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ON THE PARTITION MONOID AND SOME RELATED SEMIGROUPS

Part of: Combinatorics Ordered sets Semigroups

Published online by Cambridge University Press:  06 December 2010

D. G. FITZGERALD  and
KWOK WAI LAU
D. G. FITZGERALD*
Affiliation:
School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, TAS 7001, Australia (email: D.FitzGerald@utas.edu.au)
KWOK WAI LAU
Affiliation:
CSIRO Mathematics, Informatics and Statistics, Private Bag 5, Wembley, WA 6913, Australia (email: Rex.Lau@csiro.au)
*
✉For correspondence; e-mail: D.FitzGerald@utas.edu.au
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Abstract

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The partition monoid is a salient natural example of a *-regular semigroup. We find a Galois connection between elements of the partition monoid and binary relations, and use it to show that the partition monoid contains copies of the semigroup of transformations and the symmetric and dual-symmetric inverse semigroups on the underlying set. We characterize the divisibility preorders and the natural order on the (straight) partition monoid, using certain graphical structures associated with each element. This gives a simpler characterization of Green’s relations. We also derive a new interpretation of the natural order on the transformation semigroup. The results are also used to describe the ideal lattices of the straight and twisted partition monoids and the Brauer monoid.

Keywords

partition monoidBrauer monoidideal structurenatural order

MSC classification

Secondary: 20M20: Semigroups of transformations, etc. 05A18: Partitions of sets 06A15: Galois correspondences, closure operators
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 2 , April 2011 , pp. 273 - 288
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

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