Academic Staff

Professor Peter M Higgins

Position in department	First Year Director (Spring Term), Exams Officer, 1st Year (Spring Term)
Staff position	Professor
E-mail	peteh (non Essex users should add @essex.ac.uk)
Telephone	3019 (non Essex users should add 01206-87 to the beginning of this number)
Room	6.317
Biography	I was born in Australia in 1956 and together with my English wife, Theresa, our son and three daughters we have lived in Colchester since 1990. I was promoted to Professor in 2000 and was Head of the Dept of Mathematical Sciences from January 2005 until December 2010. My research is in algebraic semigroups which is the algebraic type that arises, quite literally, from the stringing together of words and the composition of functions. The former gives rise to connections with the theory of computing and automata while the latter leads to abstract algebra and sometimes to combinatorial problems, that is to say questions concerning counting and enumeration. Like most mathematicians, I try to maintain a wider interest in things mathematical. I have published over 50 research papers and seven books, the first of which was a specialist research monograph, Techniques of Semigroup Theory. I have four popular science books with Oxford University Press, 'Matheamtics for the Curious', Mathematics for the Imagination', 'Nets, Puzzles and Postmen, an exploration of mathematical connections', (in hardback and paperback) and 'Numbers, A Very Short Introduction', which have been translated into Italian, Spanish, Ukrainian, Chinese, Japanese and Arabic. With Copernicus Press I have the hardback, 'Number Story, from Counting to Cryptography' now translated into Korean. In 2005 I invented 'Circular Sudoku', which now appears in magazines, newspapers and books around the world and also as a handheld computer game and an iPod Ap. The 'Official Book of Circular Sudoku' was published in 2006 by Plume Press in the States and now has translations in Japanese and Russian.
Qualifications	B.A. (ANU. Canberra, 1976), BSc (First Class, University of Tasmania, 1980), PhD, (Monash University, Melbourne, 1983).
Research interests	Algebraic Semigroups (primary). Formal Language Theory, Combinatorics and Graph Theory (secondary). Linked with the School of Computer Science and Electronic Engineering. Member of the Computational Combinatorics Group. Member of the Crypto Group. An editor of the international journal Communications in Algebra. You may also be interested in Circular Sudoku, a new adaptation of the popular Sudoku puzzles.
Teaching responsibilities	In the calendar years 2011-2012 I will be on Research Leave.
Publications	Publications in Refereed Journals 'Epis are onto for generalised inverse semigroups', Semigroup Forum, Vol. 23, 255-259, 1981. 'The determination of all absolutely closed varieties of semigroups', Proc. Amer. Math. Soc., Vol. 87, 419-421, 1983. 'The varieties of commutative semigroups for which epis are onto', Proc. Roy. Edinburgh Math. Soc., Vol. 94A, 1-7, 1983. 'A semigroup with an epimorphically embedded subband', Bull. Austral. Math. Soc., Vol. 27, No. 2, 231-242, 1983. 'Saturated and epimorphically closed varieties of semigroups', J. Austral. Math. Soc., Vol. 36, 153-175, 1984. 'Epimorphisms, permutation identities and finite semigroups', Semigroup Forum, Vol. 29, No. 1-2, 87-97, 1984. 'Epimorphisms and semigroup varieties', (Ph. D. summary), Bull. Austral. Math. Soc., Vol. 29, No. 3, 415-417, 1984. 'A new determination of the permutation identities which ensure that a semigroup variety is finitely based', J. Pure and Appl. Algebra, 38, 65-69, 1985. 'Epimorphisms, dominions and semigroups', Algebra Universalis, 21, 225-233, 1985. 'A method of constructing square roots in finite full transformation semigroups', Canad. Math. Bull., Vol. 29 (3), 344-351, 1986. 'Completely semisimple semigroups and epimorphisms', Proc. Amer. Math. Soc., Vol. 96, No. 4, 387-390, 1986. 'Dense subsets of some common classes of semigroup', Semigroup Forum, Vol. 34, 5-19, 1986. 'The naming of popes and a Fibonacci sequence in two non-commuting indeterminates', Fibonacci Quart., Vol. 25, 57-61, February 1987. 'Semigroup classes defined by sequence conditions', Indian J. of Maths. (Ramanujan Centenary Edition), Vol. 29, No. 1, 1-8, 1987. 'Digraphs and the semigroup of all mappings on a finite set', Glasgow Math. J., 30, 41-57, 1988. 'Finite full transformations semigroups as collections of random functions'; (joint with B.M. Brown), Glasgow Math. J., 30, 203-211, 1988. 'The range order of a product of i transformations from a finite full transformation semigroup', Semigroup Forum, Vol. 37, 31-36, 1988. 'Epimorphisms and amalgams', Colloquium Mathematicum, Vol. LVI, No. 1, 1-17, 1988. 'Random functions on a finite set', (joint with E. Williams), Ars Combinatoria, Vol. 26A, 93-102, 1988. 'Embedding in bisimple semigroups', Semigroup Forum, Vol. 40, No. 1, 105-107, 1990. 'An algebraic proof that pseudovarieties are defined by pseudoidentities', Algebra Universalis, 27, 597-99, 1990. 'A short proof of Isbell's zigzag theorem', Pac. J. Math., 144, No. 1, 47-50, 1990. 'Combinatorial results for semigroups of order-preserving mappings', Math. Proc. Camb. Phil. Soc, 113, 281-296, 1993. 'A class of eventually regular semigroups determined by pseudo-random sets', J. London Math. Soc. (2) 48 87-102, 1993. 'The natural partial order of Mitsch', Semigroup Forum, Vol 49, 261-266, 1994. 'Idempotent depth in semigroups of order-preserving mappings', Proc. Roy. Soc. Edin. Vol 124A, 1045-1058, 1994. 'Divisors of the semigroup of order-preserving mappings on a finite chain', International Journal of Algebra and Computation,Vol 5, No. 6 (1995), 725-742. 'Embedding semigroups in semibands of depth two', Semigroup Forum, Vol 51 (1995), 395-96. 'Epis from locally inverse semigroups are onto', Semigroup Forum, Vol 52, (1996) 49-53. 'Ramsey's Theorem in algebraic semigroups', Springer Lecture Note Series, Proc. Tainan - Moscow Algebra Workshop (Walter de Grugter, publishers), (1996) pp 1-36. 'A new proof of Simon's Theorem on piecewise testable languages', Journal of Theoretical Computer Science, 178 257-64, (1997). 'Monoids respecting n-chains of intervals' (joint with J. Almeida) Journal of Algebra, 187, 183-202, (1997). 'Generators and factorizations of transformation semigroups', (joint with J.M. Howie and N. Ruskuc), Proc. Roy. Edin. Math. Soc. 128A , 1355-1369 (1998). 'On relative ranks of full transformation semigroups', (joint with J. Howie and N. Ruskuc), Communications in Algebra, (26)(3), 733-48, (1998). 'The monoid of all orientation-preserving mappings of a cycle', (joint with P. Catarino). Semigroup Forum, 58, (1999) 190-206. "Finite aperiodic semigroup with commuting idempotents and generalizations", (joint with S.W. Margolis). Israel J. of Mathematics, 116, (2000) 367-380. "A new proof of Schutzberger's Theorem", International J. of Alg. & Comp., 10, No. 2 (2000), 217-220. "Remarks concerning Lallements's Lemma and coextentions of eventually regular semigroups", (joint with S.J. Kopamu and P.M. Edwards), Acta Scientarum Mathematicarum, (Szeged) 67 (2001), 385-400. "The gap between partial and full: an addendum", (joint with J. Almeida and M.V. Volkov), International J. of Alg. & Comp., 11, No. 1 (2001) 131-135.. "Unavoidable Sets of Words of Uniform Length", (joint with C.J. Saker), Computation and Information, 173, 1-5 (2002). "The pseudovariety generated by the semigroup of all orientation-preserving transformations on a chain", (joint with P. Catarino), International J. of Alg. & Comp., 12, No. 3 (2002), 387-405. "Set products in transformation semigroups" (joint with J.M. Howie and N. Ruskuc) Proc. Roy. Soc. of Edinburgh, 133A, (2003), 1121-1135. "Generating the full transformation semigroup using order preserving mappings" (joint with J.D. Mitchell and N. Ruskuc) Glasgow Mathematical Journal, 45, (2003), 557-566. “Countable versus uncountable ranks in infinite semigroups of transformations and relations”, joint with J.M. Howie, J.D. Mitchell and N. Ruskuc, Proc. Edin. Math. Soc. (2003), 46, 531-544. “Rank properties of endomorphisms of infinite partially ordered sets”, joint with JD Mitchell, N. Ruskuc and M Morayne Bull. Bulletin of London Math. Soc., 38 (2006), no.2, 177-191. “The semigroup of conjugates of a word”, International Journal of Algebra and Computation, Vol. 16 No. 6 (2006) 1015-1029. Unavoidable Sets, Theoretical Computer Science, joint with CJ Saker, Vol. 359, No. 1, (August 2006), 231-238. Generalized Green's Equivalences on the subsemigroups of the bicyclic monoid, joint with L. Descalco, Communications in Algebra, 38, 4597-4612, (2010). The map numbering problem, IEEE Transactions, Proceedings CEE'10, 2nd Computer Science and Electronic Engineering Conference, 2010, UK, to appear. The length of short words in unavoidable sets, International Journal of Algebra and Computation, to appear. Books `Techniques of Semigroup Theory', Oxford University Press, Oxford, 1992, (x + 258pp). 'Mathematics for the Curious', Oxford University Press, Oxford & New York 1997 'Mathematics for the Imagination'. Oxford University Press, Oxford & New York, 2002 'The Official Book of 'Cricular Sudoku', Plume Press, 2006. `Nets, Puzzles and postmen, an exploration of mathematical connections', Oxford University Press 2007. ''Number Story, from counting to cryptography', Springer-Verlag, 2008. `Numbers, A Very Short Introduction', Oxford University Press, 2011.
Conferences/presentations	Papers in Proceedings of International Conferences 'Semigroup epimorphisms', in P.R. Jones et al. (eds.), Proceedings of the Marquette Conference on Semigroups, 51-64, September 1984. 'Random products in semigroups of mappings', in J. Almeida et al. (eds.), 89-100, Lattices, Semigroups and Universa Algebra, Plenum Press, New York, 1990. 'On eventually regular semigroups', in J.M. Howie et al. (eds.), 170-189, Semigroups with Applications, (Oberwolfach Conference) World Scientific, Singapore, 1992. 'Combinatorial Aspects of Semigroups of Order-preserving and Decreasing Functions', 103-110, Proceedings of the Milano International Conference on Semigroups, June 1992, World Scientific Publishers, 1993. 'Pseudovarieties generated by classes of transformation semigroups', Proc. St. Petersburgh Semigroup Conference Russian State Hydrometeorological Inst., 85-94 (1999). 'The Converse of Lallement's Lemma', Proc. St. Andrews Semigroups Conference, World Scientific, 78-86, (1998). 'On Special amalgamation Bases', (joint with Kunitaka Shoji, Shimane University, Japan). Proc. St. Andrews Semigroups Conference, World Scientific, 87-97, (1998). "Deciding some embeddability problems for semigroups of mappings", (joint with D.H. Fremlin), Proc. Braga Semigroups Conference, World Scientific, 87-95 (2000).
Additional information	PhD Research Students Previous: 2004. Chris Saker (Further Mathematics Centre Manager and Teaching Fellow, University of Essex) - 'Unavoidable Sets and Aperiodic Unavoidable Sets', external examiner Dr MArk Lawson (Heriot-Watt). 1999. Paula Catarino, (Lecturer University of Tras-os Montes, Portugal) - 'The monoid of orientation-preserving mappings on a chain', external examiner Prof. J.M. Howie (St. Andrews). 1999. Alexei Vernitski (from Ekaterinburg, Russia, Lecturer, DMS) - 'Classes of abstract semigroups closed under the formation of subsemigroups and finitary direct products', external examiner, Prof. J.M. Howie (St. Andrews). 2011, Imtiaz Ahmad, 'Bandwidth labellings of graphs and their associated semigroups', external examiner, Dr Robert Gray (Lisbon).

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