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GRAPH PRODUCTS OF RIGHT CANCELLATIVE MONOIDS

Part of: Semigroups

Published online by Cambridge University Press:  09 October 2009

JOHN FOUNTAIN  and
MARK KAMBITES
JOHN FOUNTAIN*
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, UK (email: jbf1@york.ac.uk)
MARK KAMBITES
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK (email: Mark.Kambites@manchester.ac.uk)
*
✉For correspondence; e-mail: jbf1@york.ac.uk
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Abstract

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Our first main result shows that a graph product of right cancellative monoids is itself right cancellative. If each of the component monoids satisfies the condition that the intersection of two principal left ideals is either principal or empty, then so does the graph product. Our second main result gives a presentation for the inverse hull of such a graph product. We then specialize to the case of the inverse hulls of graph monoids, obtaining what we call ‘polygraph monoids’. Among other properties, we observe that polygraph monoids are F*-inverse. This follows from a general characterization of those right cancellative monoids with inverse hulls that are F*-inverse.

Keywords

graph productright cancellative monoidgraph monoidinverse hullF*-inverse monoid

MSC classification

Secondary: 20M10: General structure theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 87 , Issue 2 , October 2009 , pp. 227 - 252
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1]Beauregard, R. A., ‘Right LCM domains’, Proc. Amer. Math. Soc. 30 (1971), 1–7.Google Scholar
[2]Brieskorn, E. and Saito, K., ‘Artin-Gruppen und Coxeter-Gruppen’, Invent. Math. 17 (1972), 245–271.CrossRefGoogle Scholar
[3]Bulman-Fleming, S., Fountain, J. and Gould, V., ‘Inverse semigroups with zero: covers and their structure’, J. Aust. Math. Soc. 67 (1999), 15–30.Google Scholar
[4]Charney, R., ‘An introduction to right-angled Artin groups’, Geom. Dedicata 125 (2007), 141–158.CrossRefGoogle Scholar
[5]Cherubini, A. and Petrich, M., ‘The inverse hull of right cancellative semigroups’, J. Algebra 111 (1987), 74–113.CrossRefGoogle Scholar
[6]Clifford, A. H., ‘A class of d-simple semigroups’, Amer. J. Math. 75 (1953), 547–556.Google Scholar
[7]Clifford, A. H. and Preston, G. B., The Algebraic Theory of Semigroups, Vol. I (American Mathematical Society, Providence, RI, 1961).CrossRefGoogle Scholar
[8]Crisp, J. and Laca, M., ‘On the Toeplitz algebras of right-angled and finite-type Artin groups’, J. Aust. Math. Soc. 72 (2002), 223–245.CrossRefGoogle Scholar
[9]Dehornoy, P., ‘Groupes de Garside’, Ann. Sci. École Norm. Sup. 35 (2002), 267–306.Google Scholar
[10]Dehornoy, P., ‘Alternating normal forms for braids and locally Garside monoids’, J. Pure Appl. Algebra 212 (2008), 2413–2439.CrossRefGoogle Scholar
[11]Deligne, P., ‘Les immeubles des groupes de tresses généralisés’, Invent. Math. 17 (1972), 273–302.CrossRefGoogle Scholar
[12]Diekert, V., Combinatorics on Traces, Lecture Notes in Computer Science, 454 (Springer, Berlin, 1990).CrossRefGoogle Scholar
[13]Fohry, E. and Kuske, D., ‘On graph products of automatic and biautomatic monoids’, Semigroup Forum 72 (2006), 337–352.CrossRefGoogle Scholar
[14]Green, E. R., Graph Products of Groups. PhD Thesis, University of Leeds, 1990.Google Scholar
[15]Hermiller, S. and Meier, J., ‘Algorithms and geometry for graph products of groups’, J. Algebra 171 (1995), 230–257.CrossRefGoogle Scholar
[16]Howie, J. M., Fundamentals of Semigroup Theory (Oxford University Press, Oxford, 1995).CrossRefGoogle Scholar
[17]Knox, N., ‘The inverse hull of the free semigroup on a set X’, Semigroup Forum 16 (1978), 345–354.CrossRefGoogle Scholar
[18]Lawson, M. V., Inverse Semigroups: The Theory of Partial Symmetries (World Scientific, Singapore, 1998).CrossRefGoogle Scholar
[19]Lawson, M. V., ‘The structure of 0-E-unitary inverse semigroups: I. The monoid case’, Proc. Edinburgh Math. Soc. 42 (1999), 497–520.CrossRefGoogle Scholar
[20]Lawson, M. V., ‘E *-unitary inverse semigroups’, in: Semigroups, Algorithms, Automata and Languages (eds. G. M. S. Gomes, J.-E. Pin and P. Silva) (World Scientific, Singapore, 2002), pp. 155–194.Google Scholar
[21]McAlister, D. B., ‘One-to-one partial right translations of a right cancellative semigroup’, J. Algebra 43 (1976), 231–251.CrossRefGoogle Scholar
[22]McAlister, D. B., ‘A random ramble in inverse semigroups II: an introduction to E *-unitary inverse semigroups—from an old fashioned perspective’, in: Semigroups and Languages, Lisbon 2002 (eds. I. M. Araújo, M. J. J. Branco, V. H. Fernandes and G. M. S. Gomes) (World Scientific, Singapore, 2004), pp. 133–150.Google Scholar
[23]McAlister, D. B. and McFadden, R., ‘Zig-zag representations and inverse semigroups’, J. Algebra 32 (1974), 178–206.CrossRefGoogle Scholar
[24]Margolis, S. W., ‘Embedding semigroups in groups’, Talk at University of York Algebra Seminar, 21 May 2003.Google Scholar
[25]Meakin, J. and Sapir, M., ‘Congruences on free monoids and submonoids of polycyclic monoids’, J. Aust. Math. Soc. Ser. A 54 (1993), 236–253.Google Scholar
[26]Nivat, M. and Perrot, J.-F., ‘Une généralisation du monoïde bicyclique’, C. R. Acad. Sci. Paris Sér. A–B 271 (1970), A824–A827.Google Scholar
[27]Paris, L., ‘Artin monoids inject in their groups’, Comment. Math. Helv. 77 (2002), 609–637.CrossRefGoogle Scholar
[28]Rees, D., ‘On the group of a set of partial transformations’, J. London Math. Soc. 22 (1948), 281–284.Google Scholar
[29]Steinberg, B., ‘The uniform word problem for groups and finite Rees quotients of E-unitary inverse semigroups’, J. Algebra 266 (2003), 1–13.CrossRefGoogle Scholar
[30]Szendrei, M. B., ‘A generalization of McAlister’s P-theorem for E-unitary regular semigroups’, Acta Sci. Math. (Szeged) 51 (1987), 229–249.Google Scholar
[31]Veloso da Costa, A., ‘Graph products of monoids’, Semigroup Forum 63 (2001), 247–277.CrossRefGoogle Scholar
[32]Veloso da Costa, A., ‘On graph products of automatic monoids’, Theor. Inform. Appl. 35 (2001), 403–417.CrossRefGoogle Scholar