Lie-Markov models derived from finite semigroups

TL;DR

This paper introduces a method to derive Lie-Markov models from finite semigroups, resulting in continuous-time Markov chains with multiplicative closure, generalizing group-based models.

Contribution

It presents a novel general construction of Lie-Markov models from finite semigroups, extending the framework of group-based models.

Findings

Models are continuous-time Markov chains with multiplicative closure.

Construction generalizes and extends group-based models.

Provides a systematic way to derive new Lie-Markov models.

Abstract

We present and explore a general method for deriving a Lie-Markov model from a finite semigroup. If the degree of the semigroup is $k$, the resulting model is a continuous-time Markov chain on $k$ states and, as a consequence of the product rule in the semigroup, satisfies the property of multiplicative closure. This means that the product of any two probability substitution matrices taken from the model produces another substitution matrix also in the model. We show that our construction is a natural generalization of the concept of group-based models.