The authors of Berg et al. [J. Algebra 348 (2011) 446–461] provide an algorithm for finding a complete system of primitive orthogonal idempotents for ℂℳ, where ℳ is any finite ℛ-trivial monoid. Their method relies on a technical result stating that ℛ-trivial monoid are equivalent to so-called weakly ordered monoids. We provide an alternative algorithm, based only on the simple observation that an ℛ-trivial monoid may be realized by upper triangular matrices. This approach is inspired by results in the field of coupled cell network dynamical systems, where ℒ-trivial monoids (the opposite notion) correspond to so-called feed-forward networks. We first show that our algorithm works for ℤℳ, after which we prove that it also works for Rℳ where R is an arbitrary ring with a known complete system of primitive orthogonal idempotents. In particular, our algorithm works if R is any field. In this respect our result constitutes a considerable generalization of the results in Berg et al. [J. Algebra 348 (2011) 446–461]. Moreover, the system of idempotents for Rℳ is obtained from the one our algorithm yields for ℤℳ in a straightforward manner. In other words, for any finite ℛ-trivial monoid ℳ our algorithm only has to be performed for ℤℳ, after which a system of idempotents follows for any ring with a given system of idempotents.

中文翻译：

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一种计算ℛ-平凡幺半群幂等的新算法

伯格的作者等。[J.代数 348(2011) 446–461] 提供了一种算法，用于找到完整的原始正交幂等系统ℂℳ， 在哪里ℳ是任何有限的ℛ-平凡的幺半群。他们的方法依赖于一项技术结果，该结果表明ℛ-平凡的幺半群等价于所谓的弱有序幺半群。我们提供了一种替代算法，仅基于简单的观察，即ℛ-平凡的幺半群可以通过上三角矩阵来实现。这种方法的灵感来自耦合细胞网络动态系统领域的结果，其中ℒ-平凡的幺半群（相反的概念）对应于所谓的前馈网络。我们首先证明我们的算法适用于ℤℳ, 之后我们证明它也适用于Rℳ在哪里R是具有已知完整的原始正交幂等系统的任意环。特别是，如果我们的算法有效R是任何领域。在这方面，我们的结果构成了对 Berg 研究结果的相当概括等。[J.代数 348(2011) 446–461]。此外，幂等系统对于Rℳ是从我们的算法产生的ℤℳ以直截了当的方式。换句话说，对于任何有限ℛ-平凡的幺半群ℳ我们的算法只需要执行ℤℳ, 之后，对于具有给定幂等系统的任何环，都遵循幂等系统。