# Joachim Lambek

**Joachim "Jim" Lambek** FRSC (5 December 1922 – 23 June 2014)[2] was Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his Ph.D. degree in 1950 with Hans Zassenhaus as advisor.

Joachim Lambek | |
---|---|

Joachim Lambek in Philadelphia, May 2008 | |

Born | |

Died | June 23, 2014 91) | (aged

Citizenship | Canadian |

Alma mater | McGill University |

Known for | Lambek–Moser theorem, Lambek calculus, Curry–Howard–Lambek correspondence, multicategories |

Awards | Jeffery–Williams Prize (1988)[1] |

Scientific career | |

Fields | Mathematics |

Institutions | Department of Mathematics and Statistics McGill University |

Thesis | A: Biquaternion Vectorfields over Minkowski's Space B: The Immersibility of a Semigroup into a Group (1950) |

Doctoral advisor | Hans Zassenhaus |

Doctoral students | Israel Kleiner (1967) William Schelter (1972) |

## Biography

Lambek was born in Leipzig, Germany, where he attended a Gymnasium.[3] He came to England in 1938 as a refugee on the *Kindertransport*.[2] From there he was interned as an enemy alien and deported to a prison work camp in New Brunswick, Canada. There, he began in his spare time a mathematical apprenticeship with Fritz Rothberger, also
interned, and wrote the McGill Junior Matriculation in fall of 1941.[3] In the spring of 1942, he was released and settled in Montreal, where he entered studies at McGill University, graduating with an honours mathematics degree in 1945 and an M.Sc. a year later.[4] In 1950, he completed his doctorate under Hans Zassenhaus becoming McGill's first Ph.D. in mathematics.

Lambek became assistant professor at McGill; he was made a full professor in 1963. He spent his sabbatical year 1965–66 in at the Institute for Mathematical Research at ETH Zurich, where Beno Eckmann had gathered together a group of researchers interested in algebraic topology and category theory, including Bill Lawvere. There Lambek reoriented his research into category theory.[5]

Lambek retired in 1992 but continued his involvement at McGill's mathematics department. In 2000 a festschrift celebrating Lambek's contributions to mathematical structures in computer science was published.[6] On the occasion of Lambek's 90th birthday, a collection *Categories and Types in Logic, Language, and Physics* was produced in tribute to him.[7]

## Scholarly work

Lambek's PhD thesis investigated vector fields using the biquaternion algebra over Minkowski space, as well as semigroup immersion in a group. The second component was published by the Canadian Journal of Mathematics.[8] He later returned to biquaternions when in 1995 he contributed "If Hamilton had prevailed: Quaternions in Physics", which exhibited the Riemann–Silberstein bivector to express the free-space electromagnetic equations.

Lambek supervised 17 doctoral students, and has 73 doctoral descendants as of 2019. He has over 100 publications listed in the Mathematical Reviews, including 6 books. His earlier work was mostly in module theory, especially torsion theories, non-commutative localization, and injective modules. One of his earliest papers, Lambek & Moser (1954), proved the Lambek–Moser theorem about integer sequences. In 1963 he published an important result, now known as Lambek's theorem, on character modules characterizing flatness of a module.[9] His more recent work is in pregroups and formal languages; his earliest works in this field were probably Lambek (1958) and Lambek (1979). He is noted, among other things, for the Lambek calculus, an effort to capture mathematical aspects of natural language syntax in logical form, and a work that has been very influential in computational linguistics, as well as for developing the connections between typed lambda calculus and cartesian closed categories (see Curry–Howard–Lambek correspondence). His last works were on pregroup grammar.

## Selected works

### Books

- Fine, N. J.; Gillman, Leonard; Lambek, Joachim (1966).
*Rings of quotients of rings of functions*. McGill University Press. MR 0200747. - Lambek, Joachim (2009) [1966].
*Lectures on rings and modules*(3rd ed.). Blaisdell Publishing. ISBN 9780821849002. MR 0206032. - Lambek, Joachim (1966).
*Completions of categories*. Seminar lectures given in 1966 in Zürich. Lecture Notes in Mathematics, No. 24. Berlin, New York: Springer-Verlag. MR 0209330. - Lambek, Joachim (1971).
*Torsion theories, additive semantics, and rings of quotients*. Lecture Notes in Mathematics.**177**. Berlin, New York: Springer-Verlag. MR 0284459. - Lambek, J.; Scott, P. J. (1986).
*Introduction to Higher Order Categorical Logic*. Cambridge University Press. ISBN 978-0-521-35653-4. MR 0856915. - Anglin, W. S.; Lambek, Joachim (1995).
*The heritage of Thales*. Undergraduate Texts in Mathematics. Berlin, New York: Springer-Verlag. ISBN 978-0-387-94544-6. MR 1369087. - Casadio, Claudia; Lambek, Joachim (2008).
*Computational Algebraic Approaches to Natural Language*. Polimetrica. ISBN 978-88-7699-125-7. - Lambek, J. (2008).
*From word to sentence: a computational algebraic approach to grammar*. Polimetrica. ISBN 978-88-7699-117-2.

### Articles

- Lambek, J. (1951), "The immersibility of a semigroup into a group",
*Canadian Journal of Mathematics*,**3**: 34–43, doi:10.4153/CJM-1951-005-8 - Lambek, Joachim; Moser, L. (1954), "Inverse and Complementary Sequences of Natural Numbers",
*The American Mathematical Monthly*,**61**(7): 454–458, doi:10.2307/2308078, ISSN 0002-9890, JSTOR 2308078, MR 0062777 - Lambek, J. (1958), "The Mathematics of Sentence Structure" (PDF),
*The American Mathematical Monthly*,**65**(3): 154–170, doi:10.2307/2310058, ISSN 0002-9890, JSTOR 1480361 - Lambek, Joachim (1961), "How to program an infinite abacus",
*Canadian Mathematical Bulletin*,**4**(3): 295–302, doi:10.4153/CMB-1961-032-6 - Lambek, Joachim (1969). "Deductive systems and categories II. Standard constructions and closed categories".
*Lecture Notes in Mathematics*.**86**. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 76–122. doi:10.1007/bfb0079385. ISBN 978-3-540-04605-9. ISSN 0075-8434. - Lambek, Joachim (1972), "Bicommutators of nice injectives",
*Journal of Algebra*,**21**: 60–73, doi:10.1016/0021-8693(72)90034-8, ISSN 0021-8693, MR 0301052 - Lambek, Joachim (1972), "Localization and completion",
*Journal of Pure and Applied Algebra*,**2**(4): 343–370, doi:10.1016/0022-4049(72)90011-4, ISSN 0022-4049, MR 0320047 - Lambek, Joachim (1979), "A mathematician looks at Latin conjugation",
*Theoretical Linguistics*,**6**(2): 221–234, doi:10.1515/thli.1979.6.1-3.221, ISSN 0301-4428, MR 0589163 - J. Lambek (1995) "If Hamilton had prevailed: Quaternions and Physics", Mathematical Intelligencer 17(4): 7–15, reprinted in
*Mathematical Conversations*(2001) Robin Wilson & Jeremy Gray editors, Springer books, ISBN 978-1-4613-0195-0 - J Lambek & Michael Barr (2013) In Praise of Quaternions from McGill University

## References

- "The recipients of the Jeffery-Williams Prize".
*Canadian Mathematical Society*. Retrieved 1 November 2018. - "Joachim Lambek".
*Montreal Gazette*. Legacy.com. 27 June 2014. Retrieved 31 October 2018. - Darmon, Henri (2 December 2014).
*Resolution on the Death of Emeritus Professor Joachim (Jim) Lambek, Department of Mathematics and Statistics*(PDF). Faculty of Science: Meeting of Faculty. Montreal: McGill University. p. 2. - Müller-Hoissen, Folkert; Pallo, Jean Marcel; Stasheff, Jim, eds. (2012).
*Associahedra, Tamari Lattices and Related Structures*. Progress in Mathematics.**299**. Birkhäuser. p. 18. doi:10.1007/978-3-0348-0405-9. ISBN 978-3-0348-0404-2. LCCN 2012942603. - Barr, Michael (5 December 1997).
*An appreciation of Jim Lambek at McGill*. LambekFest. Montreal. - Barr, Michael; Scott, P. J.; Seely, R. A. G., eds. (2000),
*The Lambek Festschrift: mathematical structures in computer science*, Cambridge University Press, MR 1770227 - Casadio, Claudia; Coeke, Bob; Moortgat, Michael; Scott, Philip, eds. (2014),
*Categories and Types in Logic, Language, and Physics: Essays Dedicated to Jim Lambek on the Occasion of His 90th Birthday*, Springer-Verlag - Lambek, J. (1951). "The immersibility of a semigroup into a group".
*Canadian Journal of Mathematics*.**3**: 34–43. doi:10.4153/CJM-1951-005-8. - Lambek, J. (1964). "A Module is Flat if and Only if its Character Module is Injective".
*Canadian Mathematical Bulletin*.**7**(2): 237–243. doi:10.4153/CMB-1964-021-9. ISSN 0008-4395.

## External links

- Joachim Lambek at the Mathematics Genealogy Project
- Faculty profile of Joachim Lambek at McGill University
- Lambek festival (80th anniversary)