Bulletin of the Australian Mathematical Society

Table of Contents - 15 March 2011 - Volume 83, Issue 02  

Research Article

ON THE PARTITION MONOID AND SOME RELATED SEMIGROUPS

D. G. FITZGERALDa1 c1 and KWOK WAI LAUa2

a1 School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, TAS 7001, Australia (email: D.FitzGerald@utas.edu.au)

a2 CSIRO Mathematics, Informatics and Statistics, Private Bag 5, Wembley, WA 6913, Australia (email: Rex.Lau@csiro.au)

Abstract

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.

(Received June 16 2010)

(Online publication December 06 2010)

2010 Mathematics subject classification

  • primary 20M20; secondary 20M17;
  • 05A18;
  • 06A15

Keywords and phrases

  • partition monoid;
  • Brauer monoid;
  • ideal structure;
  • natural order

Correspondence:

c1 For correspondence; e-mail: D.FitzGerald@utas.edu.au