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

ARAÚJO, JOÃO; Araújo, João; BENTZ, WOLFRAM; Bentz, Wolfram; MITCHELL, JAMES D.; Mitchell, J.D.; SCHNEIDER, CSABA; Schneider, Csaba

Authors

JOÃO ARAÚJO

João Araújo

JAMES D. MITCHELL

J.D. Mitchell

CSABA SCHNEIDER

Csaba Schneider

Abstract

Let P be a partition of a finite set X. We say that a full transformation f:X→X preserves (or stabilizes) the partition P if for all P∈P there exists Q∈P such that Pf⊆Q. Let T(X,P) denote the semigroup of all full transformations of X that preserve the partition P.
In 2005 Huisheng found an upper bound for the minimum size of the generating sets of T(X,P), when P is a partition in which all of its parts have the same size. In addition, Huisheng conjectured that his bound was exact. In 2009 the first and last authors used representation theory to completely solve Hisheng's conjecture.
The goal of this paper is to solve the much more complex problem of finding the minimum size of the generating sets of T(X,P), when 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.

Journal Article Type Article
Publication Date 2015-09
Journal Mathematical Proceedings of the Cambridge Philosophical Society
Print ISSN 0305-0041
Electronic ISSN 1469-8064
Publisher Cambridge University Press (CUP)
Peer Reviewed Peer Reviewed
Volume 159
Issue 02
Pages 339-353
Institution Citation Araújo, J., Bentz, W., Mitchell, J., & Schneider, C. (2015). The rank of the semigroup of transformations stabilising a partition of a finite set. Mathematical proceedings of the Cambridge Philosophical Society, 159(02), 339-353. doi:10.1017/S0305004115000389
DOI https://doi.org/10.1017/S0305004115000389
Publisher URL https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/rank-of-the-semigroup-of-transformations-stabilising-a-partition-of-a-finite-set/A50C234AF3AE9C6E0D0A074B82060EE9