The rank of the semigroup of transformations stabilising a partition of a finite set
The rank of the semigroup of transformations stabilising a partition of a finite set
复制标题
稳定有限集划分的变换半群的秩
DOI：
10.1017/s0305004115000389
复制
发表时间：
2014
影响因子：
0.8
通讯作者：
Csaba Schneider
中科院分区：
文献类型：
--
作者：
J. Araújo;W. Bentz;J. Mitchell;Csaba Schneider
设$\mathcal{P}$是有限集合X的一个划分。我们说变换f：X → X保持（或稳定）划分$\mathcal{P}$，如果对所有P ∈ $\mathcal{P}$存在Q ∈ $\mathcal{P}$使得Pf <$Q。设T（X，$\mathcal{P}$）表示X的所有保持划分$\mathcal{P}$的全变换的半群。在2005年，裴惠生发现了T（X，$\mathcal{P}$）的生成集的最小尺寸的上界，当$\mathcal {P}$是一个划分，其中它的所有部分具有相同的尺寸。此外，裴惠生还保证他的界限是准确的。2009年，第一个和最后一个作者使用表示论解决了裴惠生猜想。本文的目的是解决一个更复杂的问题：当$\mathcal{P}$是一个任意划分时，求T（X，$\mathcal {P}$）的生成集的最小尺寸。再次，我们使用表示理论，以找到所需的最小数量的元素，以产生的圈积的100多个对称群，然后使用这个结果来解决这个问题。本文最后与专家组和半群理论的一些问题。
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.