Professor Peter Higgins

Email
peteh@essex.ac.uk 
Telephone
+44 (0) 1206 873019

Location
3A.539, Colchester Campus
Profile
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. I served as a Colchester Borough Councillor from 200216. 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 60 research papers and nine books, the first of which was a specialist research monograph, 'Techniques of Semigroup Theory', which has 455 citations in the literature (Google Scholar). I have six popular science books with Oxford University Press. The first three were, 'Mathematics for the Curious', Mathematics for the Imagination', and 'Nets, Puzzles and Postmen, an exploration of mathematical connections', (in hardback and paperback). I was invited to write two volumes in the Very Short Introduction series, they being 'Numbers', and 'Algebra'. In 2017 OUP published 'Professor Higgins's Problem Book'. My work has 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. In 2011 I was invited to become a Fellow of the Institute of Mathematics and its Applications (F.I.M.A.) In 2013 I was awarded the Premio Peano Prize for the best book on mathematics written in Italian for the translation of 'Nets, Puzzles and Postmen' entitled La matematica dei social network'. To keep uptodate with my mathematical activities, see my Web page: http://www.professorpeterhiggins.com/ As of February 2019 I have 1414 citations in Google Scholar with a hindex of 18 and an i10 index of 33. Four of my papers which have been cited over 50 times in the literature. Previous PhD students: 1999. Paula Catarino, (Lecturer University of Trasos Montes, Portugal)  'The monoid of orientationpreserving 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). 2004 Chris Saker (Senior Lecturer, Essex), 'Unavoidable Sets and Aperiodic Unavoidable Sets', External Examiner, Dr Mark Lawson (Bangor) 2011, Imtiaz Ahmad, 'Bandwidth labellings of graphs and their associated semigroups', external examiner, Dr Robert Gray (Lisbon). 2014 Suhear Alwan 'Languages that require full scanning of words to determine membership' external examiner Prof Mark Lawson (HeriotWatt) Research interests: Linked with the School of Computer Science and Electronic Engineering in previous REF cycle An editor of the international journal Communications in Algebra (1999  2004).
Qualifications

B.A. (ANU. Canberra, 1976), BSc (First Class, University of Tasmania, 1980), PhD, (Monash University, Melbourne, 1983).

Fellow of the Institute of Mathematics and its Applications, (2011  )
Appointments
University of Essex

Head of Dept, Mathematical Sciences, University of Essex (1/1/2005  31/12/2010)

Senate member, Mathematics/Mathematical Sciences, University of Essex (1/9/1999  31/12/2010)
Research and professional activities
Research interests
Algebraic Semigroups (primary)
Formal Language Theory, Combinatorics and Graph Theory (secondary)
Conferences and presentations
Semigroup classes defined by equations
Keynote presentation, Conference on Theoretical and Computational Algebra, Pocinho Portugal., 4/7/2023
The map numbering problem, IEEE transactions, Proc CEEC 2010, 2nd Computer Science and Electronic Engineering Conference, joint with I. Ahmad. Winner of Prize for Best Paper.
United Kingdom, 2010
Deciding some embeddability problems for semigroups of mappings, (joint with D.H. Fremlin), Proc. Braga Semigroups Conference, World Scientific, 8795 (2000).
2000
'Pseudovarieties generated by classes of transformation semigroups', Proc. St. Petersburgh Semigroup Conference Russian State Hydrometeorological Inst., 8594 (1999).
1999
'The Converse of Lallement's Lemma', Proc. St. Andrews Semigroups Conference, World Scientific, 7886, (1998).
1998
'On Special amalgamation Bases', (joint with Kunitaka Shoji, Shimane University, Japan). Proc. St. Andrews Semigroups Conference, World Scientific, 8797, (1998).
1998
'On eventually regular semigroups', in J.M. Howie et al. (eds.), 170189, Semigroups with Applications, (Oberwolfach Conference) World Scientific, Singapore, 1992.
Singapore, 1992
'Combinatorial Aspects of Semigroups of Orderpreserving and Decreasing Functions', 103110, Proceedings of the Milano International Conference on Semigroups, June 1992, World Scientific Publishers, 1993.
1992
'Random products in semigroups of mappings', in J. Almeida et al. (eds.), 89100, Lattices, Semigroups and Universa Algebra, Plenum Press, New York, 1990.
New York, United States, 1990
'Semigroup epimorphisms', in P.R. Jones et al. (eds.), Proceedings of the Marquette Conference on Semigroups, 5164, September 1984.
1984
Papers in Proceedings of International Conferences
Teaching and supervision
Previous supervision
Degree subject: Mathematics
Degree type: Doctor of Philosophy
Awarded date: 1/4/2014
Degree subject: Mathematics
Degree type: Master of Science (by Dissertation)
Awarded date: 27/4/2012
Publications
Publications (2)
Higgins, PM., (2023). Finite regular semigroups with permutations that map elements to inverses
Higgins, PM., (2023). Production optimization by agents of differing work rates
Journal articles (64)
Higgins, P., (2024). The Bikerhiker problem. Journal of Combinatorics. 15 (1), 105134
Higgins, PM., (2024). Finite regular semigroups with permutations that map elements to inverses. Semigroup Forum
Higgins, P., (2023). Limiting behaviour of pairs with equal images for polynomial functions. College Mathematics Journal. 54 (4), 391393
Higgins, P. and Jackson, M., (2023). Equationally defined classes of semigroups. Semigroup Forum. 107 (2), 459477
Higgins, P. and Vernitski, A., (2022). Orientationpreserving and orientationreversing mappings: a new description. Semigroup Forum. 104 (2), 509514
Alam, N., Higgins, PM. and Khan, NM., (2020). Epimorphisms, dominions and Hcommutative semigroups. Semigroup Forum. 100 (2), 349363
Higgins, P. and Jackson, M., (2020). Algebras defined by equations. Journal of Algebra. 555, 131156
East, J. and Higgins, PM., (2020). Green's relations and stability for subsemigroups. Semigroup Forum. 101 (1), 7786
Higgins, PM., (2019). Involution matchings, the semigroup of orientationpreserving and orientationreversing mappings, and inverse covers of the full transformation semigroup. Semigroup Forum. 98 (3), 669689
Higgins, PM., (2017). Embedding in a Finite 2Generator Semigroup. Glasgow Mathematical Journal. 59 (01), 6175
Higgins, PM., (2017). Orthodox semigroups and permutation matchings. Semigroup Forum. 94 (1), 167175
Higgins, PM. and Alwan, S., (2017). Languages with membership determined by single letter factors. Theoretical Computer Science. 680, 1524
Higgins, PM., (2017). THE JOY OF QUIZ. TLSTHE TIMES LITERARY SUPPLEMENT (5966), 2929
Higgins, PM., (2017). THE GCHQ PUZZLE BOOK. TLSTHE TIMES LITERARY SUPPLEMENT (5966), 2929
Higgins, PM., (2017). CAN YOU SOLVE MY PROBLEMS? A casebook of ingenious, perplexing and totally satisfying puzzles. TLSTHE TIMES LITERARY SUPPLEMENT (5966), 2929
Higgins, PM., (2016). THE PERFECT BET How science and maths are taking the luck out of gambling. TLSTHE TIMES LITERARY SUPPLEMENT (5923), 3030
Ahmad, I. and Higgins, PM., (2015). On the bandwidth of Mobius graphs. Bulletin of the Iranian Mathematical Society. 41 (3), 545550
Catarino, P., Higgins, PM. and Levi, I., (2015). On inverse subsemigroups of the semigroup of orientationpreserving or orientationreversing transformations. Algebra and Discrete Mathematics. 19 (2), 162171
Higgins, PM., (2014). Permutations of a semigroup that map to inverses. Semigroup Forum. 89 (1), 169182
Higgins, PM. and Alwan, S., (2013). Languages that require full scanning of words to determine membership. Journal of Automata Languages and Combinatorics. 18 (2), 7196
Higgins, PM., (2012). The product of the idempotents and an $\mathcal{H}$ class of the finite full transformation semigroup. Semigroup Forum. 84 (2), 203215
Higgins, PM., (2012). Burrows–Wheeler transformations and de Bruijn words. Theoretical Computer Science. 457, 128136
HIGGINS, PM., (2011). THE LENGTH OF SHORT WORDS IN UNAVOIDABLE SETS. International Journal of Algebra and Computation. 21 (06), 951960
Higgins, PM., (2011). The Length Of Short Words In Unavoidable Sets. International Journal of Algebra and Computation. 21 (06), 951960
Descalço, L. and Higgins, PM., (2010). Generalized Green'S Equivalences on the Subsemigroups of the Bicyclic Monoid. Communications in Algebra. 38 (12), 45974612
HIGGINS, PM., (2006). THE SEMIGROUP OF CONJUGATES OF A WORD. International Journal of Algebra and Computation. 16 (06), 10151029
Higgins, PM. and Saker, CJ., (2006). Unavoidable sets. Theoretical Computer Science. 359 (13), 231238
HIGGINS, PM., MITCHELL, JD., MORAYNE, M. and RUŠKUC, N., (2006). RANK PROPERTIES OF ENDOMORPHISMS OF INFINITE PARTIALLY ORDERED SETS. Bulletin of the London Mathematical Society. 38 (02), 177191
Higgins, PM., Howie, JM., Mitchell, JD. and Ruškuc, N., (2003). COUNTABLE VERSUS UNCOUNTABLE RANKS IN INFINITE SEMIGROUPS OF TRANSFORMATIONS AND RELATIONS. Proceedings of the Edinburgh Mathematical Society. 46 (3), 531544
Higgins, PM., Howie, JM. and Ruškuc, N., (2003). Set products in transformation semigroups. Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 133 (5), 11211135
HIGGINS, PM., MITCHELL, JD. and RUŠKUC, N., (2003). GENERATING THE FULL TRANSFORMATION SEMIGROUP USING ORDER PRESERVING MAPPINGS. Glasgow Mathematical Journal. 45 (3), 557566
Saker, CJ. and Higgins, PM., (2002). Unavoidable Sets of Words of Uniform Length. Information and Computation. 173 (2), 222226
CATARINO, PM. and HIGGINS, PM., (2002). THE PSEUDOVARIETY GENERATED BY ALL SEMIGROUPS OF ORIENTATIONPRESERVING TRANSFORMATIONS ON A FINITE CYCLE. International Journal of Algebra and Computation. 12 (03), 387405
ALMEIDA, J., HIGGINS, PM. and VOLKOV, MV., (2001). THE GAP BETWEEN PARTIAL AND FULL: AN ADDENDUM. International Journal of Algebra and Computation. 11 (01), 131135
Higgins, PM. and Margolis, SW., (2000). Finite aperiodic semigroups with commuting idempotents and generalizations. Israel Journal of Mathematics. 116 (1), 367380
HIGGINS, PM., (2000). A NEW PROOF OF SCHUTZENBERGER'S THEOREM. International Journal of Algebra and Computation. 10 (02), 217220
Higgins, PMCPM., (1999). The monoid of orientationpreserving mappings on a chain. Semigroup Forum. 58 (2), 190206
Howie, JM., Ruškuc, N. and Higgins, PM., (1998). On relative ranks of full transformation semigroups. Communications in Algebra. 26 (3), 733748
Higgins, PM., Howie, JM. and Ruškuc, N., (1998). Generators and factorisations of transformation semigroups. Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 128 (6), 13551369
Higgins, PM., Howie, JM. and Ruškuc, N., (1998). Generators and factorisations of transformation semigroups. Royal Society of Edinburgh  Proceedings A. 128 (6), 13551369
Higgins, PM., (1997). A proof of Simon's theorem on piecewise testable languages. Theoretical Computer Science. 178 (12), 257264
Almeida, J. and Higgins, PM., (1997). Monoids RespectingnChains of Intervals. Journal of Algebra. 187 (1), 183202
Higgins, PM., (1996). Epis from locally inverse semigroups are onto. Semigroup Forum. 52 (1), 4953
Higgins, PM., (1995). Embedding semigroups in semibands of depth two. Semigroup Forum. 51 (1), 395396
Higgins, PM., (1994). Idempotent depth in semigroups of orderpreserving mappings. Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 124 (5), 10451058
Higgins, PM., (1994). The Mitsch order on a semigroup. Semigroup Forum. 49 (1), 261266
Higgins, PM., (1993). A Class of Eventually Regular Semigroups Determined by PseudoRandom Sets. Journal of the London Mathematical Society. s248 (1), 87102
Higgins, PM., (1993). Combinatorial results for semigroups of orderpreserving mappings. Mathematical Proceedings of the Cambridge Philosophical Society. 113 (2), 281296
Higgins, P., (1990). A short proof of Isbell’s zigzag theorem. Pacific Journal of Mathematics. 144 (1), 4750
Higgins, PM., (1990). An algebraic proof that pseudovarieties are defined by pseudoidentities. Algebra Universalis. 27 (4), 597599
Higgins, PM., (1990). Embedding in bisimple semigroups. Semigroup Forum. 40 (1), 105107
Brown, B. and Higgins, PM., (1988). Finite full transformation semigroups as collections of random functions. Glasgow Mathematical Journal. 30 (2), 203211
Higgins, PM., (1988). Digraphs and the semigroup of all functions on a finite set. Glasgow Mathematical Journal. 30 (1), 4157
Higgins, PM., (1988). The range order of a product of i transformations from a finite full transformation semigroup. Semigroup Forum. 37 (1), 3136
Higgins, PM., (1986). Completely semisimple semigroups and epimorphisms. Proceedings of the American Mathematical Society. 96 (3), 387390
Higgins, PM., (1986). Dense subsets of some common classes of semigroup. Semigroup Forum. 34 (1), 519
Higgins, PM., (1985). Epimorphisms, dominions and semigroups. Algebra Universalis. 21 (23), 225233
Higgins, PM., (1985). A new determination of the permutation identities which ensure that a semigroup variety is finitely based. Journal of Pure and Applied Algebra. 38 (1), 6569
Higgins, PM., (1984). Saturated and epimorphically closed varieties of semigroups. Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics. 36 (2), 153175
Higgins, PM., (1984). Epimorphisms, permutation identities and finite semigroups. Semigroup Forum. 29 (1), 8797
Higgins, PM., (1983). A semigroup with an epimorphically embedded subband. Bulletin of the Australian Mathematical Society. 27 (2), 231242
Higgins, PM., (1983). The varieties of commutative semigroups for which epis are onto. Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 94 (12), 17
Higgins, PM., (1983). The determination of all varieties consisting of absolutely closed semigroups. Proceedings of the American Mathematical Society. 87 (3), 419421
Higgins, PM., (1981). Epis are onto for generalised inverse semigroups. Semigroup Forum. 23 (1), 255259
Books (3)
Higgins, PM., (2017). Professor Higgins's Problem Collection. Oxford University Press. 0198755473. 9780198755470
Higgins, PM., (2008). Nets, puzzles and postmen, an exploration of mathematical connections. Oxford University Press. 9780199218424
Higgins, PM., (2008). Number Story, from Counting to Cryptography. Copernicus Press, SpringerVerlag. 9781848000001
Conferences (2)
Higgins, PM., Howie, JM. and Ruškuc, N., (2003). Set products in transformation semigroups
Higgins, PM., (1996). Ramsey's theorem in algebraic semigroups
Reports and Papers (3)
Higgins, PM., (2021). The Bikerhiker problem
Higgins, PM. and Vernitski, A., (2020). A new formulation of the semigroup of orientationpreserving and orientationreversing mappings
Higgins, PM. and Jackson, M., (2018). Algebras defined by equations