有限变换半群中的直字与极小置换子
On Straight Words and Minimal Permutators in Finite Transformation Semigroups
摘要：
受计算机科学中出现的问题的启发，我们研究了具有固定生成元集的有限变换半群的Cayley图中的单位变换和相应直字的无环路径。特别感兴趣的是置换状态集合的给定子集的单词。某些这样的话，所谓的最小置换子，被证明是包括一个代码，和直的包括一个有限的代码。因此，置换一个给定子集的单词是唯一可分解为该子集的最小置换子的乘积，并且这些可以进一步简化为直接最小置换子。这导致洞察结构的局部池的可逆性变换半群的一组字置换一个给定的子集。这些发现可以在有限自动机的层次分解的实际计算中利用。作为一个例子，我们考虑生物系统中出现的群体。
英文摘要：
Motivated by issues arising in computer science, we investigate the loop-free paths from the identity transformation and corresponding straight words in the Cayley graph of a finite transformation semigroup with a fixed generator set. Of special interest are words that permute a given subset of the state set. Certain such words, called minimal permutators, are shown to comprise a code, and the straight ones comprise a finite code. Thus, words that permute a given subset are uniquely factorizable as products of the subset's minimal permutators, and these can be further reduced to straight minimal permutators. This leads to insight into structure of local pools of reversibility in transformation semigroups in terms of the set of words permuting a given subset. These findings can be exploited in practical calculations for hierarchical decompositions of finite automata. As an example we consider groups arising in biological systems.
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