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Lie-Markov Models Derived from Finite Semigroups

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.

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Acknowledgements

This research was supported by Australian Research Council (ARC) Discovery Grant DP150100088. We would like to thank Des FitzGerald for helpful comments on an early draft, and the anonymous reviewers for their careful reading and suggestions that lead to a substantially improved manuscript.

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Correspondence to Jeremy G. Sumner.

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Sumner, J.G., Woodhams, M.D. Lie-Markov Models Derived from Finite Semigroups. Bull Math Biol 81, 361–383 (2019). https://doi.org/10.1007/s11538-018-0455-x

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  • DOI: https://doi.org/10.1007/s11538-018-0455-x

Keywords

  • Lie algebras
  • Continuous-time Markov chains
  • Group-based models
  • Phylogenetics