# A new algorithm for computing idempotents of R-trivial monoids

## TL;DR

This paper introduces a new, matrix-based algorithm for computing primitive orthogonal idempotents in R-trivial monoids, simplifying previous methods and applicable over any field.

## Contribution

An alternative, matrix-based algorithm for R-trivial monoids that is simpler and works over any ring with known idempotents, inspired by dynamical systems.

## Key findings

- Algorithm works for any field R.
- Efficient computation of idempotents using upper triangular matrices.
- Applicable to rings with known primitive orthogonal idempotents.

## 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.