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<meta content="FitzGerald, D.G." name="eprints.creators_name" />
<meta content="Leech, Jonathan" name="eprints.creators_name" />
<meta content="D.FitzGerald@utas.edu.au" name="eprints.creators_id" />
<meta content="leech@westmont.edu" name="eprints.creators_id" />
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<meta content="2007-09-17" name="eprints.datestamp" />
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<meta content="There is a substantial theory (modelled on permutation representations of groups) of representations of an
inverse semigroup S in a symmetric inverse monoid I_X , that is, a monoid of partial one-to-one self-maps
of a set X. The present paper describes the structure of a categorical dual I*_X 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 I_X and I*_X, 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." name="eprints.abstract" />
<meta content="1998" name="eprints.date" />
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<meta content="Journal of the Australian Mathematical Society, Series A" name="eprints.publication" />
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<meta content="345-367" name="eprints.pagerange" />
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<meta content="[1] D. A. Bredikhin, 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. P. M. Higgins) (Department of Mathematics,
University of Essex, 1994) pp. 1-10.
[2] B. Fichtner, Ueber die zu Gruppen gehoerigen induktiven Gruppoide, I, Math. Nachr. 44 (1970),
313-339.
[3] B. Fichtner-Schultz,
Ueber die zu Gruppen gehoerigen induktiven Gruppoide, II, Math. Nachr. 48
(1971), 275-278.
[4] P. A. Grillet, Semigroups: an introduction to the structure theory (Marcel Dekker, New York,
1995).
[5] P. J. Hilton and U. Stammbach, A course in homological algebra, Graduate Texts in Math. 4
(Springer, New York, 1971).
[6] J. Leech, Constructing inverse monoids from small categories, Semigroup Forum 36 (1987),
89-116.
[7] J. Leech, Inverse monoids with a natural semilattice ordering, Proc. London Math. Soc. 70 (1995),
146-182.
[8] J. Leech, On the foundations of inverse monoids and inverse algebras, Proc. Edinburgh Math. Soc.
41 (1998), 1-21.
[9] S. Mac Lane, Categories for the working mathematician, Graduate Texts in Math. 5 (Springer,
New York, 1971).
[10] M. Petrich, Inverse semigroups (Wiley, New York, 1984).
[11] G. B. Preston, Representations of inverse semigroups by one-to-one
partial transformations of a
set, Semigroup Forum 6 (1973), 240-245; Addendum, Semigroup Forum 8 (1974), 277.
[12] J. Riguet, Relations binaires, fermetures, correspondances de Galois, Bull. Soc. Math. France 76
(1948), 114-132.
[13] B. M. Schein, Representation of inverse semigroups by local automorphisms and multiautomorphisms
of groups and rings, Semigroup Forum 32 (1985), 55-60.
[14] B. M. Schein, Multigroups, J. Algebra 111 (1987), 114-132.
[15] V. V. Wagner, Theory of generalised grouds and generalised groups, Mat. Sb. (NS) 32 (1953),
545-632." name="eprints.referencetext" />
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<meta content="There is a substantial theory (modelled on permutation representations of groups) of representations of an
inverse semigroup S in a symmetric inverse monoid I_X , that is, a monoid of partial one-to-one self-maps
of a set X. The present paper describes the structure of a categorical dual I*_X 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 I_X and I*_X, 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." name="DC.description" />
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<h1 class="ep_tm_pagetitle">Dual Symmetric Inverse Monoids and Representation Theory</h1>
<p style="margin-bottom: 1em" class="not_ep_block"><span class="person_name">FitzGerald, D.G.</span> and <span class="person_name">Leech, Jonathan</span> (1998) <xhtml:em>Dual Symmetric Inverse Monoids and Representation Theory.</xhtml:em> Journal of the Australian Mathematical Society, Series A, 64 (3). pp. 345-367. ISSN 0263-6115</p><p style="margin-bottom: 1em" class="not_ep_block"></p><table style="margin-bottom: 1em" class="not_ep_block"><tr><td valign="top" style="text-align:center"><a href="http://eprints.utas.edu.au/1911/1/FL98.pdf"><img alt="[img]" src="http://eprints.utas.edu.au/style/images/fileicons/application_pdf.png" border="0" class="ep_doc_icon" /></a></td><td valign="top"><a href="http://eprints.utas.edu.au/1911/1/FL98.pdf"><span class="ep_document_citation">PDF</span></a> - Full text restricted - Requires a PDF viewer<br />158Kb</td><td><form method="get" accept-charset="utf-8" action="http://eprints.utas.edu.au/cgi/request_doc"><input value="2405" name="docid" accept-charset="utf-8" type="hidden" /><div class=""><input value="Request a copy" name="_action_null" class="ep_form_action_button" onclick="return EPJS_button_pushed( '_action_null' )" type="submit" /> </div></form></td></tr></table><p style="margin-bottom: 1em" class="not_ep_block">Official URL: <a href="http://www.austms.org.au/Publ/Jamsa/V64P3/pdf/e07.pdf">http://www.austms.org.au/Publ/Jamsa/V64P3/pdf/e07.pdf</a></p><div class="not_ep_block"><h2>Abstract</h2><p style="padding-bottom: 16px; text-align: left; margin: 1em auto 0em auto">There is a substantial theory (modelled on permutation representations of groups) of representations of an
inverse semigroup S in a symmetric inverse monoid I_X , that is, a monoid of partial one-to-one self-maps
of a set X. The present paper describes the structure of a categorical dual I*_X 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 I_X and I*_X, 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.</p></div><table style="margin-bottom: 1em" border="0" cellpadding="3" class="not_ep_block"><tr><th valign="top" class="ep_row">Item Type:</th><td valign="top" class="ep_row">Article</td></tr><tr><th valign="top" class="ep_row">Keywords:</th><td valign="top" class="ep_row">dual symmetric inverse monoid, representations of inverse semigroups</td></tr><tr><th valign="top" class="ep_row">Subjects:</th><td valign="top" class="ep_row"><a href="http://eprints.utas.edu.au/view/subjects/230105.html">230000 Mathematical Sciences > 230100 Mathematics > 230105 Group Theory And Generalisations (Incl. Topological Groups And Lie Groups)</a></td></tr><tr><th valign="top" class="ep_row">Collections:</th><td valign="top" class="ep_row">UNSPECIFIED</td></tr><tr><th valign="top" class="ep_row">ID Code:</th><td valign="top" class="ep_row">1911</td></tr><tr><th valign="top" class="ep_row">Deposited By:</th><td valign="top" class="ep_row"><span class="ep_name_citation"><span class="person_name">Dr D. G. FitzGerald</span></span></td></tr><tr><th valign="top" class="ep_row">Deposited On:</th><td valign="top" class="ep_row">17 Sep 2007</td></tr><tr><th valign="top" class="ep_row">Last Modified:</th><td valign="top" class="ep_row">07 Feb 2008 12:22</td></tr><tr><th valign="top" class="ep_row">ePrint Statistics:</th><td valign="top" class="ep_row"><a target="ePrintStats" href="/es/index.php?action=show_detail_eprint;id=1911;">View statistics for this ePrint</a></td></tr></table><p align="right">Repository Staff Only: <a href="http://eprints.utas.edu.au/cgi/users/home?screen=EPrint::View&eprintid=1911">item control page</a></p>
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