Groups and actions in transformation semigroups
Groups and actions in transformation semigroups
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变换半群中的群和动作
DOI：
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发表时间：
1998
期刊：
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通讯作者：
N. Ruškuc
中科院分区：
文献类型：
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作者：
S. Linton;G. Pfeiffer;E. Robertson;N. Ruškuc
抽象的。让
$S$是度变换半群
$n$。到每个元件
$sin S$我们关联一个置换群
$G_R（s）$作用于
$s$，我们为这个群找到一个自然生成集。事实证明
$mathcal{R}$-类
$s$是某些集合的不相交并集，每个集合的大小等于
$G_R（s）$。因此，我们表明，两个
$mathcal{R}$-包含具有相等图像的元素的类具有相同的大小，即使它们不属于相同的
$mathcal{D}$-class。通过某种二元性过程，
$s$另一个置换群
的图像上的$G_L（s）$
$s$，并证明了类似的结果
$mathcal{L}$-类
$S$。最后，我们证明了Schützenberger群的
$mathcal{H}$-类
$s$同构于
$G_R（s）$和
$G_L（s）$。本文的结果也可以应用于新的算法调查变换半群，这将在即将出版的论文中描述。
Abstract. Let
$S$ be a transformation semigroup of degree
$n$. To each element
$sin S$ we associate a permutation group
$G_R(s)$ acting on the image of
$s$, and we find a natural generating set for this group. It turns out that the
$mathcal{R}$-class of
$s$ is a disjoint union of certain sets, each having size equal to the size of
$G_R(s)$. As a consequence, we show that two
$mathcal{R}$-classes containing elements with equal images have the same size, even if they do not belong to the same
$mathcal{D}$-class. By a certain duality process we associate to
$s$ another permutation group
$G_L(s)$ on the image of
$s$, and prove analogous results for the
$mathcal{L}$-class of
$S$. Finally we prove that the Schützenberger group of the
$mathcal{H}$-class of
$s$ is isomorphic to the intersection of
$G_R(s)$ and
$G_L(s)$. The results of this paper can also be applied in new algorithms for investigating transformation semigroups, which will be described in a forthcoming paper.