/Algebra Conferences, which continue to attract hundreds of mathematicians from around the globe every four years.

6. Collaborations and Mentorship

Howie was a pillar of the international mathematical community.

Research Partnerships: He collaborated extensively with international scholars, particularly in Portugal (a country that became a second home for his research) and Australia.
Students: He supervised numerous doctoral students who went on to become leaders in the field, including Nikola Ruškuc and James Mitchell, both of whom became professors at St Andrews, ensuring the continuity of his research lineage.
The SUMS Series: As an editor for the Springer Undergraduate Mathematics Series, he worked with dozens of authors to standardize and improve the quality of mathematics teaching materials globally.

7. Lesser-Known Facts

Linguistic Talent: Howie was a gifted linguist. He was fluent in French and German and often gave lectures in the native languages of his host universities in Europe.
Musical Interest: He was an accomplished pianist and had a deep love for classical music, which he often related to the structural beauty of mathematics.
A "Mathematical Knight": Although the Regius Chair is a royal appointment, Howie was known for his humility. He often walked to the university from his home, engaging with students and junior faculty as peers rather than subordinates.
The St Andrews Archive: He was a major supporter of the MacTutor History of Mathematics Archive, created by his colleagues at St Andrews, which is now the world’s most-visited resource for the history of math.

John Mackintosh Howie passed away on December 26, 2011. He is remembered as a man who combined the rigor of a world-class mathematician with the civic-mindedness of a great educator, leaving behind a "semigroup" of students and theories that continue to evolve.