Proceedings of the Edinburgh Mathematical Society (Series 2)

Table of Contents - October 2007 - Volume 50, Issue 03  

Research Article

LARGEST 2-GENERATED SUBSEMIGROUPS OF THE SYMMETRIC INVERSE SEMIGROUP

J. M. Andréa1a2, V. H. Fernandesa1a2 and J. D. Mitchella3

a1 Centro de Álgebra da Universidade de Lisboa, Avenida Professor Gama Pinto 2, 1649003 Lisboa, Portugal

a2 Departamento de Matemática, Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa, Monte da Caparica, 2829516 Caparica, Portugal (jmla@fct.unl.pt; vhf@fct.unl.pt)

a3 Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, UK (jdm3@st-and.ac.uk)

Abstract

The symmetric inverse monoid $\mathcal{I}_{n}$ is the set of all partial permutations of an $n$-element set. The largest possible size of a $2$-generated subsemigroup of $\mathcal{I}_{n}$ is determined. Examples of semigroups with these sizes are given. Consequently, if $M(n)$ denotes this maximum, it is shown that $M(n)/|\mathcal{I}_{n}|\rightarrow1$ as $n\rightarrow\infty$. Furthermore, we deduce the known fact that $\mathcal{I}_{n}$ embeds as a local submonoid of an inverse $2$-generated subsemigroup of $\mathcal{I}_{n+1}$.

(Online publication January 08 2008)

(Received November 14 2005)

Keywords

  • Primary 20M20;
  • Secondary 20M18;
  • inverse semigroups;
  • generators;
  • subsemigroups