A new algorithm for computing idempotents of ℛ-trivial monoids

Abstract

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 [Formula: see text], where [Formula: see text] is any finite [Formula: see text]-trivial monoid. Their method relies on a technical result stating that [Formula: see text]-trivial monoid are equivalent to so-called weakly ordered monoids. We provide an alternative algorithm, based only on the simple observation that an [Formula: see text]-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 [Formula: see text]-trivial monoids (the opposite notion) correspond to so-called feed-forward networks. We first show that our algorithm works for [Formula: see text], after which we prove that it also works for [Formula: see text] where [Formula: see text] is an arbitrary ring with a known complete system of primitive orthogonal idempotents. In particular, our algorithm works if [Formula: see text] 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 [Formula: see text] is obtained from the one our algorithm yields for [Formula: see text] in a straightforward manner. In other words, for any finite [Formula: see text]-trivial monoid [Formula: see text] our algorithm only has to be performed for [Formula: see text], after which a system of idempotents follows for any ring with a given system of idempotents.