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The rank of the semigroup of transformations stabilising a partition of a finite set

Published online by Cambridge University Press:  06 July 2015

JOÃO ARAÚJO ,
WOLFRAM BENTZ ,
JAMES D. MITCHELL  and
CSABA SCHNEIDER
JOÃO ARAÚJO
Affiliation:
Universidade Aberta, R. Escola Politécnica, 147, 1269-001 Lisboa, Portugal & CEMAT-CIÊNCIAS, Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, 1749-016, Lisboa, Portugal. e-mail: jaraujo@lmc.fc.ul.pt
WOLFRAM BENTZ
Affiliation:
CEMAT-CIÊNCIAS, Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, 1749-016, Lisboa, Portugal. e-mail: wfbentz@fc.ul.pt
JAMES D. MITCHELL
Affiliation:
School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, Scotland. e-mail: jdm3@st-andrews.ac.uk
CSABA SCHNEIDER
Affiliation:
Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, Caixa Postal 702, 31270-901 Belo Horizonte, MG, Brazil. e-mail: csaba@mat.ufmg.br
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Abstract

Let $\mathcal{P}$ be a partition of a finite set X. We say that a transformation f : X → X preserves (or stabilises) the partition $\mathcal{P}$ if for all P ∈ $\mathcal{P}$ there exists Q ∈ $\mathcal{P}$ such that Pf ⊆ Q. Let T(X, $\mathcal{P}$) denote the semigroup of all full transformations of X that preserve the partition $\mathcal{P}$.

In 2005 Pei Huisheng found an upper bound for the minimum size of the generating sets of T(X, $\mathcal{P}$), when $\mathcal{P}$ is a partition in which all of its parts have the same size. In addition, Pei Huisheng conjectured that his bound was exact. In 2009 the first and last authors used representation theory to solve Pei Huisheng's conjecture.

The aim of this paper is to solve the more complex problem of finding the minimum size of the generating sets of T(X, $\mathcal{P}$), when $\mathcal{P}$ is an arbitrary partition. Again we use representation theory to find the minimum number of elements needed to generate the wreath product of finitely many symmetric groups, and then use this result to solve the problem.

The paper ends with a number of problems for experts in group and semigroup theories.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 2 , September 2015 , pp. 339 - 353
Copyright
Copyright © Cambridge Philosophical Society 2015 

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