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

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 P f ⊆ 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.

Cellular automata and finite groups

“…Ranks of various finite monoids have been determined in the literature before (e.g. see [1,2,12,13,16]).…”

Ranks of finite semigroups of one-dimensional cellular automata

“…For p > 3 and q ∈ {5, 6, 8}, we may replace (1,2) and z q by generators of Sym q of orders 2 and 3, respectively (see Theorem 3 (i)), so w 2 = (3, 3, 0, . .…”

“…Ranks of various finite semigroups have been determined in the literature before (e.g. see [1,2,8,9,11]). In order to hopefully bring more attention to the study of finite semigroups of CA, we shall propose the following problem.…”

Orbits of primitive $k$-homogenous groups on $(n-k)$-partitions with applications to semigroups

“…any p(k) Regarding claim (6), observe that by (1) the semigroup a, G contains all the constant maps and hence, by [90,Theorem 1], its automorphisms are…”