Bulletin of the Australian Mathematical Society

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

Research Article

THE IDEAL STRUCTURE OF SEMIGROUPS OF TRANSFORMATIONS WITH RESTRICTED RANGE

SUZANA MENDES-GONÇALVESa1 c1 and R. P. SULLIVANa2

a1 Centro de Matemática, Universidade do Minho, 4710 Braga, Portugal (email: smendes@math.uminho.pt)

a2 School of Mathematics and Statistics, University of Western Australia, Nedlands, 6009, Australia (email: bob@maths.uwa.edu.au)

Abstract

Let Y be a fixed nonempty subset of a set X and let T(X,Y ) denote the semigroup of all total transformations from X into Y. In 1975, Symons described the automorphisms of T(X,Y ). Three decades later, Nenthein, Youngkhong and Kemprasit determined its regular elements, and more recently Sanwong, Singha and Sullivan characterized all maximal and minimal congruences on T(X,Y ). In 2008, Sanwong and Sommanee determined the largest regular subsemigroup of T(X,Y ) when |Y |≠1 and Y ≠ X; and using this, they described the Green’s relations on T(X,Y ) . Here, we use their work to describe the ideal structure of T(X,Y ) . We also correct the proof of the corresponding result for a linear analogue of T(X,Y ) .

(Received June 16 2010)

(Online publication December 09 2010)

2010 Mathematics subject classification

  • primary 20M20; secondary 15A04

Keywords and phrases

  • transformation semigroups;
  • ideals;
  • maximal;
  • reductive

Correspondence:

c1 For correspondence; e-mail: smendes@math.uminho.pt

Footnotes

The authors acknowledge the support of the Portuguese ‘Fundação para a Ciência e a Tecnologia’ through its Multi-Year Funding Program for ‘Centro de Matemática’ at the University of Minho, Braga, Portugal.