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The complexity of properties of transformation semigroups
International Journal of Algebra and Computation ( IF 0.8 ) Pub Date : 2019-10-30 , DOI: 10.1142/s0218196720500125
Lukas Fleischer 1, 2 , Trevor Jack 3
Affiliation  

We investigate the computational complexity for determining various properties of a finite transformation semigroup given by generators. We introduce a simple framework to describe transformation semigroup properties that are decidable in [Formula: see text]. This framework is then used to show that the problems of deciding whether a transformation semigroup is a group, commutative or a semilattice are in [Formula: see text]. Deciding whether a semigroup has a left (respectively, right) zero is shown to be [Formula: see text]-complete, as are the problems of testing whether a transformation semigroup is nilpotent, [Formula: see text]-trivial or has central idempotents. We also give [Formula: see text] algorithms for testing whether a transformation semigroup is idempotent, orthodox, completely regular, Clifford or has commuting idempotents. Some of these algorithms are direct consequences of the more general result that arbitrary fixed semigroup equations can be tested in [Formula: see text]. Moreover, we show how to compute left and right identities of a transformation semigroup in polynomial time. Finally, we show that checking whether an element is regular is [Formula: see text]-complete.

中文翻译:

变换半群性质的复杂性

我们研究了确定由生成器给出的有限变换半群的各种属性的计算复杂度。我们引入了一个简单的框架来描述在 [公式:见正文] 中可确定的变换半群性质。然后用这个框架来说明决定一个变换半群是群、交换还是半格的问题在[公式:见正文]中。确定一个半群是否具有左(分别为右)零被证明是[公式:见文本]-完全,测试变换半群是否是幂零的问题也是如此,[公式:见文本]-微不足道或具有中心幂等者。我们还给出 [公式:见正文] 算法,用于测试变换半群是否是幂等的、正统的、完全正则的、克利福德的或具有可交换幂等的。其中一些算法是更一般的结果的直接结果,即可以在 [公式:参见文本] 中测试任意固定半群方程。此外,我们展示了如何在多项式时间内计算变换半群的左右恒等式。最后,我们展示了检查一个元素是否规则是[公式:见文本]-完成的。
更新日期:2019-10-30
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