Green's Relations in Finite Transformation Semigroups
Green's Relations in Finite Transformation Semigroups
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DOI：
10.1007/978-3-319-58747-9_12
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发表时间：
2017-03
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影响因子：
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通讯作者：
Lukas Fleischer;Manfred Kufleitner
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文献类型：
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作者：
Lukas Fleischer;Manfred Kufleitner
当半群由有限集上的变换给出时，我们考虑了绿色关系的复杂性。绿色关系可以用（右/左/双侧）Cayley图的可达性来定义。等价类则对应于强连通分量。不难证明，在最坏的情况下，等价类的数量与元素的数量处于同一数量级。另一个重要的参数是组件链的最大长度。我们的主要贡献是这个参数的指数下界。有一个简单的构造任意一组生成器。然而，常数字母表的证明是相当复杂的。我们的结果也适用于自动机及其句法半群。
We consider the complexity of Green’s relations when the semigroup is given by transformations on a finite set. Green’s relations can be defined by reachability in the (right/left/two-sided) Cayley graph. The equivalence classes then correspond to the strongly connected components. It is not difficult to show that, in the worst case, the number of equivalence classes is in the same order of magnitude as the number of elements. Another important parameter is the maximal length of a chain of components. Our main contribution is an exponential lower bound for this parameter. There is a simple construction for an arbitrary set of generators. However, the proof for constant alphabet is rather involved. Our results also apply to automata and their syntactic semigroups.