A new algorithm for computing idempotents of R-trivial monoids
Abstract
The authors of [Primitive orthogonal idempotents for R-trivial monoids, Journal of Algebra] 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 monoids 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 realised 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 also show that our algorithm only has to be performed for ZM, after which a complete system of primitive orthogonal idempotents may be obtained for RM, where R is any ring with a known complete system of primitive orthogonal idempotents. In particular, the algorithm works if R is any field.
Cite
@article{arxiv.1906.02844,
title = {A new algorithm for computing idempotents of R-trivial monoids},
author = {Eddie Nijholt and Bob Rink and Sören Schwenker},
journal= {arXiv preprint arXiv:1906.02844},
year = {2019}
}