Skip to main content Accessibility help
×
Home

Dual symmetric inverse monoids and representation theory

  • D. G. Fitzgerald (a1) and Jonathan Leech (a2)

Abstract

There is a substantial theory (modelled on permutation representations of groups) of representations of an inverse semigroup S in a symmetric inverse monoid Ix, that is, a monoid of partial one-to-one selfmaps of a set X. The present paper describes the structure of a categorical dual Ix* to the symmetric inverse monoid and discusses representations of an inverse semigroup in this dual symmetric inverse monoid. It is shown how a representation of S by (full) selfmaps of a set X leads to dual pairs of representations in Ix and Ix*, and how a number of known representations arise as one or the other of these pairs. Conditions on S are described which ensure that representations of S preserve such infima or suprema as exist in the natural order of S. The categorical treatment allows the construction, from standard functors, of representations of S in certain other inverse algebras (that is, inverse monoids in which all finite infima exist). The paper concludes by distinguishing two subclasses of inverse algebras on the basis of their embedding properties.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Dual symmetric inverse monoids and representation theory
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Dual symmetric inverse monoids and representation theory
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Dual symmetric inverse monoids and representation theory
      Available formats
      ×

Copyright

References

Hide All
[1]Bredikhin, D. A., ‘Representations of inverse semigroups by difunctional multipermutations’, in: Transformation Semigroups: Proceedings of the International Conference held at the University of Essex, Colchester, England, August 3rd–6th, 1993 (ed. Higgins, P.M.) (Department of Mathematics, University of Essex, 1994) pp. 110.
[2]Fichtner, B., ‘Über die zu Gruppen gehörigen induktiven Gruppoide, I’, Math. Nachr. 44 (1970), 313339.
[3]Fichtner-Schultz, B., ‘Über die zu Gruppen gehörigen induktiven Gruppoide, II’, Math. Nachr. 48 (1971), 275278.
[4]Grillet, P. A., Semigroups: an introduction to the structure theory (Marcel Dekker, New York, 1995).
[5]Hilton, P. J. and Stammbach, U., A course in homological algebra, Graduate Texts in Math. 4 (Springer, New York, 1971).
[6]Leech, J., ‘Constructing inverse monoids from small categories’, Semigroup Forum 36 (1987), 89116.
[7]Leech, J., ‘Inverse monoids with a natural semilattice ordering’, Proc. London Math. Soc. 70 (1995), 146182.
[8]Leech, J., ‘On the foundations of inverse monoids and inverse algebras’, Proc. Edinburgh Math. Soc. 41 (1998), 121.
[9]MacLane, S., Categories for the working mathematician, Graduate Texts in Math. 5 (Springer, New York, 1971).
[10]Petrich, M., Inverse semigroups (Wiley, New York, 1984).
[11]Preston, G. B., ‘Representations of inverse semigroups by one-to-one partial transformations of a set’, Semigroup Forum 6 (1973), 240245; Addendum, Semigroup Forum 8 (1974), 277.
[12]Riguet, J., ‘Relations binaires, fermetures, correspondances de Galois’, Bull. Soc. Math. France 76 (1948), 114132.
[13]Schein, B. M., ‘Representation of inverse semigroups by local automorphisms and multiautomorphisms of groups and rings’, Semigroup Forum 32 (1985), 5560.
[14]Schein, B. M., ‘Multigroups’, J. Algebra 111 (1987), 114132.
[15]Wagner, V. V., ‘Theory of generalised grouds and generalised groups’, Mat. Sb. (NS) 32 (1953), 545632.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

MSC classification

Dual symmetric inverse monoids and representation theory

  • D. G. Fitzgerald (a1) and Jonathan Leech (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed